One of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan.

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1 PROBLEM SOLVING One of our primar goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. Step Step Step Step Understand the problem. Read the problem carefull. Organize the given information and decide what ou need to find. Check for unnecessar or missing information. Suppl missing facts, if possible. Make a plan to solve the problem. Choose a problem-solving strateg. (See the net page for a list.) Choose the correct operations. Decide if ou will use a tool such as a calculator, a graph, or a spreadsheet. Carr out the plan to solve the problem. Use the strateg and an tools ou have chosen. Estimate before ou calculate, if possible. Do an calculations that are needed. Answer the question that the problem asks. 4 Check to see if our answer is reasonable. Reread the problem. See if our answer agrees with the given information and with an estimate ou calculated. EXAMPLE Eight people can be seated evenl around a rectangular table, with one person on each end. How man people can be seated around three of these tables placed end-to-end? You know each table is rectangular, seats eight people, and can fit one person on each end. You need to find the number of people that can be seated at three tables placed end-to-end. An appropriate strateg is to draw a diagram. Draw one table, with an X for each person seated. Notice that three people can be seated at each long side of the table. Then draw three tables placed end-to-end. Draw Xs and count them. There are 0 Xs on the diagram, so 0 people can be seated. 4 At three individual tables ou can seat 4 people. Since seats are lost when tables are placed end-to-end, 0 is a reasonable answer.

2 In Step of the problem-solving plan on the previous page ou select a strateg. Here are some problem-solving strategies to consider. Guess, check, and revise. Draw a diagram or a graph. Make an organized list or table. Use an equation or a formula. Use a proportion. Look for a pattern. Break into simpler parts. Solve a simpler problem. Work backward. Act out the situation. Use when ou do not seem to have enough information. Use when words describe a visual representation. Use when ou have data or choices to organize. Use when ou know a relationship between quantities. Use when ou know that two ratios are equal. Use when ou can eamine several cases. Use when ou have a multi-step problem. Use when smaller numbers help ou understand the problem. Use when ou look for a fact leading to a known result. Use when visualizing the problem is helpful.. During the month of Ma, Rosa made deposits of $8.50 and $65.9 into her checking account. She wrote checks for $97.46 $5, $55., and $8.98. If her account balance at the end of Ma was $7.05, what was her balance at the beginning of Ma? 5 d. A rectangular room measures 9 feet b 5 feet. How man square ards of carpet are needed to cover the floor of this room?. You make 0 silk flower arrangements and plan to sell them at a craft show. Each arrangement costs $ in materials, and our at least 6 flower arrangements booth at the show costs $0. If ou price the arrangements at $4 each, how man must ou sell to make at least $00 profit? 4. A store sells sweatshirts in small, medium, large, and 0 kinds etra large. The color choices are red, white, blue, gra, and black. How man different kinds of sweatshirts are sold at the store? $ If 4.6 pounds of ham cost $6.77, what would.75 pounds cost? five 4 stamps and si 0 stamps 6. Roger bought some 4 stamps and some 0 stamps, and spent $.90. How man of each tpe of stamp did Roger bu? 4 different orders 7. Abigail, Bonnie, Carla, and Dominique are competing in a race. In how man different orders can the athletes finish the race? 8. Five bos are standing in Charlie, a line. Sam Eric, Ale, is before Sam, Mark Mark and immediatel after Ale. Eric is net to Charlie and Ale. List the order of the bos in line. 0 diagonals 9. How man diagonals can be drawn on a stop sign?

3 DECIMALS The steps for adding, subtracting, multipling, and dividing with decimals are like those for computing with whole numbers. Add or subtract. a b Write in vertical form, lining up the decimal points. Use zeros as placeholders. a. 5.0 placeholder b..80 placeholder placeholders 0.99 ANSWER ANSWER Multipl or divide. a b a. Write in vertical form. b. Write in long division form: The number of decimal places Move the decimal points the same in the product is equal to the number of places so that the divisor sum of the number of decimal is a whole number: places in the factors. Then divide decimal places decimal place decimal places 8 0 Line up the decimal point in the quotient with the decimal point in the dividend. ANSWER ANSWER Evaluate

4 FRACTION CONCEPTS Multipl or divide the numerator and denominator of a fraction b the same nonzero number to write an equivalent fraction. To write a fraction in simplest form, divide the numerator and denominator b their greatest common factor. Two numbers are reciprocals if their product is. 4 EXAMPLE Write two fractions equivalent to. 0 Divide the numerator and Multipl the numerator denominator b : and denominator b : EXAMPLE Write the fraction in simplest form. The greatest common factor of 8 and is Divide the numerator and denominator b 4: 4. Find the reciprocal of the number. a. 4 b. 0 5 a. Switch the numerator and the b. Write 0 as a fraction, 0. denominator. The reciprocal is 5 4. The reciprocal is. 0 Write two fractions equivalent to the given fraction , 9 4 0, ,, , 5 50 Write the fraction in simplest form Find the reciprocal of the number. 5. Sample answers are given

5 FRACTIONS AND DECIMALS Divide to write a fraction as a decimal. If the remainder is ever zero, the result is a terminating decimal. If the quotient has a digit or group of digits that repeats, the result is a repeating decimal. Write the fraction as a decimal. a. 5 8 b. Divide the numerator b the denominator. a terminating decimal b repeating decimal ANSWER ANSWER A bar indicates repeating digits. Round for an approimation. Write the decimal as a fraction in simplest form. a. 0.5 b. 0.8 a. 0.5 is fifteen hundredths. b Write an equation Multipl each side b Find Use simplest form ANSWER 0.5 ANSWER Solve and simplif. Write the fraction as a decimal. For repeating decimals, also round to the nearest hundredth for an approimation Write the decimal as a fraction in simplest form

6 ADDING AND SUBTRACTING FRACTIONS To add or subtract two fractions with the same denominator, add or subtract the numerators. Write the result in simplest form. Add or subtract. a. 8 8 b a b To add or subtract two fractions with different denominators, write equivalent fractions with a common denominator. Then add or subtract and write the result in simplest form. Add or subtract. 4 a. 5 5 b. 4 6 a. Write 5 as 6. b. Write 5 4 as 9 and 6 as Add or subtract. Write the answer in simplest form

7 MULTIPLYING AND DIVIDING FRACTIONS To multipl two fractions, multipl the numerators and multipl the denominators. Then write the result in simplest form. Multipl. a b a b To divide b a fraction, multipl b its reciprocal and write the product in simplest form. Divide. a b a b Multipl or divide. Write the answer in simplest form

8 RATIO AND PROPORTION The ratio of a to b is a. The ratio of a to b can also be written as a to b b or as a : b. Because a ratio is a quotient, its denominator cannot be zero. A geometr class consists of 6 female students, male students, and teachers. Write each ratio in simplest form. a. male students : female students b. students : teachers a. 6 4 b Simplif the ratio. a. cm 6 ft b. 4 m 8 in. Epress both quantities in the same units of measure so that the units divide out. Write the fraction in simplest form. a. cm cm 4 m 4 p 00 cm ft b. 6 p i 8 in. 8 n. in. 4 A proportion is an equation showing that two ratios are equal. If the ratio a b is equal to the ratio c, then the following d proportion can be written: a b c where a, b, c, and d are not equal to zero d The numbers a and d are the etremes of the proportion. The numbers b and c are the means of the proportion. Here are two properties that are useful when solving a proportion. Cross Product Propert the product of the means. The product of the etremes equals If a b c, then ad bc. d Reciprocal Propert If two ratios are equal, then their reciprocals are also equal. If a b c b d, then a d c.

9 Solve the proportion. 5 a. 6 9 b a. 6 9 b (5) Use cross products. Use reciprocals or 9 p 7 4 or An algebra class consists of 0 female students, 5 male students, and teachers. Write the ratio in simplest form female students : teachers. students : teachers. female students : male students 4. teachers : male students 5. teachers : students 6. male students : female students 5 5 Write the ratio of length to width for the rectangle cm 4 cm ft 6 ft 8 in. 8 in. Simplif the ratio. d lb.. 4 0cm 0 ft 0 0 oz 56 m kg in g ft 5. 6 mm 90 cm 5 5 Solve the proportion or

10 INEQUALITIES AND ABSOLUTE VALUE When ou compare two numbers a and b, a must be less than, equal to, or greater than b. You can compare two whole numbers or positive decimals b comparing the digits of the numbers from left to right. Find the first place in which the digits are different. a is less than b. a is equal to b. a is greater than b. a < b a b a > b Compare the two numbers. a..9 and. b. 4 and Use a number line. The numbers increase from left to right. a...9 b is to the right of., so.9 >.. 4 is to the left of, so 4 <. Also 9 >, so.9 >.. The absolute value of a number is its distance from zero on a number line. The smbol a represents the absolute value of a. Evaluate. a. b. Use a number line. a. b. 0 0 is units from 0, so. is units from 0, so. Compare the two numbers. Write the answer using <, >, or.. 6 and 6 6 > and 47. and 8 > and 5 5 < < and and and and < < < 0..4 >.4 Write the numbers in order from least to greatest. 9., 0, 9, 8, 4, 6, ,,.5,.5, 0.05,,.5 8, 6,, 0,, 4, 9.5,,.5, 0.5, 0.05,,.5. 54, 54, 54, 54, , 0.4, 0.6, 0.4, 0. 54, 54, 54, 54, , 0.4, 0.9, 0., 0.6 Evaluate

11 INTEGERS You can use a number line to add two integers. Move to the right to add a positive integer. Move to the left to add a negative integer. Add. a. 4 b. ( ) Use the number lines below. a. b Start at 4. Go units to the right. End at. So, 4. Start at. Go units to the left. End at 5. So, ( ) 5. Subtract. a. 7 9 b. ( 6) To subtract an integer, add its opposite. a ( 9) b. ( 6) 6 8 When ou multipl or divide integers, use these rules. The product or quotient of two integers with the same sign is positive. The product or quotient of two integers with opposite signs is negative. Multipl or divide. a. ( )(4) b. 7 ( 9) a. ( )(4) negative product b. 7 ( 9) positive quotient opposite signs same signs. 8 ( ) 0. 0 ( ) ( 9) ( ) ( 5) ( ) ( 4) 6 0. ( 7)( ) 7. ( )( 0) 60. ( 0)(5)() ( ) ( 8) 6. 8 ( ) ( 9) ( 4) 5 9. ( 5) 6 0. ( 9)( )(4) 7. ( 4)(5) ( 7) 7 5. ( 7) ( )( ) ( )

12 THE COORDINATE PLANE A coordinate plane is formed b two number lines that intersect at the origin. The horizontal number line is the -ais, and the vertical number line is the -ais. Each point in a coordinate plane corresponds to an ordered pair of real numbers. The ordered pair for the origin is (0, 0). Point W(, ), shown on the graph at the right, has an -coordinate of and a -coordinate of. From the origin, point W is located units to the right and units down. vertical or -ais Z 4 origin (0, 0) 4 horizontal or -ais O 4 4 W(, ) EXAMPLE Use the graph above to name the coordinates of point Z. From the origin, point Z is located 4 units to the left and unit up. The coordinates of point Z are ( 4, ). Plot each point in a coordinate plane. a. P( 5, ) b. Q(, 0) P( 5, ) a. Start at the origin. Move 5 units left and units up. b. Start at the origin. Move units right and 0 units up. P(, 0) Name the coordinates of each point.. A (, 0). B (, ). C (, ) 4. D ( 5, ) 5. E ( 5, ) 6. F (, ) 7. G (5, ) 8. H (4, ) 9. J (, 5) 0. K ( 4, ). M (, ). N ( 4, ) E D K N J A F B M C H G Plot the points in a coordinate plane. -8. See margin.. A(4, 6) 4. B(, ) 5. C(, ) 6. D(0, 4) 7. E( 6, 6) 8. F(5, 5) 9. G(4, 0) 0. H(, 4). J(0, 5). K(, ). L( 6, ) 4. M(, ) 5. N( 4, 6) 6. P(, 5) 7. Q( 5, ) 8. R(, 0)

13 8. N D A F L E P B R H J M G K C P

14 SLOPE OF A LINE The slope of a line is the ratio of the vertical rise to the horizontal run between an two points on the line. You subtract coordinates to find the rise and the run. If a line passes through the points (, ) and (, ), then slope r ise. run (, ) (, ) run rise EXAMPLE Find the slope of the line that passes through the points (, ) and (5, 4). Let (, ) (, ) and (, ) (5, 4). slope 4 5 ( ) 7 (, ) 7 (5, 4) The slope of a line can be positive, negative, zero, or undefined. Positive slope Negative slope Zero slope Undefined slope rising line falling line horizontal line vertical line Find the slope of the line that passes through the points.. (0, ) and (6, ). (, ) and (8, 4) 5. (, 0) and (, 4) 4. (, ) and (5, ) 0 5. (, ) and ( 4, 4) 6. (, 0) and (, 5) 7. (4, ) and (, ) 8. (4, ) and (, 6) 4 9. ( 4, 5) and (0, 5) 0 Plot the points and draw the line that passes through them. Determine whether the slope is positive, negative, zero, or undefined Check graphs. 0. (, ) and (, 5). (, 4) and (4, ) negative. (, ) and (, 4) positive undefined. (0, ) and (5, ) zero 4. (, 0) and (, ) undefined 5. (, ) and (, ) positive

15 GRAPHING LINEAR EQUATIONS Equations like 6 and 5 are linear equations. A solution of a linear equation is an ordered pair (, ) that makes the equation true. The graph of all solutions of a linear equation is a line. EXAMPLE Graph the equation 4. Solve the equation for : 4, so 4. Make a table of values to graph the equation. (, 6) 4 (, ) 4( ) (, ) 0 4(0) (0, ) 4() 6 (, 6) (, ) (0, ) Plot the points in the table and draw a line through them. You can use intercepts (points where the graph crosses the -ais and -ais) to graph linear equations. EXAMPLE Graph the equation 4. To find the -intercept, To find the -intercept, Plot (4, 0) and (0, ) and substitute 0 for. substitute 0 for. draw a line through them (0) (0) 4 (4, 0) 4 So, (4, 0) is a solution. So, (0, ) is a solution. (0, ) Graph the equation using a table of values or intercepts. See margin

16 See Additional Answers beginning on page AA.

17 SLOPE-INTERCEPT FORM A linear equation m b is written in slope-intercept form. The slope of the line is m and the -intercept is b. EXAMPLE Graph the equation 4. Write the equation in slope-intercept Plot the point (0, ). Use the slope to form. locate other points on the line The slope is and the -intercept is (0, ) 4 Graphs of linear equations of the form b are horizontal lines with slope 0. Graphs of linear equations of the form a are vertical lines with undefined slope. Graph the equation. a. b. a. b. Graph the equation. See margin Lines with the same slope are parallel. Tell whether the graphs of the equations are parallel or not parallel parallel not parallel not parallel parallel

19 POWERS AND SQUARE ROOTS An epression like 5 is called a power. The eponent represents the number of times the base 5 is used as a factor: 5 5 p 5 p 5 5. Evaluate. a. 4 5 b. ( 0) a p 4 p 4 p 4 p 4 04 b. ( 0) ( 0)( 0) 00 If b a, then b is a square root of a. Ever positive number has two square roots, one positive and one negative. The two square roots of 6 are 4 and 4 because 4 6 and ( 4) 6. The radical smbol indicates the nonnegative square root, so 6 4. Find all square roots of the number. a. 5 b. 8 a. Since 5 5 and ( 5) 5, b. Since 8 is negative, it has the square roots are 5 and 5. no square roots. There is no real number ou can square to get 8. The square of an integer is a perfect square, so the square root of a perfect square is an integer. Integer, (n) Perfect square, (n ) You can approimate the square root of a positive number that is not a perfect square b using a calculator and rounding. Evaluate. Give the eact value if possible. Otherwise, approimate to the nearest tenth. a. 4 9 b. 5 a. Since 49 is a perfect square with 7 49, b. Since 5 is not a perfect square, use a calculator and round: 5..

20 A number or epression inside a radical smbol is called a radicand. The simplest form of a radical epression is an epression that has no perfect square factors other than in the radicand, no fractions in the radicand, and no radicals in the denominator of a fraction. You can use the following properties to simplif radical epressions. Product Propert of Radicals a b a p b where a 0 and b 0 Quotient Propert of Radicals a a where a 0 and b > 0 b b Simplif. a. 8 b. 9 4 c. a. 8 9 p 9 p p Factor using perfect square factor. 9 4 b. 9 4 Use the quotient propert and simplif. c. p Write an equivalent fraction that has no radicals in the denominator. Evaluate ( ) 9. ( ) , ( 9) Find all square roots of the number or write no real square roots , ,. 49 7, 7 no real square roots , 8 no real square roots no real square roots Evaluate. Give the eact value if possible. If not, approimate to the nearest tenth Simplif

21 EVALUATING EXPRESSIONS To evaluate a numerical epression involving more than one operation, follow the order of operations. First do operations that occur within grouping smbols. Then evaluate powers. Then do multiplications and divisions from left to right. 4 Finall, do additions and subtractions from left to right. EXAMPLE Evaluate the epression (4 7) ( 6). (4 7) ( 6) ( ) ( 6) Evaluate within parentheses. 9 ( 6) Evaluate the power: ( ) 9. (.5).5 Do the division 9 ( 6). Do the subtraction (.5). To evaluate a variable epression, substitute a value for each variable and use the order of operations to simplif. EXAMPLE Evaluate the epression 8 when Substitute 5 for each Evaluate the power: 5 5. Add and subtract from left to right. Evaluate the epression.. 80 (0 45) 05. (8 ) p p 4. 8 ( 6) (7 ) 5 7. (00 74) p p 4 4 Evaluate the epression when n. 0. n 7. ( n) 8. n(n 7). n 4. n 7n 6 5. (n ) p n n 6. n n (n )(n ) 5

22 THE DISTRIBUTIVE PROPERTY Here are four forms of the distributive propert. a(b c) ab ac a(b c) ab ac (b c)a ba ca (b c)a ba ca Use the distributive propert to write the epression without parentheses. a. ( 4) b. 5 (n ) a. ( 4) () (4) 4 b. 5 (n ) 5 ( )(n ) 5 ( )(n) ( )( ) 5 n 7 n When an epression is written as a sum, the parts that are added are the terms of the epression. Like terms are terms in an epression that have the same variable raised to the same power. Numbers are also considered to be like terms. You can use the distributive propert to combine like terms. Simplif the epression. a. 4 7 b. ( ) (4 ) a. 4 7 ( 4 7) b. ( ) (4 ) 4 Write without parentheses. ( 4) Group and combine like terms. Simplif within parentheses. Use the distributive propert to write the epression without parentheses.. (a 4) a 8. (k )7 4k 7. ( ) 4. (7 z)z 7z z 5. ( 9) 9 6. (j )( ) j 7. 4b(b ) 4b b 8. (n 6) n Simplif the epression. 9. m 4 7m 6m a a a 9. ( 7) 4. 8 ( 4) a 0 4 6a h h(h ) h h 6. ( 4) ( 6) 6

23 SOLVING ONE-STEP EQUATIONS A solution of an equation is a value for the variable that makes a true statement. You can solve an equation b writing an equivalent equation (an equation with the same solution) that has the variable alone on one side. Here are four was to solve a one-step equation. Add the same number to each side of the equation. Subtract the same number from each side of the equation. Multipl each side of the equation b the same nonzero number. Divide each side of the equation b the same nonzero number. Solve the equation. a. 7 b. 6 c. n 0 d. 4c 5 Choose an operation to perform that will leave the variable alone on one side. Check our solution b substituting it back into the original equation. a. Add 7 to each side. b. Subtract 6 from each side CHECK CHECK 8 6 c. Multipl each side b 5. d. Divide each side b 4. n 0 5 4c 5 p n 5 5 p 0 4c 4 4 n 50 c CHECK CHECK 4( ) Solve the equation.. k r ( 9) 0. w n ( 4) 8. 6 c n 6 8. a 5 5. d a 4 4. h 4 5. b 6. 4z m w 4 5 6

24 SOLVING MULTI-STEP EQUATIONS Sometimes solving an equation requires more than one step. Use the techniques for solving one-step equations given on the previous page. Simplif one or both sides of the equation first, if needed, b using the distributive propert or combining like terms. Solve the equation. a. 5 b a. 5 b Add Subtract 5. 8 Simplif. 9 Simplif. 8 Divide b. 9 Subtract. 4 Simplif. 0 Simplif. CHECK (4) 5 CHECK 6( 0) 5( 0) 9 Solve the equation z a n m 4. r (r ) 4. 5(z ) g 8 g 6. 4(t 7) d (6 d) (6 ) a 5 7a n 9 n 4. 4r 8 r. 4 6p p j (j ) 6. ( ) (r 6) n (b ) 8b. c 7(9 c). 4z (z ) 0 6

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