Study Guide and Intervention Using a Problem-Solving Plan

Transcription

1 1-1 Using a Problem-Solving Plan FOUR-STEP PROBLEM-SOLVING PLAN When solving problems, it is helpful to have an organized plan to solve the problem. The following four steps can be used to solve an math problem. 1 Explore get a general understanding of the problem Plan make a plan to solve the problem and estimate the solution 3 Solve use our plan to solve the problem 4 Examine check the reasonableness of our solution Example HEALTH According to a recent stud, 1 out of ever 10 people is left-handed. If there are 17 people in the eighth grade, predict the number of students who are left-handed. Explore We know that 1 out of 10 people is left-handed. We also know that there are 17 people in the eighth grade. We need to predict how man of the students are left-handed. Lesson 1-1 Plan Make a table to organize the information and look for a pattern. Solve B extending the pattern, we can predict that 17 students will be left-handed. Number of people Number who are left-handed Examine For ever 10 students in the class, 1 is left-handed. There are 17 groups of 10 in a class of 17 and The answer is correct. 1. SODA POP James needs to bu one can of orange soda for ever three cans of cola. If James bus 4 cans of cola, how man cans of orange soda should he bu?. VIDEOS Bob s Video Venue has a membership fee of $5.00 and tape rentals are $1.50 each. Video Heaven has no membership fee and tape rentals are $.00 each. How man tapes must be rented in order for Bob s Video Venue to be more economical? 3. COOKIES A cookie shop offers 6 varieties of cookies and bakes 5 dozen of each kind ever da, Monda Frida. How man cookies are baked in four weeks? 4. PATTERNS Find the next term in, 6, 18, 54, 16, TYPING Jerem needs to tpe a 500-word report for science class. He knows he can tpe about 19 words per minute. About how long will it take Jerem to tpe his report? Glencoe/McGraw-Hill 1 Glencoe Pre-Algebra

2 1- Numbers and Expressions Use the order of operations to evaluate expressions. Step 1 Simplif the expressions inside grouping smbols. Step Do all multiplications and/or divisions from left to right. Step 3 Do all additions and/or subtractions from left to right. Example (3 6) Multipl 6 and 5. 4(3 6) 11 Evaluate (3 6) Divide 10 b. 4(9) 11 Multipl 4 and 9, and and Subtract 5 from Add 36 and Translate verbal phrases into numerical expressions. Example 3 Phrase Ke Word Write and evaluate a numerical expression for the product of seventeen and three. the product of seventeen and three product Expression 17 3 Find the value of each expression (6 ) (6 1) [(13 4) ()] 10. 4[(10 6) 6()] 11. ( 67 13) 1. 6(4 ) 8 ( 34 9) 13. 3[( 7) 9] (8 7) ( 18) ( 9) 16. (9 8) (100 5) Write a numerical expression for each verbal phrase. 17. eleven less than twent 18. twent-five increased b six 19. sixt-four divided b eight 0. the product of seven and twelve Glencoe/McGraw-Hill 6 Glencoe Pre-Algebra

3 1-3 Variables and Expressions An algebraic expression is a combination of variables, numbers, and at least one operation. To evaluate an algebraic expression, replace the variable(s) with numbers and follow the order of operations. ALGEBRA Evaluate each expression if r 6 and s. a. 8s r b. 3(r s) 8s r 8 6 Replace r with 6 and s with. 3(r s) 3( 6) 16 1 Multipl Subtract. 4 Replace r with 6 and s with. Evaluate the parentheses. Multipl. Example a. Assuming no other points, write an expression for a team s total points. Words Variables Expression 3f 6t three points for field goals and six points for touchdowns Let f number of field goals and t number of touchdowns. The total points for the team is 3f 6t. b. Find the total score if a team scored two field goals and three touchdowns. 3f 6t FOOTBALL Teams earn three points for field goals and six points for touchdowns. Replace f with and t with 3. Multipl. Add. The team scored a total of 4 points. ALGEBRA Evaluate each expression if x 10, 5, and z 1. Lesson x z. x 3. x 4z 4. x z x( z) 7. x 8. (x ) 1 0z z Translate each phrase into an algebraic expression. 9. eight inches taller than Mcala s height 10. twelve more than four times a number 11. the difference of sixt and a number 1. three times the number of tickets sold Glencoe/McGraw-Hill 11 Glencoe Pre-Algebra

4 1-4 Properties In algebra, there are certain statements called properties that are true for an numbers. Propert Explanations Example Commutative Propert a b b a of Addition 9 9 Commutative Propert a b b a of Multiplication 0 0 Associative Propert (a b) c (3 4) 7 3 (4 7) of Addition a (b c) Associative Propert (a b) c ( 5) 8 (5 8) of Multiplication a (b c) Additive Identit a 0 0 a a Multiplicative Identit a 1 1 a a Multiplicative Propert of Zero a 0 0 a Example Simplif 3 (x 5). 3 (x 5) 3 (5 x) Commutative Propert of Multiplication (3 5) x Associative Propert of Multiplication 15 x Multipl 3 and 5. Name the propert shown b each statement (3 4) ( 3) p (5 m) (6 5) m 6. (6) 6() Simplif each expression (x 6) 8. 3 (4a) 9. 9 (1 c) d 0 Glencoe/McGraw-Hill 16 Glencoe Pre-Algebra

5 1-5 Variables and Equations An equation that contains a variable is called an open sentence. When the variable is replaced with a number, ou can determine whether the sentence is true or false. A value that makes the sentence true is called a solution. ALGEBRA Find the solution of 7 p 14. Is it 11, 13, or 15? Value for p 7 p 14 True or False? false true false Verbal sentences can be translated into equations and then solved. Example ALGEBRA The sum of a number and six is twent-one. Find the number. Let n the number. Words Variables The sum of a number and six is twent-one. Let n the number. Equation n n 15 Write the equation. Think: What number added to 6 is 1? The solution is 15. ALGEBRA Find the solution of each equation from the list given. 1. b 11 9; 16, 18, 0. h 7 4; 35, 37, x 4; 9, 11, m 18; 6, 8, v 6 5; 7, 9, r 48; 6, 8, ; 7, 9, k 16 15; 31, 33, p; 9, 11, 13 a x 15; 70, 75, n 1 7; 3, 4, ; 10, 11, 1 ALGEBRA Define the variable. Then write the equation and solve. 13. The product of seven and a number is fift-six. 14. The quotient of eight-two and a number is two. 15. The difference between a number and four is twelve. Lesson 1-5 Glencoe/McGraw-Hill 1 Glencoe Pre-Algebra

6 1-6 Ordered Pairs and Relations In mathematics, a coordinate sstem is used to locate points. The horizontal number line is called the x-axis and the vertical number line is called the -axis. The point where the two axes intersect is the origin (0, 0). An ordered pair of numbers is used to locate points in the coordinate plane. The point (4, 3) has an x-coordinate of 4 and a -coordinate of 3. Step 1 Step Step 3 Graph A(4, 3) on the coordinate sstem. Start at the origin. Since the x-coordinate is 4, move 4 units to the right. Since the -coordinate is 3, move 3 units up. Draw a dot. A set of ordered pairs is called a relation. The set of x-coordinates is called the domain. The set of -coordinates is called the range. O A x Example Express the relation {(0, 0), (, 1), (4, ), (3, 5)} as a table and as a graph. Then determine the domain and range. x O x The domain is {0,, 4, 3}, and the range is {0, 1,, 5}. Graph each point on the coordinate sstem. 1. A(4,1). B(,0) 3. C(1,3) 4. D(5,) 5. E(0,3) 6. F(6,4) 6 5 F 4 E C 3 D A 1 B O x 7. Express the relation {(4,6), (0,3), (1,4)} as a table and a graph. Then determine the domain and range. x O x Glencoe/McGraw-Hill 6 Glencoe Pre-Algebra

7 1-7 Scatter Plots A scatter plot is a graph that shows the relationship between two sets of data. In a scatter plot two sets of data are graphed as ordered pairs on a coordinate sstem. A scatter plot ma show a pattern or relationship of the data. The relation ma be positive or negative, or there ma be no relationship. Example SCHOOL The table shows Miranda s math quiz scores for the last five weeks. Make a scatter plot of the data. Since the points are showing an upward trend from left to right, the data suggest a positive relationship. Week Score Miranda's Quiz Scores 90 Score Lesson O Week FOOD For 1 3, use the table below which shows the fat grams and calories for several snack foods. Food Fat grams Calories per serving per serving doughnut corn chips pudding cake snack crackers ice cream (light) ogurt 70 Calories Snack Food Data O Fat (grams) x cheese pizza Make a scatter plot of the data in the table.. What do the x-coordinates represent? -coordinates? 3. Is there a relationship between fat and calories? Explain. Glencoe/McGraw-Hill 31 Glencoe Pre-Algebra

8 -1 Integers and Absolute Value The set of integers can be written {, 3,, 1, 0, 1,, 3, } where means continues indefinitel. Two integers can be compared using an inequalit, which is a mathematical sentence containing or. Use the integers graphed on the number line below for each question Replace each with or to make a true sentence. a. 6 b. 3 4 is greater since it lies to the right of 6. So write 6. 3 is greater since it lies to the right of 4. So write 3 4. Lesson -1 Numbers on opposite sides of zero and the same distance from zero have the same absolute value. C A The smbol for absolute value is two vertical bars on either side of the number. and Example Evaluate each expression. a. 4 b , 6 6 Simplif. Replace each with,, or to make a true sentence Evaluate each expression Glencoe/McGraw-Hill 57 Glencoe Pre-Algebra

9 - Adding Integers Adding Integers with the Same Sign Add their absolute values. Give the result the same sign as the integers. 3 ( 4) 7 Find the sum 3 ( 4). Add 3 and 4. Both numbers are negative so the sum is negative. Adding Integers with Different Signs Subtract their absolute values. Give the result the same sign as the integer with the greater absolute value. Example a b. 6 ( ) Find each sum. 5 4 or ( ) 6 6 or 4 4 Find each sum. Subtract 4 from 5. Simplif. The sum is negative because 5 4. Subtract from 6. Simplif. The sum is positive because ( 3) 3. 3 ( 5) ( 3) ( 4) ( 10) 7. 6 ( 13) ( 6) ( 5) ( 8) ( 11) ( 9) ( 5) ( 5) ( 18) ( 15) ( 14) ( 5) ( 10) ( 3) 14 Glencoe/McGraw-Hill 6 Glencoe Pre-Algebra

10 -3 Subtracting Integers Subtracting Integers To subtract an integer, add its additive inverse. a ( 17) 8 Find each difference. To subtract 17, add 17. Simplif. b ( 3) 10 To subtract 3, add 3. Simplif. Example a. 4 ( 5) 4 ( 5) Find each difference. Find each difference. To subtract 5, add 5. Simplif. b. 6 ( ) 6 ( ) 6 4 To subtract, add. Simplif ( 4) ( 9) ( ) ( 9) ( 0) 4 Lesson ( 6) ( 6) ( 17) ( 16) ( 9) ( 9) ( 6) ( 5) ( 6) 35 Glencoe/McGraw-Hill 67 Glencoe Pre-Algebra

11 -4 Multipling Integers Multipling Integers with Different Signs The product of two integers with different signs is negative. Find each product. a. 4( 3) b. 8(5) 4( 3) 1 8(5) 40 Multipling Integers with the Same Sign The product of two integers with the same sign is positive. Example a. 6(6) b. 7( 4) 6(6) 36 7( 4) 8 Example 3 6( 3)( ) [6( 3)]( ) 18( ) 36 Find each product. Find each product. Find 6( 3)( ). Use the Associative Propert. 6( 3) 18 18( ) (7) 35. 6( 9) ( 11) ( 4) () (14) (10) ( 6) ( 11) ( 4)(5) ( 7)() ( 4)( 6) ( 3)() (4)() ( 4)( 1) (3)( ) ( 6)(7) ( 4)( 8) 96. (3)( 3) (10)( ) (5)( 9) 70 Glencoe/McGraw-Hill 7 Glencoe Pre-Algebra

12 -5 Dividing Integers Dividing Integers with the Same Sign The quotient of two integers with the same sign is positive. a Find each quotient. The dividend and the divisor have the same sign. The quotient is positive. b ( 5) 5 5 The dividend and divisor have the same sign. The quotient is positive. Dividing Integers with Different Signs The quotient of two integers with different signs is negative. Example a. 36 ( 4) 36 ( 4) 9 Find each quotient. Find each quotient. The signs are different. The quotient is negative. b The signs are different. The quotient is negative. Simplif ( 4) ( ) ( ) ( 5) ( 9) ( ) ( 1) ( 11) ( 4) ( ) ( 5) ( 0) Lesson -5 Glencoe/McGraw-Hill 77 Glencoe Pre-Algebra

13 -6 The Coordinate Sstem The x-coordinate is negative. The -coordinate is positive. Both coordinates are negative. (, ) O 1 (, ) x (, ) 3 4 (, ) Both coordinates are positive. The x-coordinate is positive. The -coordinate is negative. Example Graph and label each point on a coordinate plane. Name the quadrant in which each point lies. a. M(, 5) Start at the origin. Move units left. Then move 5 units up and draw a dot. Point M(, 5) is in Quadrant II. b. N(4, 4) Start at the origin. Move 4 units right. Then move 4 units down and draw a dot. Point N(4, 4) is in Quadrant IV. M O 1 3 N x Graph and label each point on the coordinate plane. Name the quadrant in which each point is located. 1. A(, 6) I. B( 1, 4) II 3. C(0, 5) none 4. D( 4, 3) III 5. E(, 0) none 6. F(3, ) IV 7. G( 4, 4) II 8. H(, 5) IV 9. I(6, 3) I 10. J( 5, 8) III 11. K(3, 5) IV 1. L( 7, 3) III G B O L D 3 4 C J 8 A I E x F H K Glencoe/McGraw-Hill 8 Glencoe Pre-Algebra

14 3-1 NAME DATE PERIOD The Distributive Propert The expressions (1 5) and 1 5 are equivalent expressions because the have the same value, 1. The Distributive Propert combines addition and multiplication. Smbols a(b c) ab ac (b c)a ba ca Model b c b a a a c The Distributive Propert also combines subtraction and multiplication. Rewrite subtraction expressions as addition expressions. Use the Distributive Propert to write each expression as an equivalent expression. Then evaluate the expression. a. (6 3) b. 5(9 3) (6 3) 6 3 5(9 3) 5 [9 ( 3)] ( 3) ( 15) 30 The Distributive Propert can also be used with algebraic expressions containing variables. Lesson 3-1 Example Use the Distributive Propert to write each expression as an equivalent algebraic expression. a. 7(m 5) b. 3(n 8) 7(m 5) 7m 7 5 3(n 8) 3[n ( 8)] 7m 35 3 n ( 3)( 8) 3n 4 Use the Distributive Propert to write each expression as an equivalent expression. Then evaluate the expression, if possible. 1. 3(8 ). (9 11) 3. 5(19 6) ; ; ( 6); (3 14) 5. (17 4)3 6. (5 3)7 6 3 ( 6)(14); ( 4) 3; ; (d 4) 8. (w 5)4 9. (c 7) 3d 3 4; 3d 1 w 4 ( 5) 4; 4w 0 c ( )(7); c 14 Glencoe/McGraw-Hill 107 Glencoe Pre-Algebra

15 3- NAME DATE PERIOD Simplifing Algebraic Expressions term: a number, a variable, or a product of numbers and variables coefficient: the numerical part of a term that also contains a variable constant: term without a variable like terms: terms that contain the same variables Identif the terms, like terms, coefficients, and constants in the expression 4m 5m n 7. 4m 5m n 7 4m ( 5m) n ( 7) Definition of subtraction 4m ( 5m) 1n ( 7) Identit Propert terms 4m, 5m, 1n, 7; like terms: 4m, 5m; coefficients: 4, 5, 1; constants: 7 When an algebraic expression has no like terms and no parentheses, we sa that it is in simplest form. Example Simplif 6x 5 x 7. 6x 5 x 7 6x ( 5) ( x) 7 6x ( x) ( 5) 7 [6 ( )]x ( 5) 7 4x Definition of subtraction Commutative Propert Distributive Propert Simplif. Identif the terms, like terms, coefficients, and constants in each expression. 1. 6a 4a. m 4m m c 4d c, 6a, 4a; 6a, 4a; m, 4m, m, 5; 3c, 4d, c, ; 6, 4; m, 4m, m; 1,4,;5 3c, c; 3,4, 1; 4. 5h 3g g h 5. 3w 4u r 5s 5s r 5h, 3g,g, h;5h, h and 3w, 4u, 6; none; 4r, 5s, 5s, r ; 3g, g; 5, 3,, 1; none 3, 4; 6 4r, r and 5s,5s; 4, 5, 5, ; none Simplif each expression. 7. 9m 3m 8. 5x x m 3m 1m 4x 13 m a 7a a d 4 d s s 3 a 7 6 7d 1 7 s Glencoe/McGraw-Hill 11 Glencoe Pre-Algebra

16 3-3 Step 1 Step Step 3 NAME DATE PERIOD Solving Equations b Adding or Subtracting Identif the variable. To isolate the variable, add the same number to or subtract the same number from each side of the equation. Check the solution. Solve x 6. x 6 x 6 x 4 Check: x The solution is 4. Subtract from each side. Example x 9 13 x x 4 Check: x The solution is 4. Solve x Add 9 to each side. Solve each equation. Graph the solution of each equation on the number line. 1. x w k m a b Lesson h r b k g Glencoe/McGraw-Hill 117 Glencoe Pre-Algebra

17 3-4 NAME DATE PERIOD Solving Equations b Multipling or Dividing Step 1 Step Step 3 Identif the variable. To isolate the variable, multipl or divide each side of the equation b the same nonzero number to get the variable b itself. Check the solution. 7x 4 7x x 6 The solution is 6. Divide each side b 7. Check our solution. Example Solve 7x 4. Solve. ( ) Multipl each side b. 4 Check our solution. The solution is 4. Solve each equation. Graph the solution of each equation on the number line. 1. 3a t n r h m a 7. 11b d r p w Glencoe/McGraw-Hill 1 Glencoe Pre-Algebra

18 3-5 NAME DATE PERIOD Solving Two-Step Equations A two-step equation contains two operations. To solve two-step equations, use inverse operations to undo each operation in reverse order. First, undo addition/subtraction. Then, undo multiplication/division. Example c 13 7 Solve c c Add 13 to each side. c 0 c (0) Multipl each side b. c 40 Check: c The solution is 40. For some problems, it ma be necessar to combine like terms before solving. Solve each equation. Check our solution. 1. 5t 7. x u m w d k 8. 4a c m h x m p g 6 3 n m k a c v x 16 8x a 14a c 8 c r 4r Lesson 3-5 Glencoe/McGraw-Hill 17 Glencoe Pre-Algebra

19 3-6 NAME DATE PERIOD Writing Two-Step Equations You can use two-step equations to represent situations in which ou start with a given amount and then increase it at a certain rate. Example PRINTING: A laser printer prints 9 pages per minute. Liza refilled the paper tra after it had printed 9 pages. In how man more minutes will there be a total of 45 pages printed? EXPLORE You know the number of pages printed and the total number of pages to be printed. You need to find the number of minutes required to print the remaining pages. PLAN Let m = the number of minutes. Write and solve an equation. The remaining pages to print is 9m. remaining pages pages printed total pages 9m 9 45 SOLVE 9m m m 153 9m m 17 EXAMINE The remaining 153 pages will print in 17 minutes. Since = 9, the answer is correct. Solve each problem b writing and solving an equation. 1. METEOROLOGY During one da in 1918, the temperature in Granville, North Dakota, began at 33 and rose for 1 hours. The high temperature was about 51. About how man degrees per hour did the temperature rise? 33 1d 51; about 7 degrees. SAVINGS John has $85 in his savings account. He has decided to deposit $65 per month until he has a total of $1800. In how man months will this occur? 65m ; 15 months 3. SKYDIVING A skdiver jumps from an airplane at an altitude of 1,000 feet. After 4 seconds, she reaches 4608 feet and opens her parachute. What was her average velocit during her descent? 1,000 4f 4608; 176 feet per second 4. FLOODING The water level of a creek has risen 4 inches above its flood stage. If it continues to rise steadil at inches per hour, how long will it take for the creek to be 1 inches above its flood stage? 4 h 1; 4 hours Glencoe/McGraw-Hill 13 Glencoe Pre-Algebra

20 3-7 NAME DATE PERIOD Using Formulas The formula d = rt relates distance d, rate r, and time t, traveled. d rt Find the distance traveled if ou drive at 40 miles per hour for 3 hours. d 40 3 Replace r with 40 and t with 3. d 10 The distance traveled is 10 miles. The formula P ( w) relates perimeter P, length, and width w for a rectangle. The formula A w relates area A, length, and width w for a rectangle. Example P ( w) P (7 ) P (9) P 18 The perimeter is 18 feet. Find the perimeter and area of a rectangle with length 7 feet and width feet. A w A 7 A 14 The area is 14 square feet. Lesson TRAIN TRAVEL How far does a train travel in 1 hours at 48 miles per hour? 576 miles. TRAVEL How long does it take a car traveling 40 miles per hour to go 00 miles? 5 hours 3. BICYCLING What is the rate, in miles per hour, of a bicclist who travels 56 miles in 4 hours? 14 miles per hour 4. RACING How long will it take a driver to finish a 980-mile rall race at 70 miles per hour? 14 hours Find the perimeter and area of each rectangle. 5. P 0 in.; 3 in. A 1 in. 7 in ft P 30 ft; 1 ft A 36 ft 6. 6 m P 30 m; 9 m A 54 m 8. P 16 cm; 3 cm A 15 cm 5 cm 9. P 50 d; 10 d A 150 d 15 d 10. P 76 cm; 18 cm A 360 cm 0 cm Glencoe/McGraw-Hill 137 Glencoe Pre-Algebra

21 4-1 NAME DATE PERIOD Factors and Monomials Finding Factors Two or more numbers that are multiplied to form a product are called factors. An number is divisible b its factors. The following rules can be used to determine mentall whether a number is divisible b, 3, 5, 6, or 10. A number is divisible b: if the ones digit is divisible b. 3 if the sum of the digits is divisible b 3. 5 if the ones digit is 0 or 5. 6 if the number is divisible b and b if the ones digit is 0. Example Determine whether 108 is divisible b, 3, 5, 6, or 10. Number Divisible? Reason es The ones digit is 8, and 8 is divisible b. 3 es The sum of the digits is 9, and 9 is divisible b 3. 5 no The ones digit is 8, not 0 or 5. 6 es 108 is divisible b and b no The ones digit is not is divisible b, 3, and 6. A monomial is a number, a variable, or a product of numbers and/or variables. So, 108 is a monomial. The expression 5q is also a monomial since it is the product of a number and a variable, 5 q. However, x 1 is not a monomial since it is the sum of two terms. Lesson 4-1 Use divisibilit rules to determine whether each number is divisible b, 3, 5, 6, or , , 3, 5, 6, , 3, none List all the factors of each number ,, 4, 19, 38, ,, 3, 6, 7, 14, 1, ,, 7, 13, 14, 6, 91, ,, 4, 5, 8, 10, 16, 0, 40, 80 Determine whether each expression is a monomial. Explain wh or wh not es; a number 10. x no; sum of two terms 11. 3(x 1) no; 1. 5st es; product of a number difference of two terms and variables Glencoe/McGraw-Hill 165 Glencoe Pre-Algebra

22 4- NAME DATE PERIOD Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how man times the base is used as a factor. So, 4 3 has a base of 4 and an exponent of 3, and Write each expression using exponents. a The base is 10. It is a factor 5 times, so the exponent is b. ( p )( p )(p ) The base is p. It is a factor 3 times, so the exponent is 3. ( p )( p )( p ) ( p ) 3 Expressions involving powers are evaluated using order of operations. Powers are repeated multiplications. The are evaluated after an grouping smbols and before other multiplication or division operations. Example x 4 = ( 6) 4 Evaluate x 4 if x 6. Replace x with 6. = ( 6)( 6) 4 6 is a factor times. = 36 4 Multipl. = 3 Subtract. Write each expression using exponents ( 7)( 7)( 7) ( 7) 3 3. d d d d d 4 4. x x x 5. (z 4)(z 4) (z 4) 6. 3( t)( t)( t) 3( t ) 3 Evaluate each expression if g 3, h 1, and m g g g m hm g 3 h 5 1. m hg 3 18 Glencoe/McGraw-Hill 170 Glencoe Pre-Algebra

23 4-3 NAME DATE PERIOD Prime Factorization A prime number is a whole number that has exactl two factors, 1 and itself. A composite number is a whole number that has more than two factors. Zero and 1 are neither prime nor composite. Determine whether 9 is prime or composite. Find the factors of The onl factors of 9 are 1 and 9, therefore 9 is a prime number. An composite number can be written as a product of prime numbers. A factor tree can be used to find the prime factorization. Example Find the prime factorization of is the number to be factored. Find an pair of whole number factors of 48. Continue to factor an number that is not prime. The factor tree is complete when there is a row of prime numbers. The prime factorization of 48 is 3 or 4 3. In algebra, monomials can be factored as a product of prime numbers and variables with no exponent greater than 1. So, 8x factors as x x. Determine whether each number is prime or composite composite. 151 prime composite 4. 5 composite Lesson 4-3 Write the prime factorization for each number. Use exponents for repeated factors Factor each monomial. 9. 6m 3 3 m m m 10. 0x 1 5 x 11. a b c 3 a a b b c c c 1. 5h 5 5 h Glencoe/McGraw-Hill 175 Glencoe Pre-Algebra

24 4-4 NAME DATE PERIOD Greatest Common Factor (GCF) The greatest number that is a factor of two or more numbers is the greatest common factor (GCF). Two was to find the GCF are shown below. Find the GCF of 4 and 3. Method 1 List the factors. factors of 4: 1,, 3, 4, 6, 8, 1, 4 Look for factors common to both lists, 1,, 4, and 8. factors of 3: 1,, 4, 8, 16, 3 The greatest common factor of 4 and 3 is 8. Method Use prime factorization. 4 3 Find the common prime factors of 4 and 3. 3 Multipl the common prime factors. The greatest common factor of 4 and 3 is or8. In algebra, greatest common factors are used to factor expressions. Example Factor 5x 10. First, find the GCF of 5x and 10. 5x 5 x 10 5 The GCF is 5. Now write each term as a product of the GCF and its remaining factors. 5x 10 5(x) 5() 5(x ) So, 5x 10 5(x ). Distributive Propert Find the GCF of each set of numbers , , , , Factor each expression. 5. 4g 16 4(g 4) 6. d 6 (d 3) 7. 8a 4 8(a 3) 8. f f f (f ) j 3( j ) n 40n 8n(n 5) Glencoe/McGraw-Hill 180 Glencoe Pre-Algebra

25 4-5 NAME DATE PERIOD Simplifing Algebraic Fractions A fraction is in simplest form when the GCF of the numerator and the denominator is 1. One wa to write a fraction in simplest form is to write the prime factorization of the numerator and the denominator. Then divide the numerator and denominator b the GCF. Write 1 8 in simplest form. 4 Write the prime factorization of the numerator and the denominator Divide the numerator and denominator b the GCF, 3. 4 / 3/ or 3 4 Simplif. Algebraic fractions can also be written in simplest form. Again, ou can write the prime factorization of the numerator and the denominator, then divide b the GCF. Example Simplif 4ab. 36a ab 3 a b b 36a / / / / Divide the numerator and denominator b the GCF, 3 a. / / 3 / 3 a/ a b 3 b a or b 3 a Simplif. Simplif each fraction. If the fraction is alread in simplest form, write simplified c 7. c 3 c 1 8. r r r b 1b 10. 4w 1 6w s 1t 4 simplified simplified 3 d 1. 3 d 1 3d Lesson 4-5 Glencoe/McGraw-Hill 185 Glencoe Pre-Algebra

26 4-6 NAME DATE PERIOD Multipling and Dividing Monomials When multipling powers with the same base, add the exponents. Smbols Example a m a n a m + n or 4 7 When dividing powers with the same base, subtract the exponents. Smbols Example a m an a m n, where a or 5 4 a (3a) ( 3)(a a) Example 4 ( 8) ( 8) ( 8) 4 ( 8) Find a (3a). Express our answer using exponents. (6)(a 1 ) The common base is a. 6a 3 4 Use the Commutative and Associative Properties. Add the exponents. Find ( 8). ( 8) Express our answer using exponents. The common base is 8. Subtract the exponents. Find each product or quotient. Express our answer using exponents v 5 v 4 v 9 3. ( f 3 )( f 9 ) f h(5h 3 ) 35h x (7x 3 ) 70x ( 1) ( 1) ( 1) 0 or c c13 c 7 1. ( p) ( p) 1 ( p) u 6 ( 6u 5 ) 4u w w 15. 5m 3 (4m 6 ) 0m 9 w 16. the product of two cubed and two squared the quotient of six to the eighth power and six squared Glencoe/McGraw-Hill 190 Glencoe Pre-Algebra

27 4-7 NAME DATE PERIOD Negative Exponents Extending the pattern below shows that 4 1 = 1 4 or This suggests the following definition. a n = a 1 n, for a 0 and an integer n. Write each expression using a positive exponent. a. 3 4 b Lesson 4-7 We can evaluate algebraic expressions with negative exponents using the definition of negative exponents. Example Evaluate b if b 3. b 3 Replace b with Find 3. Definition of negative exponent Write each expression using a positive exponent ( 7) 8 1 ( 7) 8 3. b 6 1 b 6 4. n 1 n 11 or 1 n Write each fraction as an expression using a negative exponent other than Evaluate each expression if m 4, n 1, and p p m (np) m 8 1 Glencoe/McGraw-Hill 195 Glencoe Pre-Algebra

28 4-8 NAME DATE PERIOD Scientific Notation When ou deal with ver large numbers like 5,000,000 or ver small numbers like , it is difficult to keep track of place value. Numbers such as these can be written in scientific notation. A number is expressed in scientific notation when it is written as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. B definition, a number in scientific notation is written as a 10 n, where 1 a 10 and n is an integer. a Express each number in standard form ,000 Move the decimal point 5 places to the right. b Move the decimal point 6 places to the left. Example a. 6,000,00 Express each number in scientific notation. To write in scientific notation, place the decimal point after the first nonzero digit, then find the power of 10. 6,000, The decimal point moves 7 places. The power of 10 is 7. b Place the decimal point after the first nonzero digit. The power of 10 is ,10, , Express each number in scientific notation. 7. 1,000,000, , Choose the greater number in each pair , , , , Glencoe/McGraw-Hill 00 Glencoe Pre-Algebra

29 5-1 Writing Fractions as Decimals Fractions can be written as decimals b dividing the numerator b the denominator. If the division ends, or terminates, when the remainder is 0, it is a terminating decimal. If the decimal number repeats without end, it is a repeating decimal. Write each fraction as a decimal. a b Lesson is a terminating decimal or 0.4 is a repeating decimal. It ma be easier to compare numbers if the are written as decimals. Example Replace with,,or to make a true sentence , so can be written as Since , Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal Replace each with,, or to make a true sentence Glencoe/McGraw-Hill 5 Glencoe Pre-Algebra

30 5- Rational Numbers A number that can be written as a fraction is called a rational number. Mixed numbers, integers, and man decimals can be written as fractions. Onl decimals that terminate or repeat can be written as fractions. Other decimal numbers such as π = are infinite and nonrepeating. The are called irrational numbers. Example Write each number as a fraction. a. 3 b. 7 5 Write the mixed number as an improper fraction. 3 5 = = 5 1 The denominator is 1. c d is 14 hundredths. 0.5 = = or Simplif. N = N = N = (N = ) 9N = 5 9 N 5 = 9 9 N = 5 9 Let N represent the number. Multipl each side b 10 because one digit repeats. Subtract N from 10N. Divide each side b 9. Simplif. The diagram at the right illustrates the relationships among whole numbers, integers, and rational numbers. Rational Numbers Integers Whole Numbers Write each number as a fraction Identif all sets to which each number belongs (N = natural numbers, W = whole numbers, I = integers, Q = rational numbers) Glencoe/McGraw-Hill 30 Glencoe Pre-Algebra

31 5-3 Multipling Rational Numbers To multipl fractions, multipl the numerators and multipl the denominators. So a b c ac b, d d where b, d 0. The fractions ma be simplified either before or after multipling. Example 8 a Find each product. Write in simplest form = 8 5 Divide 5 and 15 b their GCF, = = 1 Multipl. Simplif. b = Rename mixed numbers as improper fractions. = Divide 15 and 3 b 3, and 8 and b = Multipl. = 0 or 0 Simplif. 1 Lesson 5-3 Find each product. Write in simplest form ( 5) x 3 z Glencoe/McGraw-Hill 35 Glencoe Pre-Algebra

32 5-4 Dividing Rational Numbers Two numbers whose product is 1 are called multiplicative inverses or reciprocals. For an fraction a b, where a, b 0, b a is the multiplicative inverse and a b b a 1. This means that 3 and 3 are multiplicative inverses because To divide b a fraction, multipl b its multiplicative inverse: a b c a d b d c a d,where b, c, d 0. bc Example Find each quotient. Write in simplest form. a Multipl b the multiplicative inverse of 5 8, Divide 4 and 8 b their GCF, or 1 1 Simplif. 5 b Rename mixed numbers as improper fractions or Find each quotient. Write in simplest form. Multipl b the multiplicative inverse of 1 1 5, Divide out common factors. Simplif Glencoe/McGraw-Hill 40 Glencoe Pre-Algebra x c 4d 3 8d a ac b b m 1. 9 m n 3

33 5-5 Adding and Subtracting Like Fractions To add fractions with like denominators, add the numerators and write the sum over the denominator. So, a c b c a b, where c 0. c Find Write the sum in simplest form (1 3) or 4 3 Add the whole numbers and fractions separatel or write as improper fractions. Add the numerators Simplif. To subtract fractions with like denominators, subtract the numerators and write the difference over the denominator. So, a c b c a b, where c 0. c Example Find Write the difference in simplest form Write mixed numbers as improper fractions. Subtract the numerators. 5 3 or 1 Simplif. 3 Find each sum or difference. Write in simplest form Lesson 5-5 Glencoe/McGraw-Hill 45 Glencoe Pre-Algebra

34 5-6 Least Common Multiple The least common multiple (LCM) is the least nonzero number that is a multiple of two or more given numbers. When finding the LCM of small numbers, list several multiples of each number until a common multiple is found. Find the LCM of 3 and 5. multiples of 3: 0, 3, 6, 9, 1, 15, 18,... multiples of 5: 0, 5, 10, 15, 0,... The least nonzero number that is a multiple of both numbers is 15. When finding the LCM of large numbers, write the prime factorization of each and multipl the greatest power of each prime factor used. Example Find the LCM of 4 and Find the prime factorization of each number. LCM Write the product of the greatest power of each prime factor. The least common denominator (LCD) of two or more fractions is the LCM of the denominators. Example 3 Find the least common multiple (LCM) of each pair of numbers or monomials. 1. 3, 6. 4, , , , , x, 6x 8. 15b,5b x, 1x 4 Find the least common denominator (LCD) of each pair of fractions , , , t, 4st Find the LCD of 7 8 and 6. Find the LCM of the denominators, 6 and LCM The LCD is , ab b 15. x, x 3 Glencoe/McGraw-Hill 50 Glencoe Pre-Algebra

35 5-7 Adding and Subtracting Unlike Fractions To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplif. Find Use 7 3 or 1 as the common denominator Add the numerators. 1 Rename each fraction with the common denominator. To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplif. Lesson 5-7 Example Find Write the mixed numbers as improper fractions Rename fractions using the LCD, 18. Simplif. 7 Subtract the numerators. 1 8 Find each sum or difference. Write in simplest form Glencoe/McGraw-Hill 55 Glencoe Pre-Algebra

36 5-8 Measures of Central Tendenc When working with numerical data, it is often helpful to use one or more numbers to represent the whole set. These numbers are called the measures of central tendenc. You will stud the mean, median, and mode. Statistic mean median mode Definition sum of the data divided b the number of items in the data set middle number of the ordered data, or the mean of the middle two numbers number or numbers that occur most often Example Jason recorded the number of hours he spent watching television each da for a week. Find the mean, median, and mode for the number of hours. Mon. Tues. Wed. Thurs. Fri. Sat. Sun sum of hours mean = n umber of das = 7 = 3 The mean is 3 hours. To find the median, order the numbers from least to greatest and locate the number in the middle The median is 3 hours. There is no mode because each number occurs once in the set. Find the mean, median, and mode for each set of data. 1. Maria s test scores 9, 86, 90, 74, 95, 100, 90, 50. Rainfall last week in inches 0, 0.3, 0, 0.1, 0, 0.5, Resting heart rates of 8 males 84, 59, 7, 63, 75, 68, 7, 63 Glencoe/McGraw-Hill 60 Glencoe Pre-Algebra

37 5-9 Solving Equations with Rational Numbers To solve rational number equations, use the same skills applied to solving equations involving integers. Example Solve 4. p 1.7. Solve p p p Write the equation. Add 1.7 to each side. Simplif Write the equation. Subtract 1 from each side. Simplif or Rename fractions using LCD and subtract. Example 3 Example 4 Solve 6z 4.. Solve x. 6z 4. 6z z 0.7 Write the equation. Divide each side b 6. Simplif. 8 3 x Write the equation (8) x 3 x 3 Multipl each side b 4 3. Simplif. 10 x Simplif. 3 Lesson 5-9 Solve each equation. Check our solution. 1. p a h v q c m d 56 8 Glencoe/McGraw-Hill 65 Glencoe Pre-Algebra

38 5-10 Arithmetic and Geometric Sequences A sequence is an ordered list of numbers. An arithmetic sequence is a sequence in which the difference between an two consecutive terms, called the common difference, is the same. In the sequence, 9, 16, 3, 30,..., the common difference is 7. A geometric sequence is a sequence in which the quotient of an two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 56,.., the common ratio is 4. Example State whether the sequence is arithmetic, geometric, or neither. If it is arithmetic or geometric, state the common difference or common ratio and write the next three terms of the sequence. a. 7, 14, 8, 56, 11,... The common ratio is, so the sequence is geometric. The next three terms are 11 or 4, 4 or 448, and 448 or 896. b. 8, 6, 4,, 0,,... The common difference is, so the sequence is arithmetic. The next three terms are ( ) or 4, 4 ( ) or 6, and 6 ( ) or 8. State whether each sequence is arithmetic, geometric, or neither. If it is arithmetic or geometric, state the common difference or common ratio and write the next three terms , 0, 40, 80, 160,.... 7, 17, 7, 37, 47, , 50, 5, 5 5,, , 9, 6, 3, 0, ,,, 3, 3, 3, , 1, 1, 1, 1,... 7., 3, 4.5, , 10, 1, 1, 3, , 3., 4.8, , 1, 36,... Glencoe/McGraw-Hill 70 Glencoe Pre-Algebra

39 6-1 NAME DATE PERIOD Ratios and Rates 7 Ratios written as 7 to 1, 7:1, and are different was to write the same ratio. Ratios should be 1 written in simplest form. 6 feet 15 inches Express the ratio 6 feet to 15 inches as a fraction in simplest form. Write the ratio as a fraction inches inches 4 5 Convert feet to inches. inches Divide the numerator and denominator b the GCF, 3. inches Lesson 6-1 Written as a fraction in simplest form, the ratio is 4 to 5. Example Express the ratio $10 for 8 fish as a unit rate. Round to the nearest tenth, if necessar. 10 dollars 8 fish 8 Write the ratio as a fraction. 10 dol 8 fi lars sh 1.5 dollars 1 Divide the numerator and denominator b 8 to get a denominator of 1. fish 8 The unit rate is $1.5 per fish. Express each ratio as a fraction in simplest form weeks to plan events 1. 9 inches to feet teaspoons to 1 forks cups to 10 servings shelves to 84 books 6. 6 teachers to 165 students Express each ratio as a unit rate. Round to the nearest tenth, if necessar. 7. $58 for 5 tickets $11.60 per 1 ticket 8. $4.19 for 4 cans of soup $1.05 per can 9. $74.90 for 6 people miles in 1 hours $45.8 per person 47.1 miles per hour Glencoe/McGraw-Hill 99 Glencoe Pre-Algebra

40 6- NAME DATE PERIOD Using Proportions A proportion is an equation stating that two ratios are equal. You can use cross products to solve a proportion. Example c 4 Solve the proportion c c 3 Cross products c Multipl c 3 3 Divide c The solution is ALGEBRA Solve each proportion. x w u a 5. s w m h g f z k r 5 d d c p q n 4.5 v Glencoe/McGraw-Hill 304 Glencoe Pre-Algebra

41 6-3 NAME DATE PERIOD Scale Drawings and Models A scale gives the relationship between the measurements on the drawing or model and the measurements of the real object. Example Let x represent the actual distance. Write and solve a proportion. map width 1 inch 4.5 inches map width actual width 6 miles x miles actual width 1 x Find the cross products. A map shows a scale of 1 inch 6 miles. The distance between two places on the map is 4.5 inches. What is the actual distance? x 5.5 The actual distance is 5.5 miles. Simplif. On a set of architectural drawings for an office building, the scale is 0.5 inch 5 feet. Find the actual length of each room. Room Drawing Distance Actual Distance 1. Lobb 1.6 inches 3 feet. CEO Office 1.35 inches 7 feet 3. Cop Room 0.55 inch 11 feet 4. CEO Secretar s Office 0.6 inch 1 feet 5. Vice President s Office 0.9 inch 18 feet 6. Librar 1.55 inches 30.5 feet 7. Storage Area.115 inches 4.5 feet 8. Personal Manager s Office inches feet 9. Manager s Office 0.65 inch 1.5 feet 10. Mail Room.65 inches 45.5 feet 11. Boiler Room 3.75 inches 74.5 feet 1. Conference Room A.6 inches 5.4 feet 13. Conference Room B 0.95 inch 18.5 feet 14. Cafeteria.3 inches 46 feet 15. Kitchen inches 40 feet Lesson 6-3 Glencoe/McGraw-Hill 309 Glencoe Pre-Algebra

42 6-4 NAME DATE PERIOD Fractions, Decimals, and Percents A percent is a ratio that compares a number to % 65 65% Express the percent as a fraction. Write as a fraction with denominator of 100. Simplif. Example 150% 150% 150% 1.5 Express the percent as a decimal. Divide b 100 and remove the %. Example or 15% Express the fraction as a percent. Write equivalent fraction with denominator of 100. Example % Express the decimal as a percent. Multipl b 100 and add the %. Express each percent as a fraction or mixed number in simplest form and as a decimal %, % 5 0, % 17, % 1 8, % 1 1 5, % 1, % 4 9, % , % 3, Express each decimal or fraction as a percent. Round to the nearest tenth percent, if necessar % % % % % % % % % 4 Glencoe/McGraw-Hill 314 Glencoe Pre-Algebra

43 6-5 NAME DATE PERIOD Using the Percent Proportion In a percent proportion, one of the numbers, called the part, is being compared to the whole quantit, called the base. The other ratio is the percent, written as a fraction, whose base is 100. Find each percent. a. Twelve is what percent of 16? a b p p p p 75 p So, twelve is 75% of 16. Replace the variables. Find the cross products. Simplif. Divide. b. What percent of 8 is 7? a b p p 100 p p 87.5 p So, 87.5% of 8 is 7. Example Find the part or the base. a. What number is 1.4% of 15? a b p 100 a a a 1 a 0.1 So, 0.1 is 1.4% of 15. Replace the variables. Find the cross products. Simplif. Divide. b. 5 is 36% of what number? a b p b b,500 36b 65 b So, 5 is 36% of 65. Use the percent proportion to solve each problem. Round to the nearest tenth is what percent of 5? 9.3%. 95 is what percent of 400? 73.8% 3. What percent of is 56? 54.5% 4. What percent of 4 is 15? 375% 5. What is 99% of 840? What is 4.5% of 38? What is 16% of 36.? is 80% of what number? is 9% of what number? is 90% of what number? 5 Lesson 6-5 Glencoe/McGraw-Hill 319 Glencoe Pre-Algebra

44 6-6 NAME DATE PERIOD Finding Percents Mentall When working with common percents like 10%, 5%, 40%, and 50%, it ma be helpful to use the fraction form of the percent. Percent-Fraction Equivalents 0% % % % % % 5 30% % % % % % % % % 3 80% % % % % 5 6 Example Find 0% of 35 mentall. 0% of of 35 Think: 0% = Think: 1 of 35 is 7. So, 0% of 35 is 7. 5 Find the percent of each number mentall % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of % of Glencoe/McGraw-Hill 34 Glencoe Pre-Algebra

45 NAME DATE PERIOD 6-7 Using Percent Equations The percent equation is an equivalent form of the percent proportion. In the percent equation, the percent is written as a decimal. Example Solve each problem using the percent equation. a. Find % of 95. n 0.(95) n 0.9 So, % of 95 is 0.9. c. 90 is 0% of what number? 90 0.n 450 n So, 90 is 0% of 450. b. 15 is what percent of 75? 15 n(75) 0. n So, 15 is 0% of 75. Lesson 6-7 Solve each problem using the percent equation. 1. Find 76% of Find 9% of Find 40% of Find 6% of Find 3.5% of Find 18.5% of Find 107% of is what percent of 800? 3% is what percent of 40? 15% is what percent of 100? 99.5% is what percent of 55? 8% 1. 7 is what percent of 350? % is what percent of 80? 16.5% is what percent of 10? 1% is 9% of what number? is 39% of what number? is 90% of what number? is 55% of what number? is 15% of what number? is 7.5% of what number? 8 Glencoe/McGraw-Hill 39 Glencoe Pre-Algebra

46 6-8 NAME DATE PERIOD Percent of Change A percent of change tells how much an amount has increased or decreased in relation to the original amount. There are two methods ou can use to find percent of change. Example Find the percent of change from 75 ards to 54 ards. Step 1 Subtract to find the amount of change new measurement original measurement Step Write a ratio that compares the amount of change to the original measurement. Express the ratio as a percent. percent of change amount of change or 8% Substitution Write the decimal as a percent. State whether each change is a percent of increase or a percent of decrease. Then find the percent of change. Round to the nearest tenth, if necessar. 1. from inches to 16 inches. from 8 ears to 10 ears D; 7.3% I; 5% 3. from $815 to $95 4. from 15 meters to 1 meters I; 13.5% D; 0% 5. from 55 people to 17 people 6. from 45 mi per gal to 4 mi per gal I; 94.5% D; 46.7% 7. from 8 cm to 3 cm 8. from 18 points to 144 points I; 14.3% I; 1.5% 9. from $8 to $ from 800 roses to 639 roses D; 68.8% D; 0.1% 11. from 8 tons to 4. tons 1. from 5 qt to 18 qt D; 47.5% I; 60% 13. from $85.75 to $ from 198 lb to 11 lb I; 5.1% D; 43.4% Glencoe/McGraw-Hill 334 Glencoe Pre-Algebra

47 6-9 NAME DATE PERIOD Probabilit and Predictions Probabilit is the chance some event will happen. P(event) (number of favorable outcomes) (number of possible outcomes) Example A bag contains 6 red marbles, 1 blue marble, and 3 ellow marbles. One marble is selected at random. Find the probabilit of each outcome. a. P(ellow) (number of favorable outcomes) P(event) b. P(blue or ellow) (number of favorable outcomes) P(event) 3 or 30% 1 0 There is a 30% chance of choosing a ellow marble. (1 3) 4 or 40% There is a 40% chance of choosing a blue or ellow marble. c. P(red, blue, or ellow) (number of favorable outcomes) P(event) d. P(black) (number of favorable outcomes) P(event) (6 1 3) 1 0 or 100% There is a 100% chance of choosing a red, blue, or ellow marble. 0 or 0% 1 0 There is a 0% chance of choosing a black marble. A bag contains 5 red marbles, 5 blue marbles, 6 green marbles, 8 purple marbles, and 1 white marble. One is selected at random. Find the probabilit of each outcome. Express each probabilit as a fraction and as a percent. Lesson P(white). P(red) 3. P(green) 1,4% 1 5 5, 0% 6, 4% 5 4. P(purple) 5. P(white, blue, green) 6. P(red or blue) 8, 3% 1 5 5, 48%, 40% 5 7. P(red or purple) 8. P(green or purple) 9. P(green, purple, or white) , 56% 3, 60% P(red, blue, green, purple, or white) 11. P(red, blue, or purple) 1 1, 100% 1 8, 7% 5 Glencoe/McGraw-Hill 339 Glencoe Pre-Algebra

48 7-1 Solving Equations with Variables on Each Side To solve equations with variables on each side, use the Addition or Subtraction Propert of Equalit to write an equivalent equation with the variable on one side. Then solve the equation. Example 1x 3 4x 13 1x 4x 3 4x 4x 13 8x x Solve the equation 1x 3 4x 13. Then check our solution. 8x 16 Write the equation. Subtract 4x from each side. Simplif. Add 3 to each side. Simplif. x Mentall divide each side b 8. Lesson 7-1 To check our solution, replace x with in the original equation. CHECK 1x 3 4x 13 Write the equation. 1() 3 4() 13 Replace x with Check to see whether the statement is true. 1 1 The statement is true. Solve each equation. Check our solution. 1. x 1 x 11. a 5 4a n 11 n c 3c b 6b d 9 3d 3 8. f 4 6f 6 9. s 3 5s a 3 8a n 1 1n x 7x a 3 4 7a 15. 3b 1 14 b 16. 1c c n 5n 1 Glencoe/McGraw-Hill 367 Glencoe Pre-Algebra

49 7- Solving Equations with Grouping Smbols Equations with grouping smbols can be solved b first using the Distributive Propert to remove the grouping smbols. Solve the equation (6m 1) 8m. Check our solution. (6m 1) 8m Write the equation. 1m 8m 1m 1m 8m 1m CHECK 4m Appl the Distributive Propert. Subtract 1m from each side. Simplif. 4m Divide each side b (6m 1) 8m 1 m Simplif ( 1 ) Replace m with 1. (3 1) 4 Simplif. 4 4 The solution checks. Some equations have no solution. The solution set is the null or empt set. Other equations have ever number as a solution. Such an equation is called an identit. Example a. (x 1) 4 x b. (x 1) x x 4 x x x 4 x x 4 Solve each equation. The solution set is Ø. Solve each equation. Check our solution. x x x x x x x x The solution set is all real numbers. 1. 8( g 3) 4. 5(x 3) (c 5) 7 4. (3d 7) 5 6d 5. (s 11) 5(s ) ( 3) 7. ( f 3) 8 f 8. (x ) 3 x (b 6) 5(b 1) 10. x 5 3(x 3) Glencoe/McGraw-Hill 37 Glencoe Pre-Algebra

50 7-3 Inequalities A mathematical sentence that contains an of the smbols listed below is called an inequalit. is less than is greater than is less than or is greater than equal to or equal to is fewer than is more than is no more than is no less than exceeds is at most is at least Write an inequalit for each sentence. a. Less than 70 students attended the last dance. b. At the store, the camera cost at least as much as the mail-order price of $9. Graph an inequalit on a number line to help visualize the values that make the inequalit true. Example Graph each inequalit on a number line. a. x 8 b. x 8 The open circle means the number 8 is not included in the graph Write an inequalit for each sentence. 1. Your age is greater than 1 ears.. A number decreased b 5 is no more than More than $1000 was collected for the charit. 4. At least 80 runners showed up for the charit race. The closed circle means the number 8 is included in the graph. Lesson 7-3 Graph each inequalit on a number line. 5. x 7 6. a 7. d w 9 9. t n Glencoe/McGraw-Hill 377 Glencoe Pre-Algebra

51 7-4 Solving Inequalities b Adding or Subtracting Use the Addition and Subtraction Properties of Inequalities to solve inequalities. When ou add or subtract a number from each side of an inequalit, the inequalit remains true. Example Solve 1 0. Check our solution. 1 0 Write the inequalit Subtract 1 from each side. 8 Simplif. To check our solution, tr an number greater than 8. CHECK? 1 0 Write the inequalit.? Replace with This statement is true. An number greater than 8 will make the statement true. Therefore, the solution is 8. Solve each inequalit. Check our solution b. t p n ( 18) 6. s w ( ) 8. j z ( 4) b g z x a ( 6) m b k w 3 Glencoe/McGraw-Hill 38 Glencoe Pre-Algebra

52 7-5 Solving Inequalities b Multipling or Dividing Use the Multiplication and Division Properties of Inequalities to solve inequalities. When ou multipl or divide each side of an inequalit b a positive number, the inequalit remains true. The direction of the inequalit sign does not change. 8x 7 Solve 8x 7. Check our solution. Write the inequalit. 8 x 7 Divide each side b x 9 Simplif. The solution is x 9. You can check this solution b substituting 9 or a number greater than 9 into the inequalit. For an inequalit to remain true when multipling or dividing each side of the inequalit b a negative number, however, ou must reverse the direction of the inequalit smbol. Example Solve 4 and check our solution. Then graph the solution on 1 a number line. 4 1 Write the inequalit (4) Multipl each side b 1 and reverse the smbol. 48 Graph the solution, 48. Check the result Solve each inequalit and check our solution. Then graph the solution on a number line d. p h t x b c n Lesson 7-5 Glencoe/McGraw-Hill 387 Glencoe Pre-Algebra

53 7-6 Solving Multi-Step Inequalities Some inequalities require more than one step to solve. For such inequalities, undo the operations in reverse order, just as in solving multi-step equations. Remember to reverse the inequalit smbol when multipling or dividing each side of the inequalit b a negative number. Example Solve 1 x 4 x and check our solution. Graph the solution on a number line. 1 x 4 x Write the inequalit. 1 x x 4 x x Subtract x from each side. 1 4x 4 Simplif x 4 1 Subtract 1 from each side. 4x 1 Simplif. 4x Divide each side b 4 and reverse the smbol. x 3 Simplif. CHECK 1 x 4 x Tr 4, a number less than 3. 1 ( 4) 4 ( 4) Replace x with Simplif The solution checks. Graph the solution x If the inequalit contains parentheses, use the Distributive Propert to begin simplifing the inequalit. Solve each inequalit and check our solution. Graph the solution on a number line. 1. 5c p c 5 4c n (d ) b Glencoe/McGraw-Hill 39 Glencoe Pre-Algebra

54 8-1 Functions Function A special relation in which each member of the domain is paired with exactl one member in the range. Vertical Line Test Since functions are relations, the can be represented using ordered pairs, tables, or graphs. Example a. {( 10, 34), (0, ), (10, 9), (0, 3)} Domain (x) Move a pencil or straightedge from left to right across the graph of a relation. If it passes through no more than one point on the graph, the graph represents a function. If it passes through more than one point on the graph, the graph does not represent a function. Determine whether each relation is a function. Explain. Range () Because each element in the domain is paired with onl one value in the range, this is a function. b. x Domain (x) Range () Because 10 in the domain is paired with 34 and in the range, this is not a function. Lesson 8-1 Determine whether each relation is a function. Explain. 1. {( 5, ), (3, 3), (1, 7), (3, 0)} No; 3 in the domain is paired with 3 and 0 in the range.. {(, 7), ( 5, 0), ( 10, 0), (, 10), (1, 0)} Yes; each x value is paired with onl one value. 3. x x Yes; each x value is paired with onl one value. No; 1 in the domain is paired with 3 and 7 in the range. Glencoe/McGraw-Hill 417 Glencoe Pre-Algebra

55 8- Linear Equations in Two Variables A function can be represented with an equation. An equation such as 1.50x is called a linear equation. A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. Linear Equations x 1, x, 1 3 x Nonlinear Equations x 1, x 3, 3, x 4 x Solutions of a linear equation are ordered pairs that make the equation true. One wa to find solutions is to make a table. Example Complete the table. Use the results to write four solutions of 4x 10. Write the solution as ordered pairs. x 4x 10 (x, ) 1 4( 1) ( 1, 14) 0 4(0) (0, 10) 1 4(1) 10 6 (1, 6) 4() 10 (, ) A linear equation can also be represented b a graph. The coordinates of all points on a line are solutions to the equation. Graph 4x 10 b plotting ordered pairs. Plot the points found in. Connect the points using a straight line. O x 4 (, ) 6 (1, 6) 8 10 (0, 10) 1 14 Find four solutions of each equation. Write the solutions as ordered pairs Sample answers are given. 1. x 4. 3x x 5 ( 1, ), (0, 4), ( 1, 4), (0, 7), ( 1, 9), (0, 5), (1, 6), (, 8) (1, 10), (, 13) (1, 1), (, 3) Graph each equation b plotting ordered pairs. 4. 4x 5. x 6 6. x 8 4x O x x O x O 4 x x Glencoe/McGraw-Hill 4 Glencoe Pre-Algebra

56 8-3 Graphing Linear Equations Using Intercepts Finding Intercepts The x-intercept is the x-coordinate of a point where a graph crosses the x-axis. The -coordinate of this point is 0. The -intercept is the -coordinate of a point where a graph crosses the -axis. The x-coordinate of this point is 0. To find the x-intercept, let 0 in the equation and solve for x. To find the -intercept, let x 0 in the equation and solve for. Find the x-intercept and the -intercept for the graph of x To find the x-intercept, let 0. x 5 10 Write the equation. x 5(0) 10 Replace with 0. x 5 Simplif. To find the -intercept, let x 0. x 5 10 Write the equation. Example O (0, ) x 5 10 (5, 0) Graph x x (0) 5 10 Replace x with 0. Simplif. Find the x-intercept and the -intercept for the graph of each equation. 1. x x 1 5; 5 none; 1 4; 6 Graph each equation using the x- and -intercepts. Lesson x 3 5. x 5 6. x 9 (1, 0) O (0, 3) x (0, 9) O 3 (9, 0) 6 9 1x (5, 0) O (0, 5) x Glencoe/McGraw-Hill 47 Glencoe Pre-Algebra

57 8-4 Slope Slope describes the steepness of a line. slope = r ise vertical change run Model (x 1, 1 ) Note that the slope is the same for an two points on a straight line. Smbols m 1 x, x where x x 1 1 rise run (x, ) O x Example m x 1 1 x Find the slope of the line. Definition of slope m 4 1 (x 9 5 1, 1 ) (5, 1), (x, ) (9, 4) m (9, 4) (5, 1) O x The slope is 3 4. Find the slope of each line (, 3) O (, ) x (1, 1) (0, ) O x (1, 1) O x (4, 3) Find the slope of the line that passes through each pair of points. 4. A(, ), B( 5, 4) 5. L(5, 5), M(4, ) 6. R(7, 4), S(7, 3) 3 undefined 7 Glencoe/McGraw-Hill 43 Glencoe Pre-Algebra

58 8-5 Rate of Change Rate of Change A change in one quantit with respect to another quantit slope change in change in x A special tpe of linear equation that describes kx, where k 0 rate of change Direct Variation x and both increase or both decrease at the same rate varies directl with x. Slope, or rate of change, k, is called the constant of variation. O x x rate of change slope Find the rate of change for the linear function represented in the table. c hange chang in e in x or 1 For each time increase of 1h, the temperature increases b C. Time (h) x Temperature ( C) Example Write an equation relating x and from. Step 1 Find the value of k. Step Use k to write an equation. kx Direct variation kx Direct variation k(1) Replace with and x with 1. x Replace k with. k Simplif. 1. Find the rate of change for the linear function. increase of 500 mph Time (h) x Distance Flown (mi) Suppose varies directl with x. Write an equation relating x and.. 14 when x when x 5 4. when x 4 x 3 5 x 1 x Lesson 8-5 Glencoe/McGraw-Hill 437 Glencoe Pre-Algebra

59 8-6 Slope-Intercept Form The slope-intercept form of a line makes it eas to graph the line: mx b Example: 3x slope m slope 3 -intercept b -intercept Example Step 1 Step Graph = 4x 3 using the slope and -intercept. Find the slope and -intercept. slope 4 -intercept 3 Graph the -intercept point at (0, 3). Step 3 Write the slope 4 as 4. Use the slope 1 to locate a second point on the line. m 4 change in : down 4 units 1 change in x: right 1 unit Step 4 Draw a line through the two points. Step 5 Check b locating another point on the line and substituting the coordinates into the original equation. Given the slope and -intercept, graph each line. 1. slope 4,. slope 6, 3. slope 1 4, -intercept 1 -intercept 4 -intercept 5 O (0, 3) (1, 7) x 4x 1 6x 4 1 x O x O x O x Graph each equation using the slope and -intercept. 4. 3x 5. x x 3 3 3x O x x O x O x 5x 3 Glencoe/McGraw-Hill 44 Glencoe Pre-Algebra

60 8-7 Writing Linear Functions If ou know the slope and -intercept, ou can write the equation of a line b substituting these values in mx b. Write an equation in slope-intercept form for each line. a. slope 1, -intercept 3 b. slope 0, -intercept 9 4 mx b Slope-intercept form mx b Slope-intercept form 1 4 x ( 3) Replace m with 1 and b with 3. 0x ( 9) Replace m with 0 and b with x 3 Simplif. 9 Simplif. Lesson 8-7 Example Write an equation in slope-intercept form for the line passing through ( 4, 4) and (, 7). Step 1 Find the slope m. m 1 x Definition of slope x1 7 4 m or 1 ( 4) (x 1, 1 ) ( 4, 4), (x, ) (, 7) Step Find the -intercept b. Use the slope and the coordinates of either point. mx b Slope-intercept form 4 1 ( 4) b Replace (x, ) with ( 4, 4) and m with 1. 6 b Simplif. An equation is 1 x 6. Write an equation in slope-intercept form for each line. 1. slope 1,. slope 3, 3. slope 0, 4 -intercept -intercept 3 -intercept 5 x 3 x Write an equation in slope-intercept form for the line passing through each pair of points. 4. (6, ) and (3, 1) 5. (8, 8) and ( 4, 5) 6. (7, 3) and ( 5, 3) 1 3 x 1 x Glencoe/McGraw-Hill 447 Glencoe Pre-Algebra

61 8-8 Best-Fit Lines A best-fit line is a line that is ver close to most of the data points. You can use best-fit lines to make predictions from real-world data. Make a scatter plot and draw a best-fit line for the data in the table. Year Percent of Population Source: U.S. Census Bureau Year Percent of Population Percent of Population U.S. Civilian Labor Force O Year Source: U.S. Census Bureau Example Use the best-fit line in example 1 to predict the percent of the population in the U.S. labor force in 010. Use the extended line to find the value for an x value of 010 about 70. A prediction for the percent of the U.S. population in the labor force in 010 is approximatel 70 percent. Use the table that shows the number of girls who participated in high school athletic programs in the United States from 1973 to Make a scatter plot and draw a best-fit line. Number of Participants (thousands) O Year Year Number of Participants (thousands) Source: U.S. Census Bureau Use the best-fit line to predict the number of female participants in 008. Sample answer: about 3,000,000 female participants Glencoe/McGraw-Hill 45 Glencoe Pre-Algebra

62 8-9 Solving Sstems of Equations Two linear equations together are called a sstem of equations. One method for solving a sstem of equations is to graph the equations on the same coordinate plane. Another wa to solve a sstem of equations is b using substitution. Solve the sstem of equations b graphing. x 1 x The graphs appear to intersect at (1, 1). Check this estimate b substituting the coordinates into each equation. CHECK: x 1 x 1 (1) x 1 x O (1, 1) x Example 3x 1 Solve the sstem of equations b substitution. Since must have the same value in both equations, ou can replace with in the first equation. 3x 1 3x 1 Write the first equation. Replace with. 1 x Solve for x. The solution of this sstem of equations is (1, ). Check b graphing. Solve each sstem of equations b graphing. 1. 3x 6. x 4 x x (, 0) x O (, 0) (, ) x 3x 6 x x 4 O x (, ) Lesson 8-9 Solve each sstem of equations b substitution. 3. x 4 4. x x 11 x 1 5 (, ) (1, 1) (3, 5) Glencoe/McGraw-Hill 457 Glencoe Pre-Algebra

63 8-10 Graphing Inequalities To graph a linear inequalit, first graph the boundar. Test an point to determine which region is the solution, or half plane. Shade this region. You can also use inequalities to solve real-world problems. Example GARDENING Marisol has 10 square feet in her garden for planting squash and peppers. How man square feet can she plant of each? Squash plus peppers is at most 10 square feet. x + 10 x 10 x 10 Write the inequalit. Subtract x from each side. Graph x 10 as a solid line since the boundar is part of the graph. The origin is part of the graph since Thus, the coordinates of all points in the shaded region are possible solutions. (7, 3) 7 square feet of squash, 3 square feet of peppers (4, 6) 4 square feet of squash, 6 square feet of peppers (, 7) square feet of squash, 7 square feet of peppers Peppers (sq ft) O x Squash (sq ft) LEISURE Use the following information for 1 3. Fred and Roa have $500 to spend on ballroom dancing classes. Each foxtrot class costs $16 per couple and each salsa lesson costs $0 per couple. 1. Write an inequalit to represent this situation. 16x Graph the inequalit. Salsa Classes Foxtrot Classes x 3. Use the graph to determine how man of each tpe of dance class Fred and Roa can take. Sample answers: (5 foxtrot, 1 salsa) (10 foxtrot, 17 salsa), (15 foxtrot, 10 salsa) Glencoe/McGraw-Hill 46 Glencoe Pre-Algebra

64 NAME DATE PERIOD 9-1 Squares and Square Roots Squares and Square Roots A perfect square is the square of a whole number. A square root of a number is one of two equal factors of the number. Ever positive number has a positive square root and a negative square root. The square root of a negative number such as 5, is not real because the square of a number is never negative. a. 144 Find each square root. 144 indicates the positive square root of 144. Since 1 144, b indicates the negative square of 11. Since 11 11, Lesson 9-1 c. 4 4 indicates both square roots of 4. Since 4, 4 and 4 Example Use a calculator to find 34 to the nearest tenth. nd ENTER Use a calculator Round to the nearest tenth. Find each square root not poss not poss. 11. ± Use a calculator to find each square root to the nearest tenth Glencoe/McGraw-Hill 489 Glencoe Pre-Algebra

65 9- NAME DATE PERIOD The Real Number Sstem The set of real numbers consists of all natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as fractions. Irrational numbers are decimals that do not repeat or terminate. Rational Numbers Integers 1 Whole 5 Numbers 0 Natural Numbers Irrational Numbers Example a. 7 b. 0.6 This number is a natural number, a whole number, an integer, and a rational number. This repeating decimal is a rational number because it is equivalent to 3. c. 71 It is not the square root of a perfect square so it is irrational. d Since 8 1 = 7, this number is an integer and a rational number. 3 e. 169 Because 169 = 13, this number is an integer and a rational number. Name all of the sets of numbers to which each real number belongs. Let N natural numbers, W whole numbers, Z integers, Q rational numbers, and I irrational numbers N, W, Z, Q. 3 7 I 3. Q Z, Q N, W, Z, Q 6. O W, Z, Q Q Q N, W, Z, Q Name all of the sets of numbers to which each real number belongs. Q 11. I Z, Q Q Q Z, Q Glencoe/McGraw-Hill 494 Glencoe Pre-Algebra

66 9-3 NAME DATE PERIOD Angles Measuring and Drawing Angles An angle is made up of two ras that have a common endpoint, called the vertex. The most common unit of measure for angles is the degree. There are 360 in a circle. Angles are measured using protractors. a. Use a protractor to measure ABC A m ABC is 65. b. Draw D having a measure of B C D Classifing Angles Angles are classified b their degree measure. Acute angles measure between 0 and 90. A right angle measures 90. An obtuse angle measures between 90 and 180. A straight angle measures 180. Example a. B b. c. G D F A C m ABC = 90 So, ABC is right. Classif each angle as acute, obtuse, right, or straight. Use protractor to find the measure of each angle. Then classif each angle as acute, obtuse, right, or straight. 1. AHB 30 acute. BHD 60 acute 3. AHD 90 right 4. AHE 98 obtuse 5. AHF 135 obtuse 6. AHG 180 straight E 30 m ABC 90 So, ABC is acute. H 150 m ABC 90 So, ABC is obtuse. A B C D H E I F G Lesson 9-3 Use a protractor to draw an angle having each measurement. Then classif each angle as acute, obtuse, right, or straight acute acute Glencoe/McGraw-Hill 499 Glencoe Pre-Algebra

67 9-4 NAME DATE PERIOD Triangles Angles of a Triangle The sum of the measures of the angles of a triangle is 180. Find the value of x in DEF. m D m E m F 180 The sum of the measures is x 180 m D 43 and m E 5 95 x 180 Simplif. x x 85 Subtract 95 from each side. E 5 D 43 x F Classifing Triangles Triangles are classified b their angles or b their sides. Acute triangles have all acute angles. Obtuse triangles have one obtuse angle. Right triangles have one right angle. Scalene triangles have no congruent sides. Isosceles triangles have at least two sides congruent. Equilateral triangles have all sides congruent. Example STU has all acute angles. Classif the triangle b its angles and b its sides. STU has no two sides that are congruent. So, STU is an acute scalene triangle. U 50 S T Find the value of x in each triangle. Then classif each triangle as acute, right, or obtuse. 11 x 1. x ; obtuse ; acute x 68 ; right Classif each triangle b its angles and b its sides right, scalene obtuse, isosceles acute, equilateral Glencoe/McGraw-Hill 504 Glencoe Pre-Algebra

68 9-5 NAME DATE PERIOD The Pthagorean Theorem Pthagorean Theorem Words If a triangle is a right triangle, then the square of the length of the hpotenuse is equal to the sum of the squares of the lengths of the legs. Model c Smbols c a b a b Example c a + b Find the length of the hpotenuse of the right triangle. Pthagorean Theorem c Replace a with 16 and b with 30. c Evaluate 16 and 30. c 1156 Add 56 and 900. c 1156 c 34 Take the square root of each side. Simplif. The length of the hpotenuse is 34 cm. 16 cm 30 cm c cm Find the length of the hpotenuse in each right triangle. Round to the nearest tenth if necessar. 15 in in. 5. c m c ft 43.0 c in. 11 m If c is the measure of the hpotenuse, find each missing measure. Round to the nearest tenth, if necessar. 4. a = 18, b = 80, c =? 8 5. a =?, b = 70, c = a = 14, b =?, c = a =?, b = 48, c = The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 8. a = 16, b = 30, c = 34 es 9. a = 5, b = 31, c = 37 no 1 m 35 ft 5 ft 10. a = 1, b = 9, c = 4 no Lesson 9-5 Glencoe/McGraw-Hill 509 Glencoe Pre-Algebra

69 9-6 NAME DATE PERIOD The Distance and Midpoint Formulas Distance and Midpoint Formulas On a coordinate plane, the distance d between two points with coordinates (x 1, 1 ) and (x, ) is given b d (x x 1 ) (. 1 ) The midpoint of a line 1 segment whose endpoints are (x 1, 1 ) and (x, ) is given b x, 1. x Example Find the distance between M(8, 1) and N (, 3). Round to the nearest tenth, if necessar. Then find the coordinates of the midpoint of M N. d (x x 1 ) ( 1 ) Distance Formula MN (8 ( )) (1 3) MN (10) ) ( (x, 1 ) (, 3), (x, ) (8, 1) Simplif. MN Evaluate 10 and ( ). MN 104 Add 100 and 4. MN 10. Take the square root. (, 3) N O (8, 1) M The distance between points M and N is about 10. units. 1 x midpoint x, 1 Midpoint Formula ( ) 8, 1 3 Substitution (3, ) Simplif. The coordinates of the midpoint of M N are (3, ). Find the distance between each pair of points. Round the nearest tenth, if necessar. 1. A (3, 1), B (,5) 4.1. C (, 4), D (3, 7) E (5, 3), F (4,) G ( 6, 5), H ( 4, 3) I ( 4, 3), J(4,4) K (5, 0), L (, 1) M (,1), N (6,5) O (0,0), P ( 5, 6) 7.8 The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. 9. Q (3, 5), R (4, ) (3.5, 3.5) 10. S ( 6, 4), T ( 5, 6) ( 5.5, 1) 11. U (, 1), V (4, 4) (3,.5) 1. W (5, 1), X (, 1) (1.5, 0) 13. Y ( 5, 3), Z (, 5) ( 1.5, 1) 14. A (8, 1), B (3, 1) (5.5, 1) 15. C (0, 0), D (, 4) (1, ) 15. E ( 5, 3), F (4, 7) ( 0.5, 5) Glencoe/McGraw-Hill 514 Glencoe Pre-Algebra

70 9-7 NAME DATE PERIOD Similar Triangles and Indirect Measurement Corresponding Parts of Similar Triangles Similar triangles are triangles that have the same shape but not necessaril the same size. If two triangles are similar, then the corresponding angles have the same measure, and the corresponding sides are proportional. AC B C D F EF The corresponding sides are proportional. Write a proportion x Replace AC with 1, DF with 36, BC with 7, and EF with x. 1 x 36 7 Find the cross products. 1x 5 If ABC DEF, what is the value of x? Simplif. x 1 Divide each side b 1. D A B 1 m E 36 m 7 m C x m F Lesson 9-7 Indirect Measurement The properties of similar triangles can be used to find measurements which are difficult to measure directl. This is called indirect measurement. Example The Chrsler Building in New York casts a foot shadow the same time a 5.8 foot tourist casts a 4.4 foot shadow. How tall is the Chrsler Building to the nearest tenth? tourist s height tourist s shadow Chrsler Building s height h Chrsler Building s shadow h 4.4 Find the cross products h Multipl h Divide each side b 4.4. The height of the Chrsler Building is 1046 feet. The triangles are similar. Write a proportion and find the value of x Q 66 5 m A B 7 m C D E 10 m x m 3. SURVEYOR A surveor needs to find the distance AB across a pond. He constructs CDE similar to CAB and measures the distances as shown on this figure. Find AB. 8.8 d F Glencoe/McGraw-Hill 519 Glencoe Pre-Algebra 17 m M N m C O 51 m P 5 d 60 d D 1 d E x m R A x d B

71 9-8 NAME DATE PERIOD Sine, Cosine, and Tangent Ratios Trigonometric Ratios Words: If A is an acute angle of a right triangle, Smbols measure of leg opposite A sin A =, sin A = a measure of hpotenuse c cos A = b c tan A = a b measure of leg adjacent to A cos A =, Model measure of hpotenuse tan A = measure of leg opposite to A measure of leg adjacent to A B leg opposite a A hpotenuse c C b A leg adjacent to A sin D Find sin D, cos D, and tan D. measure of leg opposite D measure of hpotenuse F 1 13 cos D Example 1 or measure of leg adjacent to D measure of hpotenuse tan D 5 or or Find each value to the nearest ten thousandth. a. cos 48 b. sin 3 COS 48 ENTER SIN 3 ENTER So, cos 48 is about So, sin 3 is about E 5 D measure of leg opposite to D measure of leg adjacent to D Find each sine, cosine, or tangent. Round to four decimal places, if necessar. 1. sin A 0.6. sin D 0.8 B 3. cos A cos D tan C tan F Use a calculator to find each value to the nearest ten thousandth. 7. sin sin cos cos tan tan A 105 C D 96 E F 8 Glencoe/McGraw-Hill 54 Glencoe Pre-Algebra

72 10-1 NAME DATE PERIOD Line and Angle Relationships Names of Special Angles Interior angles lie inside the parallel lines. Exterior angles lie outside the parallel lines. 3, 4, 5, 6 1,, 7, 8 Transversal Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. 3, and 5, 4 and Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. 1 and 7,, and 8 Corresponding angles are in the same position on 1 and 5, and 6, the parallel lines in relation to the transversal. 3 and 7, 4 and 8 Line and Angle Relationships Lesson 10-1 Parallel Lines Perpendicular Vertical Adjacent Compelementar Supplementar Lines Angles Angles Angles Angles n D C m a b a b m n A B mabc m1 m m1 m 90 m1 m 180 Example In the figure, f n and v is a transversal. If m 3 100, find m 1 and m 6. Since 1 and 3 are corresponding angles, the are congruent. So, m Since 3 and 6 are alternate interior angles, the are congruent. So, m Find the value of x in each figure x 135 x 40 x v 55 x f n ALGEBRA Angles A and B are complementar. If m A 3x 8 and m B 5x 10, what is the measure of each angle? 5 ; ALGEBRA Angles Q and R are supplementar. If m Q 4x 9 and m R 8x 3, what is the measure of each angle? 65 ; 115 Glencoe/McGraw-Hill 549 Glencoe Pre-Algebra

73 10- NAME DATE PERIOD Congruent Triangles Corresponding Parts of Congruent Triangles Words Model Two triangles are congruent if the have the same size and shape. If two triangles are congruent, their corresponding sides are congruent and their corresponding angles are congruent. Slash marks are used to indicate which sides are congruent. Y Q Arcs are used to indicate which angles are congruent. X Z P R Smbols Congruent Angles: X P, Y Q, Z R Congruent Sides: X Y P Q, Y Z Q R, X Z P R Example Corresponding Angles Q S, R Z, N V Corresponding Sides S Z Q R, Z V R N,V S N Q NQR VSZ Name the corresponding parts in the congruent triangles shown. Then write a congruence statement. Complete each congruence statement if DFH PWZ. N R 1. F W. P D 3. D H P Z 4. Z W H F Q S V Z Find the value of x for each pair of congruent triangles. 5. B 4 D 6. Q 9 R 7. D x C 8. J 40 7 x P 3 B A S 7 E x G x V O E 8 F Glencoe/McGraw-Hill 554 Glencoe Pre-Algebra

74 10-3 NAME DATE PERIOD Transformations on the Coordinate Plane Transformation Movement How To s Translation Reflection Rotation You slide a figure from one position to another without turning it. Ever point of the original figure is moved the same distance and in the same direction. You flip a figure over a line of smmetr. The figures are mirror images of each other. Ever point of the original figure has a corresponding point on the other side of the line of smmetr. You turn the figure around a fixed point. You can turn the figure 90 clockwise, 90 counterclockwise, or 180 about the the origin. To describe the translation using an ordered pair, add the coordinates of the ordered pair to the coordinates of the original point. To reflect a point over the x-axis, use the same x-coordinate and multipl the -coordinate b 1. To reflect a point over the -axis, use the same -coordinate and multipl the x-coordinate b 1. To rotate a figure 90 clockwise, switch the coordinates of each point and multipl the new second coordinate b 1. To rotate a figure 90 counterclockwise, switch the coordinates of each point and multipl the new first coordinate b 1. To rotate a figure 180, multipl both coordinates of each point b 1. Example vertex The vertices of figure PQRS are P(, ), Q( 1, ), R( 1, ), and S( 3, 1). Graph the figure and its image after a reflection over the -axis. Use the same -coordinate and multipl the x-coordinate b 1. reflection P(, ) ( 1, ) P (, ) Q( 1, ) ( 1 1, ) Q (1, ) R( 1, ) ( 1 1, ) R (1, ) S( 3, 1) ( 3 1, 1) S (3, 1) S P Q R Q' P' O R' S' x Lesson 10-3 For 1 3, use the graph shown. 1. Graph the image of the figure after a rotation of 90 counterclockwise.. Graph the image of the figure after a rotation of Find the vertices of the figure after a rotation of 90 clockwise. A( 1, 4), B( 1, 3), C(, ), D( 1, 1), E( 3, 1), and F ( 3, 4) E''. F'' C'' O D'' B'' A'' A D' E' x BC D B' C' 1. F E A' F' Glencoe/McGraw-Hill 559 Glencoe Pre-Algebra

75 10-4 NAME DATE PERIOD Quadrilaterals A quadrilateral is a closed figure with four sides and four vertices. The segments that make up a quadrilateral intersect onl at their endpoints. The vertices are A, B, C, and D. The angles are A, B, C, and D. D A B C The sides are AB, BC, CD, and DA. A quadrilateral can be seperated into two triangles. Since the sum of the measures of the angles of a triangle is 180, the sum of the measures of the angles of a quadrilateral is (180 ) or 360. Example ALGEBRA Find the value of x. Then find each missing angle measure. Words The sum of the measures of the angles is 360. Variable Let m A, m B, m C, and m D represent the measures of the angles. Equation m A m B m C + m D 360 Angles of a quadrilateral A 3x 3x 4x x x x 140 Substitution Combine like terms. Subtract 0 from each side. Simplif. x 0 Divide each side b 7. The value of x is 0. So, m A 3(0) or 60 and m B 4(0) or 80. B 4x 130 D C ALGEBRA Find the value of x. Then find the missing angle measures x 7 x x ; 60 5; 75 60; 60 Classif each quadrilateral using the name that best describes it square trapezoid parallelogram Glencoe/McGraw-Hill 564 Glencoe Pre-Algebra

76 10-5 NAME DATE PERIOD Area: Parallelograms, Triangles, and Trapezoids Shape Words Area Formula Model Parallelogram The area of a parallelogram can be found b A bh multipling the measures of the base and the height. h b Triangle A diagonal of a parallelogram separates the parallelogram into two congruent triangles. The area of each triangle is one-half the area of the parallelogram. A 1 bh h b Trapezoid A trapezoid has two bases. The height of a trapezoid is the distance between the bases. A trapezoid can be separated into two triangles. A 1 h(a b) a h b Example Find the area of the trapezoid. A 1 h(a b) Area of a trapezoid 7 mm A 1 17(7 6) Replace h with 17, a with 7, and b with 6. A = 33 mm 17 mm 18 mm A 561 or mm The area of the trapezoid is 80 1 mm. Find the area of each figure m 19 ft.5 cm cm.5 m 7.5 ft 10 ft 8 ft 3 cm 7 ft 3 cm 7.5 m 97.5 ft Find the area of each figure described. 4. trapezoid: height, 1 d; bases, 6 d, 8 d 84 d 5. parallelogram: base, 4.5 cm; height, 8 cm 36 cm Lesson 10-5 Glencoe/McGraw-Hill 569 Glencoe Pre-Algebra

77 10-6 NAME DATE PERIOD Polgons A polgon is a simple, closed figure formed b three or more line segments. The line segments, called sides, meet onl at their endpoints. The points of intersection are called vertices. Polgons can be classified b the number of sides the have. A diagonal is a line segment in a polgon that joins two nonconsecutive vertices, forming triangles. You can use the propert of the sum of the measures of the angles of a triangle to find the sum of the measures of the interior angles of an polgon. An interior angle is an angle inside a polgon. Number of Sides Name of Polgon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon If a polgon has n sides, then n triangles are formed. The sum of the degree measures of the interior angles of the polgon is (n )180. A regular polgon is a polgon that is equilateral (all sides are congruent) and equiangular (all angles are congruent). Since the angles of a regular polgon are congruent, their measures are equal. Example Find the measure of one interior angle of a regular 0-gon. Step 1 A 0-gon has 0 sides. Therefore, n = 0. (n )180 (0 )180 Replace n with 0. 18(180) or 340 Simplif. The sum of the measures of the interior angles is 340. Step Divide the sum b 0 to find the measure of one angle So, the measure of one interior angle in a regular 0-gon is 16. Classif each polgon. Then determine whether it appears to be regular or not regular heptagon; not regular octagon; not regular quadrilateral; regular Find the sum of the measures of the interior angles of each polgon. 4. quadrilateral 5. nonagon 6. heptagon 7. 1-gon Glencoe/McGraw-Hill 574 Glencoe Pre-Algebra

78 10-7 NAME DATE PERIOD Circumference and Area: Circles Circles A circle is the set of all points in a plane that are the same distance from a given point. The circumference of a circle is equal to its diameter times, or times its radius times. The area of a circle is equal to times the square or its radius. C d or C r A r Find the circumference of the circle to the nearest tenth. a. C r Circumference of a circle C 7 Replace r with 7. C km Simplif. Use a calculator. The circumference is about 44.0 kilometers. Example Find the area of the circle. Round to the nearest tenth. b. A r 30 in. Area of a circle A (15) Replace r with 15. A 5 Evaluate (15). A Use a calculator. The area is about square inches. Lesson 10-7 Find the circumference and area of each circle. Round to the nearest tenth m 7 m 8 cm 1 ft 31.4 m; 78.5 m.0 m; 38.5 m 50.3 cm; 01.1 cm 37.7 ft; ft Match each circle described in the column on the left with its corresponding measurement in the column on the right. 5. radius: 5 units c a. area: 63.6 units 6. diameter: 9 units a b. circumference: 50.3 units 7. diameter: 1 units d c. circumference: 31.4 units 8. diameter: 16 units b d. area: units Glencoe/McGraw-Hill 579 Glencoe Pre-Algebra

79 10-8 NAME DATE PERIOD Area: Irregular Figures To find the area of an irregular figure, separate the irregular figure into figures whose area ou know how to find. Use the area formulas ou have learned in this chapter. Triangle Trapezoid Parallelogram Circle A 1 bh A 1 h(a b) A bh A r a. Example 5 m 7 m Find the area of each figure. Round to the nearest tenth, if necessar. b. 13 d 7.5 m 9 d 10 m Area of Parallelogram Area of Triangle Area of Semicircle Area of Triangle A bh A 1 bh A = 1 r A = 1 bh A 7(7.5) or 5.5 A 1 (15 7.5) A 56.5 The area of the figure is or about m. A = 1 (4.5) A = 1 (9 13) A = 31.8 A = 58.5 The area of the figure is or about 90.3 d. Find the area of each figure. Round to the nearest tenth, if necessar. 1. What is the area of a figure formed using a rectangle with a base of 10 ards and a height of 4 ards and two semicircles, one with a radius of 5 ards and the other a radius of ards? 85.6 d. Find the area of a figure formed using a square with sides of 9 centimeters and three attached triangles with heights of 6 centimeters. 16 cm Find the area of each shaded region. Round to the nearest tenth. (Hint: Find the total area and subtract the non-shaded area.) m 4 cm cm 1 cm 1 cm 1 cm 5 in. 8 in. 8 in. 8 m 7 m 8 m 15 m 3 cm 6.9cm 110.8in 169 m Glencoe/McGraw-Hill 584 Glencoe Pre-Algebra

80 11-1 NAME DATE PERIOD Three-Dimensional Figures Identifing Three-Dimensional Figures A prism is a polhedron with two parallel bases. A pramid is a polhedron with one base. Prisms and pramids are named b the shape of their bases, such as triangular or rectangular. F E G Identif the solid. Name the bases, faces, edges, and vertices. triangular pramid An of the faces can be considered a base. faces: EFG, EGH, EFH, FGH H edges: E F, E G, E H, F G, F H, G H vertices: E, F, G, H Diagonals and Skew Lines Skew lines are lines that lie in different planes and do not intersect. A diagonal of a figure joins two vertices that have no faces in common. Lesson 11-1 Example Identif a diagonal and name all segments that are skew to it. F G diagonal: F M E H skew segments: E H, H G, J K, K L, E J, G L K L J M Identif each solid. Name the bases, faces, edges, and vertices. 1.. H M N L R S J K For 3 4, use the rectangular prism in Example. 3. Identif a diagonal that could be drawn from point E. 4. Name all segments that are skew to the new diagonal. Glencoe/McGraw-Hill 609 Glencoe Pre-Algebra

81 NAME DATE PERIOD 11- Volume: Prisms and Clinders Volume of Prisms To find the volume V of a prism, use the formula V = Bh, where B is the area of the base, and h is the height of the solid. Find the volume of each prism. a. b. 4 cm 3 cm V Bh 6 cm 4 m 5 m 9.8 m V Bh V (3 6)4 V V 7 V 98 The volume is 7 cm 3. The volume is 98 m 3. Volume of Clinders The volume V of a clinder with radius r is the area of the base, r, times the height h, or V r h. Example V r h V. 4.5 V 68.4 The volume is about 68.4 ft 3. Find the volume of each clinder. Find the volume of each solid. If necessar, round to the nearest tenth ft 4.5 ft.5 in. 10 mm 1 ft.5 in..5 in. 6 mm 1.9 mm 1 ft 5 ft 4. rectangular prism: length 9 mm, width 8. mm, height 5 mm 5. triangular prism: base of triangle 5.8 ft, altitude of triangle 5. ft, height of prism 6 ft Glencoe/McGraw-Hill 614 Glencoe Pre-Algebra

82 11-3 NAME DATE PERIOD Volume: Pramids and Cones Volume of Pramids To find the volume V of a pramid, use the formula V = 1 Bh, where B is the area 3 of the base, and h is the height of the solid. Find the volume of each pramid. a. V 1 3 Bh 6 ft b. V 1 Bh 3 1 in. 3 in. 8 in. V 1 3 (7 7) h V h 7 ft V 1 3 (7 7)6 V ft V 98 V 4 The volume is 98 ft 3. The volume is 4 in 3. Volume of Cones To find the volume V of a cone, use the formula V = 1 3 πr h, where r is the radius and h is the height of the solid. Example Find the volume of the cone. Round to the nearest tenth. V 1 3 r h V 1 3 (4.3) 11 V 13.0 m 3 The volume is about 13.0 m m 4.3 m Lesson 11-3 Find the volume of each solid. If necessar, round to the nearest tenth in. 1.6 m 9.1 ft 9 in. 9 in. 0.8 m 1.9 ft 4. square pramid: length 1. cm, height 5 cm 5. cone: diameter 4 d, height 7 d Glencoe/McGraw-Hill 619 Glencoe Pre-Algebra

83 11-4 NAME DATE PERIOD Surface Area: Prisms and Clinders Surface Areas of Prisms To find the surface area S of a rectangular prism with length, width w, and height h, use the formula S w h wh. To find the surface area of a triangular prism, find the area of each face, then add to find the total surface area.. S w h wh S (.8)(.1) (.8)(5.8) (.1)(5.8) S 68.6 The surface area is 68.6 ft. Find the surface area of the rectangular prism.8 ft.1 ft 5.8 ft Example Find the surface area of the triangular prism. A w bottom left side right side A 1 bh two bases Total surface area cm 6 cm 9 cm 10.3 cm 14 cm Surface Areas of Clinders To find the surface area S of a clinder with radius r and height h, use the formula S r rh. Example 3 S r rh S = (4.6) (4.6)(6) Find the surface area of the clinder. Round to the nearest tenth. 4.6 m S The surface area is about m. 6 m Find the surface area of each solid. If necessar, round to the nearest tenth in. 1 mm 8 mm 14 in. 13 in. in. 8 in. 15 in. 10 in. 4. cube: side length 8.3 cm 5. clinder: diameter 0 d, height d Glencoe/McGraw-Hill 64 Glencoe Pre-Algebra

84 11-5 NAME DATE PERIOD Surface Area: Pramids and Cones Surface Area of Pramids The sum of the areas of the lateral faces is the lateral area of a pramid. The surface area of a pramid is the sum of the lateral area and the base area. slant height lateral face base Area of base A s A 3 or 9 Find the surface area of the square pramid. Area of each lateral face A 1 bh A 1 (3)(4.5) or 6.75 Lateral area 4(6.75) cm 3 cm 3 cm Total surface area 7 9or36 cm. Surface Area of Cones The surface area S of a cone with radius r and slant height can be found b using the formula S r r. slant height radius Example Find the surface area of the cone. Round to the nearest tenth. S r r S (7.7)(11.) (7.7) S 457. in The surface area is about 457. in. 11. in. 7.7 in. Find the surface area of each solid. If necessar, round to the nearest tenth ft.3 m 9 cm 0. m ft ft 10 cm 10 cm mm mm 6 mm 6.8 d 10.8 mm 6 mm A 15.6 mm 8.6 d 7. mm Lesson 11-5 Glencoe/McGraw-Hill 69 Glencoe Pre-Algebra

85 11-6 NAME DATE PERIOD Similar Solids Identif Similar Solids Solids are similar if the have the same shape and their corresponding linear measures are proportional. Determine whether each pair of solids is similar a ft 6.5 ft 50(3.5) 6.5(8) Write a proportion comparing corresponding edge lengths. Find the cross products. 8 ft 3.5 ft Simplif. The corresponding measures are proportional, so the pramids are similar. b. 8 cm 1 cm Write a proportion comparing radii and heights. 10 cm 14 cm 8(14) 1(10) Find the cross products. Simplif. The radii and heights are not proportional, so the clinders are not similar. Use Similar Solids You can find missing measures if ou know solids are similar. Example Find the missing measure for the pair of similar solids Write a proportion. x 1x 0.8(6) x 4.8 Find the cross products. Simplif. The missing length is 4.8 ft. 1 ft 0.8 ft 3 ft 6 ft.4 ft x Determine whether each pair of solids is similar. 15 cm cm. 11 in..75 in. 6 cm 6 cm 4 cm 4 cm 16 in. in. 4 in. 5.5 in. Find the missing measure for each pair of similar solids. 3. x ft in. x 0.4 ft 0.5 ft 8.4 in in. Glencoe/McGraw-Hill 634 Glencoe Pre-Algebra

86 11-7 NAME DATE PERIOD Precision and Significant Digits Precision and Significant Digits Use the following rules to determine the number of significant digits in a measurement. If a number contains a decimal point, the number of significant digits is found b counting the digits from left to right, starting with the first nonzero digit and ending with the last digit. If a number does not contain a decimal point, the number of significant digits is found b counting the digits from left to right, starting with the first digit and ending with last nonzero digit. Determine the number of significant digits in each measure. a kilometers 3 significant digits b grams 4 significant digits c. 330 miles significant digits d liters 1 significant digit Compute Using Significant Digits When adding or subtracting measurements, the sum or difference should have the same precision as the least precise measurement. When multipling or dividing measurements, the product or quotient should have the same number of significant digits as the measurement with the least number of significant digits. Lesson 11-7 Example Calculate. Round to the correct number of significant digits. a ft ft 8.94 decimal places decimal places The least precise measurement, 8.94, has two decimal places. So, round to two decimal places, 5.07 ft. b m.1 m significant digits.1 significant digits significant digits The answer cannot have more significant digits than the factors, so round to significant digits, 1 m. Determine the number of significant digits in each measure m mm in g Calculate. Round to the correct number of significant digits mm 6.31 mm ft ft cl 0.45 cl km.8 km cm 0.8 cm in..05 in kl 4. kl mm 4 mm Glencoe/McGraw-Hill 639 Glencoe Pre-Algebra

87 1-1 Stem-and-Leaf Plots Stem-and- Words One wa to organize and displa data is to use a stem-and-leaf plot. Leaf Plot In a stem-and-leaf plot, numerical data are listed in ascending or descending order. Model The greatest place value of the data is used for the stems. The next greatest place value forms the leaves. Example Step 1 The least and the greatest numbers are 55 and 95. The greatest place value digit in each number is in the tens. Draw a vertical line and write the stems from 5 to 9 to the left of the line. Step Write the leaves to the right of the line, with the corresponding stem. For example, for 85, write 5 to the right of 8. Step 3 Rearrange the leaves so the are ordered from least to greatest. Then include a ke or an explanation. ZOOS Displa the data shown at the right in a stem-and-leaf plot. Stem Leaf Stem Leaf Displa each set of data in a stem-and-leaf plot acres 1. {7, 35, 39, 7,. {94, 83, 88, 77, 4, 33, 18, 19} 95, 99, 88, 87} Size of U. S. Zoos Zoo Audubon (New Orleans) Size (acres) 58 Cincinnati 85 Dallas 95 Denver 80 Houston 55 Los Angeles 80 Oregon 64 St. Louis 90 San Francisco 75 Woodland Park (Seattle) 9 Lesson 1-1 ROLLER COASTERS For 3 and 4, use the stem-and-leaf plot shown. 3. What is the speed of the fastest roller coaster? The slowest? 100 mph; 83 mph 4. What is the median speed? 9 mph The Fastest Roller Coasters Stem Leaf mph Glencoe/McGraw-Hill 667 Glencoe Pre-Algebra

88 1- Measures of Variation The range and the interquartile range describe how a set of data varies. Measures of Variation Term range median lower quartile upper quartile interquartile range Definition The difference between the greatest and the least values of the set The value that separates the data set in half The median of the lower half of a set of data The median of the upper half of a set of data The difference between the upper quartile and the lower quartile Example Find the range and interquartile range for each set of data. a. {3, 1, 17,, 1, 14, 14, 8} Step 1 List the data from least to greatest. The range is 1 or 19. Then find the median median 14 1 or 13 Step Find the upper and lower quartiles. LQ or median UQ or 15.5 The interquartile range is or 10. WEATHER For 1 and, use the data in the stem-and-leaf plot at the right. 1. Find the range, median, upper quartile, Average Extreme Jul Temperatures in World Cities lower quartile, and the interquartile Low Temps. High Temps. range for each set of data. Jul lows: range: 30 F; median: 73; UQ: 76; LQ: 59; interquartile range: 17 F; Jul highs: range: 43 F; median: 83; UQ: 91; LQ: 79; interquartile range: 1 F Write a sentence that compares the data Jul highs var more widel than F F Jul lows. Glencoe/McGraw-Hill 67 Glencoe Pre-Algebra b. Stem Leaf The stem-and-leaf plot displas the data in order. The greatest value is 56. The least value is. So, the range is 56 or 34. The median is 4. The LQ is 31 and the UQ is 48. So, the interquartile range is or 17.

89 1-3 Box-and-Whisker Plots Box-and- Whisker Plot Words Model A box and whisker plot divides a set of data into four parts using the median and quartiles. Each of these parts contains 5% of the data. lower extreme, or least value LQ median UQ upper extreme, or greatest value Example Step 1 Find the least and greatest number. Then draw a number line that covers the range of the data. 0 50,000 5, , ,000 00,000 50,000 75,000 15, ,000 5,000 Step Mark the median, the extremes, and the quartiles. Mark these points above the number line. lower extreme: 0 0 median: 1,500 50,000 5,000 FOOD The heat levels of popular chile peppers are shown in the table. Displa the data in a box-and-whisker plot. LQ: 4000 UQ: 75, , ,000 00,000 50,000 75,000 15, ,000 5,000 Step 3 Draw a box and the whiskers. upper extreme: 10,000 Heat Level of Chile Peppers Name Heat Level* Aji escabeche 17,000 Bell 0 Caenne 8,000 Habañero 10,000 Jalapeño 5,000 Mulato 1,000 New Mexico 4,500 Pasilla 5,500 Serrano 4,000 Tabasco 10,000 Tepín 75,000 Thai hot 60,000 Lesson ,000 5, , ,000 00,000 50,000 75,000 15, ,000 5,000 Source: Chile Pepper Institute *Scoville heat units Draw a box-and-whisker plot for each set of data. 1. {17, 5, 8, 33, 5, 5, 1, 3, 16, 11,, 31, 9, 11}. {$1, $50, $78, $13, $45, $5, $1, $37, $61, $11, $77, $31, $19, $11, $9, $16} Glencoe/McGraw-Hill 677 Glencoe Pre-Algebra

90 1-4 Histograms Histograms Example A histogram uses bars to displa numerical data that have been organized into equal intervals. There is no space between bars. Because the intervals are equal, all of the bars have the same width. Intervals with a frequenc of 0 have no bar. ELEVATIONS The frequenc table shows the highest elevations in each state. Displa the data in a histogram. Step 1 Draw and label the axes as shown. Include the title. Step Show the frequenc intervals on the horizontal axis and an interval of on the vertical axis. Step 3 For each elevation interval, draw a bar whose height is given b the frequenc. Highest Elevation in Each State Elevation (ft) Tall Frequenc ,999 1,000 15, ,000 19, ,000 3,999 1 Source: Highest Elevation in Each State Elevation (ft) For 1 3, use the information shown in the table below. 1. The frequenc table shows voter participation in Displa the data in a histogram. Number of States ,999 4,000 7,999 8,000 11,999 1,000 15,999 16,000 0,000 19,999 3,999. How man states had a voter turnout greater than 50 percent? How man states had fewer than 40 percent voting? 13 Voter Participation b State (1998) Percent voting Tall Frequenc Source: U.S. Census Bureau Glencoe/McGraw-Hill 68 Glencoe Pre-Algebra

91 1-5 Misleading Statistics Example The graphs below show the increase in the number of events held in the Summer Olmpics from 1948 to Number of Events Events in Summer Olmpics Graph A '48 '5 '56 '60 '64 '68 '7 '76 '80 '84 '88 '9 '96 Year Number of Events Events in Summer Olmpics Graph B '48 '5 '56 '60 '64 '68 '7 '76 '80 '84 '88 '9 '96 Year a. What causes the graphs to differ in their appearance? Different vertical scales and the lack of a zero on Graph A s axis result in different visual impressions. b. Which graph appears to show a more rapid increase in the number of events held in the Summer Olmpics? Explain. Graph A; the steeper slope of the line and the higher position of the line relative to the scale make it appear that the number of events is greater and increasing faster. For 1 3, refer to the graphs below. Source: National Oceanic and Atmospheric Administration Temperature ( F) Highest Temperatures Graph A Alaska California Florida Hawaii Texas State Source: National Oceanic and Atmospheric Administration Temperature ( F) Highest Temperatures Graph B Alaska California Florida Hawaii Texas State 1. What is the highest recorded temperature in California? In Alaska? 134 F; 100 F. In which graph does the difference between these two temperatures appear to be minimized? Graph B 3. How do the graphs create different appearances? The vertical scale on Graph A has a smaller range than the scale on Graph B. Glencoe/McGraw-Hill 687 Glencoe Pre-Algebra Lesson 1-5

92 1-6 Counting Outcomes How man different combinations of beverage and bread can be made from 3 beverage choices and 3 bread choices? Draw a tree diagram to find the number of different combinations. Coffee Tea Juice List each beverage choice. Muffin Scone Bagel Muffin Scone Bagel Muffin Scone Each beverage choice is paired with each bread choice. Beverage Bread Outcome Bagel Coffee, Bagel Coffee, Muffin Coffee, Scone Tea, Bagel Tea, Muffin Tea, Scone Juice, Bagel Juice, Muffin Juice, Scone There are 9 possible outcomes. The Fundamental Counting Principle also relates the number of outcomes to the number of choices. When ou know the number of outcomes, ou can find the probabilit that an event will occur. Example Refer to. What is the probabilit of randoml selecting coffee with a scone? Use the Fundamental Counting Principle to find the number of outcomes. beverage choices times bread choices Onl one of the 9 possible outcomes is coffee and a scone. So, the probabilit of randoml selecting coffee with a scone is 1 9. Find the number of possible outcomes for each situation. 1. one six-sided number cube is rolled, and one card is drawn from a 5-card deck 31. There are 51 juniors and 498 seniors. One junior and one senior are randoml drawn as raffle winners. 54,976 Find the probabilit of each event. 3. A coin is tossed and a card is drawn from a 5-card deck. What is the probabilit of 1 getting tails and the ten of diamonds? Four coins are tossed. What is the probabilit of four tails? 1 6 equals total number of possible outcomes Glencoe/McGraw-Hill 69 Glencoe Pre-Algebra

93 1-7 Permutations and Combinations Permutations Words Smbols An arrangement or listing in which order is important is called a permutation. P(m, n) means m number of choices taken n at a time. Example P(3, ) 3 6 Combinations Example Words SPORTS How man was can the top five finishers be arranged in a 0-person cross-countr race? Order is important. So, this arrangement is a permutation. 0 runners choose 5 P(0, 5) ,860,480 was 0 choices for 1st place 19 choices for nd place 18 choices for 3rd place 17 choices for 4th place 16 choices for 5th place SCHOOL In a science class with 4 students, how man 3-person lab teams can be formed? From 4 students, take 3 at a time. Order is not important. So, this arrangement is a combination. An arrangement or listing where order is not important is called a combination. Smbols C(m, n) P(m, n) n! Example C(6, ) P(6, )! 6 5 or 15 1 P(4, 3) C(4, 3) 3! or 11,480 lab teams 3 1 Lesson 1-7 Tell whether each situation is a permutation or combination. Then solve. 1. How man was can three people be selected from a group of seven? C; 35 was. How man was can a 6-person kickball team be chosen from 7 students? C; 96,010 was 3. How man was can 15 actors fill 6 roles in a pla? P; 3,603,600 was 4. How man was can 5 books be borrowed from a collection of 40 books? C; 658,008 was 5. JOBS A telemarketing firm has 35 applicants for 8 identical entr-level positions. How man was can the firm choose 8 emploees? C; 3,535,80 was 6. FOOD A pizza place sends neighbors a coupon for a 4 topping pizza of an size. If the pizzeria has 15 toppings and 3 sizes to choose from, how man possible pizzas could be purchased using the coupon? C; 4,095 pizzas Glencoe/McGraw-Hill 697 Glencoe Pre-Algebra

94 1-8 Odds Odds The odds in favor of an outcome is the ratio of the number of was the outcome can occur to the number of was the outcome cannot occur. Odds in favor number of successes : number of failures The odds against an outcome is the ratio of the number of was the outcome cannot occur to the number of was the outcome can occur. Odds against number of failures : number of successes Example Find the odds of a sum of 11 if a pair of number cubes are rolled. There are 6 6 or 36 sums possible for rolling a pair of number cubes. There are sums of 11. The are (5, 6) and (6, 5). There are 36 or 34 sums that are not 11. Odds of rolling a sum of 11 In a bag of marbles, 30 are blue, 15 are orange, 0 are ellow, and 10 are green. What are the odds against drawing a green marble from the bag? There are marbles that are not green. Odds against drawing a green marble numbers of was to roll a sum of 11 to : :34 or 1:17 numbers of was to roll an other sum 34 The odds of rolling an 11 are 1:17. numbers of was to draw a marble that is not green 65 : 65:10 or 13: numbers of was to draw a green marble 10 The odds against drawing a green marble are 13:. to Find the odds of each outcome if a pair of number cubes are rolled. 1. not a sum of 7 5:1. a sum less than 5 or greater than 9 1: 3. a sum that is a multiple of 3 1: 4. a double number 1:5 5. not a double number 5:1 6. double twos 35:1 7. A hocke team scores 3 goals for ever 30 shots taken. What are the odds in favor of the next shot being a goal? 1:9 8. A bookshelf holds 1 msteries, 3 histor books, 4 how-to books, and 11 self-help books. What are the odds of randoml picking a how-to book? :13 9. A professional football team finished the season with a record of 10 wins and 6 losses. What were its odds that season of winning a game? 5:3 10. After man sessions of plaing rock-paper-scissors with Joe, ou discover that he throws rock 55 percent of the time. What are the odds in favor of his next throw being rock? 11:9 Glencoe/McGraw-Hill 70 Glencoe Pre-Algebra

95 1-9 Probabilit of Compound Events Probabilit of Two Independent Events Words Smbols The probabilit of two independent events is found b multipling the probabilit of the first event b the probabilit of the second event. P(A and B) P(A) P(B) Probabilit of Two Dependent Events Words If two events, A and B, are dependent, then the probabilit of events occurring is the product of the probabilit of A and the probabilit of B after A occurs. Smbols P(A and B) P(A) P(B following A) GAMES A card is drawn from a standard deck of 5 cards. The card is replaced and another is drawn. Find the probabilit if the first card is the 3 of hearts and the second card is the of clubs. Since the first card is replaced, the events are independent. P(3 of hearts and of clubs) P(3 of hearts) P( of clubs) The probabilit is Probabilit of Mutuall Exclusive Events Example Words Smbols The spinner at the right is spun. What is the probabilit that the spinner will stop on 7 or an even number? The events are mutuall exclusive because the spinner cannot stop on both 7 and an even number at the same time. P(7 or even) = P(7) + P(even) = = 5 8 The probabilit of one or the other of two mutuall exclusive events can be found b adding the probabilit of the first event to the probabilit of the second event. P(A or B) P(A) P(B) The probabilit that the spinner will stop on 7 or an even number is Lesson 1-9 A card is drawn from a standard deck of cards. The card is not replaced and a second card is drawn. Find each probabilit P(4 and 8) P(queen of hearts and 10) A card is drawn from a standard deck of cards. Find each probabilit. 3. P(queen of clubs or a red card) P(queen of hearts or 10) 5 5 Glencoe/McGraw-Hill 707 Glencoe Pre-Algebra

96 13-1 Polnomials Polnomials are classified according to the number of terms the have. A monomial has one term, a binomial has two terms, and a trinomial has three terms. The exponent of a variable in a monomial must be a whole number, and the variable cannot be in the denominator or under a radical sign. Determine whether each expression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial. a. 3 The expression is not a polnomial because 3 has a variable in the denominator. b. 3 a 6a 3 5a 4 4 The expression is a polnomial with three terms, so it is a trinomial. A polnomial also has a degree. The degree of a polnomial is the same as that of the term with the greatest degree. The degree of a term is the sum of the exponents of its variables. Lesson 13-1 Example Find the degree of each polnomial. a. x 6 3x 4 1 The greatest degree is 6, so the degree of the trinomial is 6. b. 10b c 8bc c 10b c has degree 1 or 3. 8bc has degree 1 1 or. c has degree. The greatest degree is 3, so the trinomial has degree 3. Determine whether each expression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial. 1. 7q r 10. 8r 3. x 4 es; trinomial no es; binomial v 4w 6. a 5 b c es; monomial es; binomial es; trinomial Find the degree of each polnomial h 1 9. x p 3 6p mn 5 mn 4 m x 4x Glencoe/McGraw-Hill 733 Glencoe Pre-Algebra

97 13- Adding Polnomials Add polnomials b combining like terms, which are monomials that contain the same variables to the same power. Example Method 1 Add verticall. 8x 7x 1 ( ) x 5 9x 7x 6 Find each sum. Find (8x 7x 1) (x 5). Method Add horizontall. (8x 7x 1) (x 5) = (8x x ) 7x (1 5) = 9x 7x x 7 ( ) x 1 4x w 6w + 3 ( ) w 5 5w 6w. 6d 8 ( ) 4d 1 d a a ( ) a 5 5a a 5 5. ( m 3) (7m 1) 6. (9x 3x 1) (4x 1) 6m 9x 7x 7. (k k) (k 1) 8. (5a 6ab) ( ab b ) k 1 5a 5ab b 9. (4c 7) + (c 3c 6) 10. (x ) (x ) 5c 3c 1 x x 11. (1h 6) (h 8h 6) 1. (10x x 5) (x 10x ) h 4h x (6 1) ( 3 6) 14. ( p 3 4) (p p 3) p 3 p p (3g 3g 5) (5g 3) 16. (5r 6) ( r 4r 7) 8g 3g 4r 4r 1 Glencoe/McGraw-Hill 738 Glencoe Pre-Algebra

98 13-3 Subtracting Polnomials To subtract polnomials, subtract like terms. Example Method 1 Subtract verticall. x 3x 6 ( ) 4x 1 3x 3x 5 Find each difference. Find (x 3x 6) (4x 1). Method Add the additive inverse of 4x 1, which is ( 1)(4x 1) or 4x 1. (x 3x 6) (4x 1) (x 3x 6) ( 4x 1) (x 4x ) (3x) ( 6 1) 3x 3x c 7 ( ) 3c 3 c k 4k 5 ( ) k 5 8k 4k 10. m 5 ( ) 8m 1 10m z z ( ) 3z 5 3z 4z 5 5. ( 6r 3) (7r ) 6. (8f 7f 3) (f 4) 13r 1 8f 9f 7 7. (5n n) (3n 9) 8. (a 5ab) ( ab 3b ) 5n 5n 9 a 7ab 3b 9. (6g 8) (5g g 6) 10. (8x 3) (x 3) g g 8x x 6 Lesson (n 1) (n n 9) 1. (h h 1) (3h 7h ) n 1 8h 5h ( 1) ( 1) 14. (6p 5p 1) (p 4) 6p 7p (4q q) (q 3) 16. (6v 8) (7v v 5) 3q q 3 v v (u u 4) (5u 4) 18. (9b ) ( b b 9) 4u u 10b b 7 Glencoe/McGraw-Hill 743 Glencoe Pre-Algebra

99 13-4 Multipling a Polnomial b a Monomial The Distributive Propert can be used to multipl a polnomial b a monomial. Find 7(4x 8). 7(4x 8) 7(4x) 7(8) 8x 56 Example Find (x 5x 4)( x). (x 5x 4)( x) x ( x) 5x( x) 4( x) x 3 10x 8x Find each product. 1. 5(7 4). (3h 6)4 3. 9(q 8) h 4 9q (d ) 5. (4g 5)( ) 6. 7(4x 7) 6d 1 8g 10 8x (n 3n 9) 8. (a ab b )5 9. r(r 9) n 6n 18 5a 10ab 5b r 9r 10. (b 4)( b) 11. x(3x 6) 1. (k 9)(k ) b 3 4b 3x 6x k 3 9k 13. m(6m 1) 14. p(7p ) 15. (8 3h)( h) 6m m 7p p 3h 8h 16. w(4w w 3) 17. ab(a b) 18. x(7x ) 4w 3 w 3w a b ab 7x x 19. (m mn n)m 0. (5 1) 1. 10u(u 5) m 3 m n mn 10 10u 50u. (5r r)( 3r) 3. 8z(z 7) 4. 5b (6b ) 15r 3 6r 16z 56z 30b 3 10b 5. 4p (6p 3p) 6. (5v v 4)( v) (3 8) 4p 4 1p 3 10v 3 4v 8v m(m 4n) 9. (8gh 3h)( 3gh) 30. 5a(a 3ab b) 6m 1mn 4g h 9gh 10a 15a b 5ab Glencoe/McGraw-Hill 748 Glencoe Pre-Algebra

100 13-5 Linear and Nonlinear Functions Linear functions have constant rates of change. Their graphs are straight lines and their equations can be written in the form mx b. Nonlinear functions do not have constant rates of change and their graphs are not straight lines. Determine whether each equation represents a linear or nonlinear function. a. 9 This is linear because it can be written as 0x 9. b. x 4 This is nonlinear because the exponent of x is not 1, so the equation cannot be written in the form mx b. Tables can represent functions. A nonlinear function does not increase or decrease at a constant rate. Example Determine whether each table represents a linear or nonlinear function. a. As x increases b. x b, increases x 0 7 b 8. The rate of change is 5 1 constant, so this is a linear function As x increases b 5, decreases b a greater amount each time. The rate of change is not constant, so this is a nonlinear function. Determine whether each equation or table represents a linear or nonlinear function. Explain. 1. x 3 9 Linear; equation can. 8 Nonlinear; equation cannot x be written as 1 x 3. be written in the form mx b x(x 1) Nonlinear; equation x Linear; equation can be cannot be written in the form mx b. written as 5x x x 0 4 Linear; rate of 1 1 change is 14 constant Nonlinear; rate of change is not constant. Lesson 13-5 Glencoe/McGraw-Hill 753 Glencoe Pre-Algebra

101 13-6 Graphing Quadratic and Cubic Functions To graph a quadratic or cubic function, make a table of values and then plot the points. Example x 1 3 Graph = x O x x 3 1 Graph each function. 1. x. x O x x 4. x 3 O x 5. x 6. x 3 O x O O O x x x Glencoe/McGraw-Hill 758 Glencoe Pre-Algebra