Words to Review. Give an example of the vocabulary word. Numerical expression. Variable. Evaluate a variable expression. Variable expression

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1 1 Words to Review Give an example of the vocabulary word. Numerical expression 5 12 Variable x Variable expression 3x 1 Verbal model Distance Rate p Time Evaluate a variable expression Evaluate the expression 3x 1 when x 0. 3x 1 3(0) 1 1 Power 2 5 Base In the expression 2 5, 2 is the base. Integer..., 2, 1, 0, 1, 2,... Exponent In the expression 2 5, 5 is the exponent. Negative integer..., 3, 2, 1 Positive integer 1, 2, 3,... Absolute value 2 2 Opposite The opposite of is. Additive inverse The additive inverse of is.

2 Additive inverse property a (a) 0 Coordinate plane, x-axis, y-axis, origin, quadrant Quadrant II origin 32 y-axis y Quadrant I O x x-axis 2 3 Quadrant III Quadrant IV Ordered pair (3, ) y-coordinate In the ordered pair (3, ), is the y-coordinate. x-coordinate In the ordered pair (3, ), 3 is the x-coordinate. Scatter plot y O x Review your notes and Chapter 1 by using the Chapter Review on pages of your textbook.

3 1.1 Expressions and Variables Goal: Evaluate and write variable expressions. Vocabulary Numerical expression: A numerical expression consists of numbers and operations. Variable: A variable is a letter used to represent one or more numbers. Variable expression: Evaluate a variable expression: A variable expression consists of numbers, variables, and operations. To evaluate a variable expression, substitute a number for each variable and evaluate the resulting numerical expression. Verbal model: A verbal model describes a problem using words as labels and using math symbols to relate the words. When you write a variable expression involving multiplication, avoid using the symbol. It may be confused with the variable x. Example 1 Evaluating a Variable Expression Evaluate the expression 3 p b when b p b 3 p 90 Substitute for b. 270 Multiply. Example 2 Evaluating Expressions with Two Variables Evaluate the expression when x 9 and y 5. a. x y 9 5 Substitute for x and for y. 1 Add. b. xy 9 ( 5 ) Substitute for x and for y. 5 Multiply.

4 Checkpoint Evaluate the expression when x 8 and y x x y 3. x y 20 6 Common Words and Phrases that Indicate Operations Addition Subtraction Multiplication Division plus minus times divided by the sum of the difference of the product of divided into increased by decreased by multiplied by the quotient of total fewer than of more than less than added to subtracted from Example 3 Writing a Variable Expression Editing You have a 350-page manuscript that needs to be edited very quickly. You are going to divide the number of pages among several editors. You want to give the same number of pages to each editor. Use a verbal model to write a variable expression for the number of pages given to each editor if you know the number of editors. Let e represent the number of editors. The phrase divide indicates division. Number of pages for each editor Total number of pages Number of editors 350 e When you write a variable expression involving division, use a fraction bar instead of the division symbol. For example, write the "quotient of n and 12" as n. 12 Answer: The number of pages for each editor is 350 e, 35 0 e or.

5 1.2 Powers and Exponents Goal: Use powers to describe repeated multiplication. Vocabulary Power: Base: A power is the result of a repeated multiplication of the same factor. A base is the number or variable that is used as a factor in repeated multiplication. Exponent: Formula: An exponent is the number that represents the number of times the base is used as a factor. A formula describes a relationship between quantities. Example 1 Using Exponents Write the product using an exponent. a. 7 p 7 p 7 p 7 p The base 5 times. 7 is used as a factor b. (0.)(0.) (0.) 2 The base 2 times. 0. is used as a factor c. a p a p a p a a The base times. a is used as a factor d. r p r p r p r p r p r r 6 The base 6 times. r is used as a factor Checkpoint Write the product using an exponent p 12 p 12 p (0.2)(0.2)(0.2)(0.2)(0.2)(0.2)(0.2) 12 (0.2) 7 3. x p x p x p x p x. y p y p y x 5 y 3

6 Example 2 Evaluating Powers with Variables Evaluate the expression x 3 when x x 3 ( ) 3 Substitute for x ( )( )( ) Use 0. as a factor 3 times Multiply. Area and Volume Formulas Area is measured in square units, such as square feet (ft 2 ) or square centimeters (cm 2 ). Volume is measured in cubic units, such as cubic inches (in. 3 ) or cubic meters (m 3 ). s s Area A of a square Volume V of a cube A s 2 V s 3 Example 3 Room Size You are planning to put wall-to-wall carpeting in your room. To do this, you need to find the area of the square-shaped floor. A s 2 Write the formula. ( 11 ) 2 Substitute for s. 121 Using Powers in Formulas Evaluate power. s s s 11 ft 11 ft Answer: The area of the floor is 121 square feet. Checkpoint Evaluate the expression when n n 2 6. n 3 7. n 8. n Find the volume of a cube with the given side length meters feet 8 m 3 27 ft 3

7 1.3 Order of Operations Goal: Use order of operations to evaluate expressions. Vocabulary Order of operations: The order of operations are the rules established to evaluate an expression involving more than one operation. Grouping symbols: Grouping symbols indicate operations that should be performed first. Parentheses ( ), brackets [ ], and fraction bars are common grouping symbols. Order of Operations 1. Evaluate expressions inside grouping symbols. 2. Evaluate powers. 3. Multiply and divide from left to right.. Add and subtract from left to right. Example 1 Using Order of Operations Evaluate the expression p 20 8 p 5.8. p 20 8 p 5.8 Write expression Multiply Add. Checkpoint Evaluate the expression p

8 When grouping symbols appear inside other grouping symbols, work from the innermost grouping symbols out. Example 2 Evaluate the expression. a. 5(1 3.8) 5( 10.2 ) b ( 27 3 ) ( 2 ) Using Grouping Symbols Subtract within parentheses. Multiply. Rewrite fraction as division. Evaluate within parentheses. Divide. c. p [35 (11 8)] p [35 19 ] Add within parentheses. p 6 16 Subtract within brackets. Multiply. Example 3 Evaluating Variable Expressions Evaluate the expression when x 3 and y 6. a. 3(x y) 3( 3 ) Substitute for x and for y. 3( 9 ) Add within parentheses. 27 Multiply. b. 5(y x) 2 5( 6 ) 2 Substitute for x and for y. 5( 3 ) 2 Subtract within parentheses. 5( ) Evaluate power Multiply. Checkpoint Evaluate the expression when x and y y(19 x 2 6y ). x

9 1. Comparing and Ordering Integers Goal: Compare and order integers. Vocabulary Integers: Integers are the numbers..., 3, 2, 1, 0, 1, 2, 3,.... Negative integers: Negative integers are integers that are less than zero. The expression a is always read as "the opposite of a" and not as "negative a." If a is a positive number, then a is a negative number. If a is a negative number, then a is a positive number. Positive integers: Absolute Value: Opposites: Positive integers are integers that are greater than zero. The absolute value of a number is its distance from 0 on a number line. The absolute value of a number a is written as a. Two numbers are opposites if they have the same absolute value but different signs. Example 1 Graphing and Ordering Integers Use a number line to order these integers from least to greatest: 0, 6, 2, 8, 7, 9, Read the numbers from left to right: 9, 8, 6, 2, 0, 3, 7. Checkpoint Graph the integers on a number line. Then write the integers in order from least to greatest. 1. 2, 7, 6,, 0,, ,, 1, 0, 2,, 6

10 Example 2 Finding Absolute Value State the absolute value of the number. a. 7 b. 5 a. 7 units The distance between 7 and 0 is 7. So, 7 7. b. 5 units The distance between 5 and 0 is 5. So, 5 5. Example 3 Finding Opposites State the opposite of the number. a. 3 b. 8 a. 3 units 3 units b. The opposite of 3 is 3. 8 units 8 units The opposite of 8 is. 8 Checkpoint State the absolute value and the opposite of the number ; 9 12; 12

11 1.5 Adding Integers Goal: Add integers. Vocabulary Additive inverse: The opposite of a number is also called its additive inverse. Additive inverse property: The sum of a number and its opposite is 0. Example 1 Adding Integers Using a Number Line Use a number line to find the sum. a. 7 (10) Start at 7. Then move 10 units to the left Answer: The final position is 3. So, 7 (10) 3. b. 6 5 Start at 6. Then move 5 units to the right Answer: The final position is 1. So, Checkpoint Use a number line to find the sum (6) 2. 2 (6)

12 Adding Integers Words Numbers 1. Same Sign Add the absolute values and use the common sign. 6 () Different Signs Subtract the lesser 5 (8) absolute value from the greater absolute value and use the sign of the number with the greater absolute value Opposites The sum of a number and 7 (7) its opposite is Example 2 Adding Two Integers a. Same sign: Add 35 and (18) 53 Both integers are sum is negative. negative, so the b. 27 (13) 1 Different signs: Subtract 13 from 27. Because 27 > 13 has the same sign as 27., the sum Example 3 Adding More Than Two Integers Find the sum 7 (1) (1) Add 7 and Add 8 and 32. Checkpoint Find the sum (31)

13 1.6 Subtracting Integers Goal: Subtract integers. Subtracting Integers Words To subtract an integer, add its opposite. Numbers ( 7 ) Algebra a b a + ( b ) Example 1 Subtracting Integers a ( 9 ) To subtract 9, add its opposite, 9. Add 5 and 9. b. 3 (8) 3 8 To subtract 8, add its opposite, Add 3 and 8. c. (10) 10 To subtract 10, add its opposite, Add and 10. Checkpoint Find the difference (3)

14 Example 2 Evaluating Variable Expressions Evaluate the expression when x 8. a. x (22) 8 (22) Substitute for x To subtract 22, add Add 8 and 22. b. 9 x 9 ( ) Substitute for x To subtract 8, add Add 9 and 8. Checkpoint Evaluate the expression when y 12.. y y 6. 7 y You can use subtraction to find the change in a variable quantity such as elevation or temperature. Subtract the original value of the quantity from the value after the change. Example 3 Write a verbal model to find the change in temperature given the start temperature and the end temperature. Use the model to find the change in temperature from 5F to 12F. Change in temperature Evaluating Change End temperature Start temperature 12 ( 5 ) Substitute values To subtract 5, add Add 12 and 5. Answer: The temperature increased by 17 F. Checkpoint Find the change in temperature. 7. From 3F to 8F 8. From 15C to 2C 11F 13C

15 1.7 Multiplying and Dividing Integers Goal: Multiply and divide integers. Multiplying Integers Words Numbers The product of two integers with 2() 8 2() the same sign is positive. 8 The product of two integers with 2() 8 2() different signs is negative. The product of any integer and 2(0) 0 2(0) 0 is Example 1 Multiplying Integers a. 5(8) 0 Same sign: Product is positive. b. 8(7) 56 Different signs: Product is negative. c. 51(0) 0 The product of any integer and 0 is 0. Checkpoint Find the product. 1. 7(12) 2. 9(5) (0). (11) 0

16 Dividing Integers Words Numbers The quotient of two integers with 8 2 the same sign is positive. 8 () 2 The quotient of two integers with 8 2 different signs is negative. 8 () 2 The quotient of 0 and any nonzero 0 0 integer is 0. 0 () 0 Example 2 Dividing Integers a. 63 (9) 7 Same sign: Quotient is positive. b. 2 () 6 Different signs: Quotient is negative. c. 0 (2) 0 The quotient of 0 and any nonzero integer is 0. Checkpoint Find the quotient (3) () (19) 2

17 1.8 The Coordinate Plane Goal: Identify and plot points in a coordinate plane. Vocabulary Coordinate plane: A coordinate plane is formed by the intersection of a horizontal number line and a vertical number line. x-axis: The horizontal axis in a coordinate plane y-axis: The vertical axis in a coordinate plane Origin: The point (0, 0) in a coordinate plane at which the horizontal axis intersects the vertical axis Quadrant: One of four parts into which the axes divide the coordinate plane Ordered pair: Each point in the coordinate plane corresponds to an ordered pair of real numbers. x-coordinate: The first number in an ordered pair is the x-coordinate. y-coordinate: The second number in an ordered pair is the y-coordinate. Scatter plot: A scatter plot uses a coordinate plane to display paired data.

18 Example 1 Naming Points in a Coordinate Plane Give the coordinates of the point. a. A b. B A y 3 2 O B x a. Point A is 2 units to the left of the origin and 3 units up. The x-coordinate is 2 and the y-coordinate is 3. The coordinates are ( 2, 3 ). b. Point B is units to the right of the origin and 1 unit down. The x-coordinate is and the y-coordinate is 1. The coordinates are (, 1 ). Checkpoint Give the coordinates of the point. 1. C (2, 0) 2. D C O y x (, 3) 3. E D 2 3 E (2, )

19 Example 2 Plotting Points in a Coordinate Plane Plot the point in the coordinate plane. Describe the location of the point. a. A(2, 3) b. B(, 0) c. C(1, 3) B(, 0) O y A(2, 3) x 2 3 C(1, 3) a. Begin at the origin and move 2 units to the right, then move 3 units up. Point A is located in Quadrant I. b. Begin at the origin and move units to the left. Point B is located on the x-axis. c. Begin at the origin and move 1 unit to the right, then move 3 units down. Point C is located in Quadrant IV. Checkpoint Plot the point in the coordinate plane. Describe the location of the point.. D(0, 3) Point D is located on the y-axis. 5. E(2, ) Point E is located in Quadrant III. 6. F(1, 2) 3 F(1, 2) O 2 3 E(2, ) y D(0, 3) x Point F is located in Quadrant II.

20 Example 3 Making a Scatter Plot The number of hours you spent studying for 5 different math tests and the score you got on each test is given in the table. Make a scatter plot of the data and describe any relationship you see. Hours studying Test score Write the data as ordered pairs. Let the x-coordinate represent the hours spent studying, and let the y-coordinate represent the test score: (1, 55), (3, 78), (, 86), (, 89), (5, 98). 2. Plot the ordered pairs in a coordinate plane. You need only the first quadrant. Test score Hours studying Notice that the points rise from left to right. You can conclude as the hours spent studying increases, your test scores increase.

21 2 Words to Review Give an example of the vocabulary word. Additive identity 0 Multiplicative identity 1 Equivalent numerical expressions 3(3.99) 3( 0.01) Equivalent variable expressions 2(x 10) 2x 20 Term, coefficient, constant term, like terms 5 2x 3 x Terms: 5, 2x, 3, x Like terms: 5 and 3; 2x and x Coefficients: 2, 1 Constant terms: 5, 3 of an equation 5 is a solution of 2x 10. Inverse operations Addition and subtraction; Multiplication and division Equation 2x 10 Solving an equation x 5 2 x x 7 Equivalent equations 2x 10 and x 5 Review your notes and Chapter 2 by using the Chapter Review on pages of your textbook.

22 2.1 Properties and Operations Goal: Use properties of addition and multiplication. Commutative and Associative Properties Commutative Property of Commutative Property of Addition Multiplication Words In a sum, you can Words In a product, you can add the numbers in any multiply the numbers in any order. order. Numbers (7) 7 Numbers 8(5) 5(8) Algebra a b b a Algebra ab ba Associative Property of Associative Property of Addition Multiplication Words Changing the grouping Words Changing the grouping of the numbers in a sum does of the numbers in a product not change the sum. does not change the product. Numbers Numbers (9 6) 2 9 (6 2) (3 p 10) p 3 p (10 p ) Algebra Algebra (a b) c a (b c) (a p b) p c a p (b p c) Use properties of addition to group together distances that are easy to add mentally. Example 1 Using Properties of Addition Distance This week, you rode in a car for 2 miles, rode a bike for 5 miles, and rode in a bus for 23 miles. Find the total distance. The total distance is the sum of the three distances (2 5) 23 Use order of operations. Commutative property ( 5 2 ) 23 of addition Associative property 5 ( 2 23) of addition 5 65 Add 2 and Add 5 and 65. Answer: The total distance is 70 miles.

23 Example 2 Evaluate xy when x 8 and y 15. xy ( )( ) Substitute for x and for y. [( )]( ) Use order of operations. 8 [ ( )]( ) Commutative property of multiplication 8 [( )( )] Associative property of multiplication Using Properties of Multiplication ( ) Multiply and Multiply 8 and 60. Checkpoint Evaluate the expression when x 7 and y (2x y) 6 2. x 2 y Example 3 Using Properties to Simplify Variable Expressions Simplify the expression. a. x 5 2 (x 5) 2 Use order of operations. x (5 2) Associative property of addition x 7 Add 5 and 2. b. 3(9y) (3 p 9)y Associative property of multiplication 27y Multiply 3 and 9. Checkpoint Simplify the expression. 3. n 6 7. (r)(3) n 13 12r

24 Identity Properties Identity Property of Identity Property of Addition Multiplication Words The sum of a number Words The product of a and the additive identity, 0, number and the multiplicative is the number. identity, 1, is the number. Numbers Numbers p 1 Algebra a 0 a Algebra a p 1 a Example Identifying Properties Statement Property Illustrated a. (3 2) 3 (2 ) Associative property of addition b. 0 b b c. 1(7) 7 d. cd dc Identity property of addition Identity property of multiplication Commutative property of multiplication Checkpoint Identify the property that the statement illustrates. 5. (2 p 6) p 3 2 p (6 p 3) 6. q (r) r q Associative property of multiplication Commutative property of addition

25 2.2 The Distributive Property Goal: Use the distributive property. Vocabulary Equivalent numerical expressions: Numerical expressions that have the same value Equivalent variable expressions: Equivalent variable expressions have the same value for all values of the variable(s). The Distributive Property Algebra a(b c) ab ac Numbers (6 3) (b c)a ba ca (6 3) a(b c) ab ac 5(7 2) (6) (3) 6() 3() 5(7) 5(2) (b c)a ba ca (7 2)5 7(5) 2(5) Example 1 Using the Distributive Property Crafts You are buying beads for a craft project. You need gold, silver, and white beads. A bag of each type of bead costs $3.99. Use the distributive property and mental math to find the total cost of the beads. Total cost 3(3.99) Write expression for total cost. 3( ) Rewrite 3.99 as ( ) 3( 0.01 ) Distributive property Multiply using mental math Subtract using mental math. Answer: The total cost of the beads is $

26 Checkpoint Use the distributive property to evaluate the expression. 1. 2(9 ) 2. (12 3)3 3. ( 11)() Evaluate the expression using the distributive property and mental math.. 5(103) 5. (3.8) 6. 3(6.03) Example 2 Writing Equivalent Variable Expressions Use the distributive property to write an equivalent variable expression. a. 2(x 10) 2(x) 2(10) Distributive property 2x 20 Multiply. b. (m 3)() m() 3() Distributive property m (12) m 12 Multiply. Definition of subtraction c. 3(2y 6) 3(2y) (3)(6) Distributive property 6y (18) 6y 18 Multiply. Definition of subtraction

27 Checkpoint Use the distributive property to write an equivalent variable expression. 7. (x 7) 8. 3(m 7) Example 3 x 28 12m 21 Finding Areas of Geometric Figures Find the area of the rectangle or triangle. a. b y 3x 2 1 a. Use the formula for the b. Use the formula for the area of a rectangle. area of a triangle. A lw A 1 2 bh 1 2 ( 1 )( 12x 3y ) ( 3x 2 )( 5 ) 7 ( 12 3y ) 3x ( 5 ) 2 ( 5 ) 7 ( 12 ) 7 ( 3y ) 15x y Answer: The area is (15x 10) square units. Answer: The area is (8 21y) square units. Checkpoint Find the area of the rectangle or triangle y 5x x y

28 2.3 Simplifying Variable Expressions Goal: Simplify variable expressions. Vocabulary Terms of an expression: Coefficient of a term: The parts of an expression that are added together are called terms. The coefficient of a term with a variable is the number part of the term. Constant term: A constant term has a number but no variable. Like terms: Like terms are terms that have identical variable parts. Example 1 Identifying Parts of an Expression Identify the terms, like terms, coefficients, and constant terms of the expression 5 2x 3 x. 1. Write the expression as a sum: 5 (2x) (3) x. 2. Identify the parts of the expression. Note that because x 1 x, the coefficient of x is 1. Terms: 5, 2x, 3, x Like terms: 5 and 3; 2x and x Coefficients: 2, 1 Constant terms: 5, 3 Checkpoint Identify the terms, like terms, coefficients, and constant terms of the expression. 1. y 6 3y 2. 9 w 5 8w Terms: y, 6, 3y Like terms: y and 3y Coefficients:, 3 Constant term: 6 Terms: 9, w, 5, 8w Like terms: 9 and 5; w and 8w Coefficients: 1, 8 Constant terms: 9, 5

29 Example 2 Simplifying an Expression 5m 8 3m 7 5m 8 ( 3m ) ( 7 ) 5m ( 3m ) 8 ( 7 ) [ 5 ( 3 )]m 8 ( 7 ) Write as a sum. Commutative property Distributive property 2m 1 Simplify. Example 3 Simplifying Expressions with Parentheses a. 3(x 2) x 9 3x 6 x 9 Distributive property 3x x 6 9 2x 15 Group like terms. Combine like terms. b. 2k 5(k ) 2k 5k 20 Distributive property 3k 20 Combine like terms. c. 5a (5a 7) 5a 1 (5a 7) Identity property 5a 5a 7 Distributive property Combine like terms. Simplify. Checkpoint Simplify the expression. 3. y 6 3y. 9 w 5 8w 7y 6 7w 5. (x 1) 2x (k 3) 5k 2x 11 k 18

30 2. Variables and Equations Goal: Solve equations with variables. Vocabulary Equation: An equation is a mathematical sentence formed by placing an equal sign,, between two expressions. of an equation: A solution of an equation with a variable is a number that produces a true statement when it is substituted for the variable. Solving an equation: Finding all solutions of an equation is called solving the equation. Example 1 Writing Verbal Sentences as Equations Verbal Sentence Equation a. The sum of x and is 8. b. The difference of 7 and y is 13. c. The product of 2 and p is 2. x 8 7 y 13 2p 2 d. The quotient of n and 3 is 5. n 5 3 Example 2 Checking Possible s Tell whether 7 or 8 is a solution of x 3 5. a. Substitute 7 for x. b. Substitute 8 for x. x 3 5 x Answer: 7 is not a Answer: 8 is a solution. solution.

31 Checkpoint 1. The sum of x and 7 is 12. Write the verbal sentence as an equation. 2. The quotient of n and is 16. x 7 12 n Tell whether 8 or 10 is a solution of x 6. 8 is not a solution; 10 is a solution Example 3 Solving Equations Using Mental Math Equation Question Check What number plus a. x equals 7? 7 12 minus what b. 12 n number equals 5? 5 18 equals 3 times c. 18 3t ( 6 ) what number? y 20 What number divided d by equals 5? 5 Checkpoint Solve the equation using mental math.. x m 6. 3 c

32 2.5 Solving Equations Using Addition or Subtraction Goal: Solve equations using addition or subtraction. Vocabulary Inverse operations: Inverse operations are two operations that undo each other, such as addition and subtraction. Equivalent equations: Equivalent equations are equations that have the same solution(s). Subtraction Property of Equality Words Subtracting the same number from each side of an equation produces an equivalent equation. Numbers If x 3 5, then x , or x 2. Algebra If x a b, then x a a b a, or x b a. Example 1 Solving an Equation Using Subtraction When you solve an equation, your goal is to write an equivalent equation that has the variable by itself on one side. This process is called solving for the variable. Solve x 5 2. Use the subtraction property of equality to solve for x. x 5 2 Write original equation. x Subtract 5 from each side. x 7 Simplify. Answer: The solution is 7. Check: x 5 2 Write original equation Substitute for x. 2 2 checks.

33 Addition Property of Equality Words Adding the same number to each side of an equation produces an equivalent equation. Numbers If x 3 5, then x , or x 8. Algebra If x a b, then x a a b a, or x b a. Example 2 Solving an Equation Using Addition Solve 12 y 7. Use the addition property of equality to solve for y. 12 y 7 Write original equation y 7 7 Add 7 to each side. 19 y Simplify. Answer: The solution is 19. Checkpoint Solve the equation. Check your solution. 1. x y m

34 2.6 Solving Equations Using Multiplication or Division Goal: Solve equations using multiplication or division. Division Property of Equality Words Dividing each side of an equation by the same nonzero number produces an equivalent equation. Numbers If 3x 12, then 3x 12, or x. 3 3 Remember that you cannot divide a number or an expression by 0. Algebra If ax b and a 0, then ax b, or x b. a a a Example 1 Solve 7x 2. Solving an Equation Using Division 7x 2 Write original equation. 7x 2 Divide each side by x 6 Simplify. Answer: The solution is 6. Check: 7x 2 Write original equation. 7( 6 ) 2 Substitute for x. 2 2 checks. Checkpoint Solve the equation. Check your solution. 1. 5x y 9 7

35 Multiplication Property of Equality Words Multiplying each side of an equation by the same nonzero number produces an equivalent equation. x x Numbers If 12, then 3 p 3 3 p 12, or x x x Algebra If b and a 0, then a p a a p b, or x ab. a Example 2 Solving an Equation Using Multiplication w Solve w 5 Write original equation p 5 11 w p Multiply each side by w Simplify. Answer: The solution is 55. Checkpoint Solve the equation. Check your solution. 3. m c 6 5

36 2.7 Decimal Operations and Equations with Decimals Goal: Solve equations involving decimals. Example 1 a. Find the sum 1.7 (3.). Use the rule for adding numbers with the same sign. Add 1.7 and 3. sum is negative. Adding and Subtracting Decimals. Both decimals are negative, so the 1.7 (3.) 5.1 b. Find the difference (3.62). First rewrite the difference as a sum: Then use the rule for adding numbers with different signs. Subtract from > has the same sign as (3.62) 13.33, so the sum Checkpoint Find the sum or difference (5.9) (3.6)

37 You can use estimation to check the results of operations with decimals. For instance, notice that (1.9) 15.3 is about 30 (2), or 15. So, an answer of 15.3 is reasonable. Example 2 Multiplying and Dividing Decimals a. 0.(13.7) 5.8 Different signs: Product is negative. b. 2.5(6.75) Same signs: Product is positive. c (2.9) 8.1 Same signs: Quotient is positive. d (1.9) 9.5 Different signs: Quotient is negative. Checkpoint Find the product or quotient (5.9) (3.6) Example 3 Solving Addition and Subtraction Equations Solve the equation. a. x b. y a. x Write original equation. x Subtract 6.3 from each side. x 1.5 Simplify. b. y Write original equation. y Add 5.7 to each side. y 2.23 Simplify.

38 Checkpoint Solve the equation. Check your solution. 5. x y m Example Solving Multiplication and Division Equations Solve the equation. n a. 0.8m.8 b a. 0.8m.8 Write original equation. 0.8m.8 Divide each side by m 6 Simplify. b. 5 n 2.15 Write original equation. 5 5 n 5 (2.15) Multiply each side by 5. n Simplify. Checkpoint Solve the equation. Check your solution. y 8. 6x

39 3 Words to Review Give an example of the vocabulary word. Inequality x > 0 of an inequality The solution of m 9 12 is m 3. Equivalent inequalities < x and x > Review your notes and Chapter 3 by using the Chapter Review on pages of your textbook.

40 3.1 Solving Two-Step Equations Goal: Solve two-step equations. Example 1 Using Subtraction and Division to Solve Solve x 9 7. Check your solution. x 9 7 Write original equation. x Subtract 9 from each side. x 16 Simplify. x 16 Divide each side by. x Simplify. Answer: The solution is. Check: x 9 7 Write original equation. ( ) 9 7 Substitute for x. 7 7 checks. Checkpoint Solve the equation. Check your solution. 1. 3x x 7 6 7

41 Example 2 Using Addition and Multiplication to Solve Solve 3 x 1. Check your solution. 3 x 1 Write original equation. x 1 Add to each side. 3 3 x 3 Simplify. 3 3 x 3 ( 3 ) Multiply each side by 3. x 9 Simplify. Answer: The solution is 9. Check: x 1 3 Write original equation Substitute for x. 1 1 checks. Checkpoint Solve the equation. Check your solution. 3. x b

42 Example 3 Solving an Equation with Negative Coefficients Solve 2 3x 17. Check your solution. 2 3x 17 Write original equation. 2 3x Subtract 2 from each side. 3x 15 Simplify. 3x 15 Divide each side by x 5 Simplify. Answer: The solution is 5. Check: 2 3x 17 Write original equation. 2 3( 5 ) 17 Substitute for x checks. Checkpoint Solve the equation. Check your solution y m 8 9

43 3.2 Solving Equations Having Like Terms and Parentheses Goal: Solve equations using the distributive property. Example 1 Writing and Solving an Equation Baseball Game A group of five friends are going to a baseball game. Tickets for the game cost $12 each, or $60 for the group. The group also wants to eat at the game. Hot dogs cost $2.75 each and bottled water costs $1.25 each. The group has a total budget of $76. If the group buys the same number of hot dogs and bottles of water, how many can they afford to buy? Let n represent the number of hot dogs and the number of bottles of water. Then 2.75n represents the cost of n hot dogs and 1.25n represents the cost of n bottles of water. Write a verbal model. Cost of hot dogs Cost of Cost of bottled water tickets Total budget 2.75n 1.25n Substitute. n n Combine like terms Subtract 60 from each side. n 16 Simplify. n 16 Divide each side by. n Simplify. Answer: The group can afford to buy bottles of water. hot dogs and

44 Example 2 Solving Equations Using the Distributive Property Solve the equation. a. 2 6(2 x) b. 2(7 x) 10 a. 2 6(2 x) Write original equation x Distributive property x 12 Subtract 12 from each side. 36 6x Simplify x 6 Divide each side by 6. 6 x Simplify. b. 2(7 x) 10 Write original equation. 1 8x 10 Distributive property 1 8x Add 1 to each side. 8x 2 Simplify. 8x 2 Divide each side by x 3 Simplify.

45 Example 3 Combining Like Terms After Distributing Solve 6x (x 1) 1. 6x (x 1) 1 Write original equation. 6x x 1 Distributive property 2x 1 Combine like terms. 2x 1 Subtract from each side. 2x 10 Simplify. 2x 10 Divide each side by x 5 Simplify. Checkpoint Solve the equation. Check your solution (3 x) 2. y 1 3y (6 2x) 12. 5x 2(x 3)

46 3.3 Solving Equations with Variables on Both Sides Goal: Solve equations with variables on both sides. Example 1 Solving an Equation with the Variable on Both Sides Solve 5n 7 9n 21. 5n 7 9n 21 Write original equation. 5n 7 5n 9n 21 5n Subtract 5n from each side. 7 n 21 Simplify n Subtract 21 from each side. 28 n Simplify. 28 n Divide each side by. 7 n Simplify. Answer: The solution is 7. Example 2 An Equation with No Solve 3(2x 1) 6x. 3(2x 1) 6x Write original equation. 6x 3 6x Distributive property Notice that this statement is not true because the number 6x cannot be equal to 3 more than itself. The equation has no solution. As a check, you can continue solving the equation. 6x 3 6x 6x 6x Subtract 6x from each side. 3 0 Simplify. The statement 3 0 is not true, so the equation has no solution.

47 Example 3 Solving an Equation with All Numbers as s Solve (x 2) x 8. (x 2) x 8 Write original equation. x 8 x 8 Distributive property Notice that for all values of x, the statement true x 8. The equation has every number as a solution. x 8 is Checkpoint Solve the equation. Check your solution. 1. 3n 6 5n x (3x 1) 13 no solution 3. 3(2n ) 2(3n 6). 2x 7 2x 13 all numbers 5

48 Example Solving an Equation to Find a Perimeter Geometry Find the perimeter of the square. x 6 1. A square has four sides of equal length. Write an equation and solve for x. 3x x 6 3x Write equation. 3x x x 6 x Subtract x from each side. 2x 6 Simplify. 2x 6 Divide each side by x 3 Simplify. 2. Find the length of one side by substituting 3 for x in either expression. 3x 3( 3 ) 9 Substitute for x and multiply. 3. To find the perimeter, multiply the length of one side by. p 9 36 Answer: The perimeter of the square is 36 units. 5. Checkpoint Find the perimeter of the square. 3x 8 5x 80 units

49 3. Solving Inequalities Using Addition or Subtraction Goal: Solve inequalities using addition or subtraction. Vocabulary Inequality: An inequality is a statement formed by placing an inequality symbol, such as < or >, between two expressions. of an inequality: Equivalent inequalities: The solution of an inequality with a variable is the set of all numbers that produce true statements when substituted for the variable. Equivalent inequalities are inequalities that have the same solution. Example 1 Writing and Graphing an Inequality Air Travel An airline allows passengers to carry on-board one piece of luggage. Luggage that exceeds 0 pounds cannot be carried on-board. Write an inequality that gives the weight of luggage that cannot be carried on-board. Let w represent the weight of the luggage. Because the weight cannot exceed 0 pounds, the weight of luggage that cannot be carried on-board must be greater than 0 pounds. Answer: The inequality is w > 0. Draw the graph below

50 The addition and subtraction properties of inequality are also true for inequalities involving and. Addition and Subtraction Properties of Inequality Words Adding or subtracting the same number on each side of an inequality produces an equivalent inequality. Algebra If a < b, then a c < b c and a c < b c. If a > b, then a c > b c and a c > b c. Example 2 Solving an Inequality Using Subtraction Solve m Graph and check your solution. m 9 12 Write original inequality. m Subtract 9 from each side. m 3 Simplify. Answer: The solution is m 3. Draw the graph below Check: Choose any number less than or equal to the number into the original inequality. 3. Substitute m 9 12 Write original inequality. 0 9 <? 12 Substitute 0 for m checks You can read an inequality from left to right as well as from right to left. For instance, "2 is greater than x" can also be read "x is less than 2." Algebraically, this means that 2 x can also be written as x 2. Example 3 Solving an Inequality Using Addition Solve 7 < x 11. Graph your solution. 7 < x 11 Write original inequality < x Add 11 to each side. < x Simplify. Answer: The solution is < x, or x >. Draw the graph below

51 Checkpoint Solve the inequality. Graph and check your solution. 1. m 7 < 13 m < 6 2. a a 1 3. x x 5. 6 < z < z or z >

52 3.5 Solving Inequalities Using Multiplication or Division Goal: Solve inequalities using multiplication or division. The multiplication properties of inequality are also true for inequalities involving >,, and. Multiplication Property of Inequality Words Multiplying each side of an inequality by a positive number produces an equivalent inequality. Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. Algebra If a < b and c > 0, then ac If a < b and c < 0, then ac < > bc. bc. Example 1 Solving an Inequality Using Multiplication m Solve > 2. m > 2 m < p 2 m < 8 Write original inequality. Multiply each side by. Reverse inequality symbol. Simplify. Checkpoint Solve the inequality. Graph your solution. t b 1. < t < 15 b

53 The division properties of inequality are also true for inequalities involving >,, and. Division Property of Inequality Words Dividing each side of an inequality by a positive number produces an equivalent inequality. Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. Algebra If a < b and c > 0, then a c If a < b and c < 0, then a c < > b c. b c. Example 2 Solve 11t t 33 11t 11 t Solving an Inequality Using Division Write original inequality. Divide each side by 11. Reverse inequality symbol. Simplify. Checkpoint Solve the inequality. Graph your solution. 3. y 36. 3x > 12 y 9 x <

54 3.6 Solving Multi-Step Inequalities Goal: Solve multi-step inequalities. Example 1 Writing and Solving a Multi-Step Inequality Charity Walk You are participating in a charity walk. You want to raise at least $500 for the charity. You already have $175 by asking people to pledge $25 each. How many more $25 pledges do you need? Let p represent the number of additional pledges. Write a verbal model. Money already raised Amount Additional per pledge p pledges Minimum desired amount p 500 Substitute. 25p Subtract 175 each side. 25p 325 Simplify. from 25p Divide each side by. 25 p 13 Simplify. Answer: You need at least 13 more $25 pledges. Checkpoint 1. Look back at Example 1. Suppose you wanted to raise at least $620 and you already have raised $380 by asking people to pledge $20 each. How many more $20 pledges do you need? at least 12 more $20 pledges

55 Example 2 x 9 < 7 3 Original inequality x 9 9 < 7 9 Add 9 to each side. 3 3 Solving a Multi-Step Inequality x < 3 x 3 > x > p 2 Simplify. Multiply each side by 3. Reverse inequality symbol. Simplify. Checkpoint Solve the inequality. Then graph the solution. x 2. 2x 9 < x < 8 x x 5. 2 x 9 x > 6 x

56 Words to Review Give an example of the vocabulary word. Prime number 3 Composite number 8 Prime factorization 6 2 p 3 Factor tree Monomial 2x Common factor 2 is a common factor of and 6. Greatest common factor (GCF) The GCF of 12 and 18 is 6. Relatively prime 10 and 11 are relatively prime.

57 Equivalent fractions The fractions 1 2 and are equivalent to. 1 8 Simplest form 1 2 written in simplest form 36 is 1 3. Multiple 12 is a multiple of 6. Common multiple 10 is a common multiple of 5 and 2. Least common multiple (LCM) The LCM of 12 and 18 is 36. Least common denominator (LCD) 5 7 The LCD of and is Scientific notation Review your notes and Chapter by using the Chapter Review on pages of your textbook.

58 .1 Factors and Prime Factorization Goal: Write the prime factorization of a number. Vocabulary Prime number: Composite number: Prime factorization: Factor tree: Monomial: Example 1 Writing Factors A rectangle has an area of 18 square feet. Find all possible whole number dimensions of the rectangle. 1. Write 18 as a product of two whole numbers in all possible ways. p 18 p 18 p 18 The area of a rectangle can be found using the formula, Area length width. width length The factors of 18 are. 2. Use the factors to find all rectangles with an area of 18 square feet that have whole number dimensions. Then label the given rectangles. 56 Chapter Notetaking Guide

59 Example 2 Writing a Prime Factorization Write the prime factorization of 20. One possible factor tree: Write original number. Write 20 as 10 p. Write 10 as p. Write as p 6. Write 6 as p. Another possible factor tree: Write original number. Write 20 as 6 p. Write 6 as p. Write as p 10. Write 10 as p. Both trees give the same result: 20. Answer: The prime factorization of 20 is. Example 3 Factoring a Monomial Factor the monomial 2x y. 2x y p x y Write 2 as. p y Write x as Lesson.1 Factors and Prime Factorization 57

60 Checkpoint Write all factors of the number Tell whether the number is prime or composite. If it is composite, write its prime factorization Factor the monomial n x 2 y 3 58 Chapter Notetaking Guide

61 .2 Greatest Common Factor Goal: Find the greatest common factor of two or more numbers. Vocabulary Common factor: Greatest common factor (GCF): A common factor is a whole number that is a factor of two or more nonzero whole numbers. The GCF is the greatest whole number that is a factor of two or more nonzero whole numbers. Relatively prime: Two or more numbers are relatively prime if their greatest common factor is 1. Example 1 Finding the Greatest Common Factor Volunteers A high school asks for volunteers to help clean up local highways on one Saturday each month. The group of volunteers has 27 freshman, 18 sophomores, 36 juniors, and 5 seniors. What is the greatest number of groups that can be formed if each group is to have the same number of each type of student? How many freshman, sophomores, juniors, and seniors will be in each group? Method 1 List the factors of each number. Identify the greatest number that is on every list. Factors of 27: 1, 3, 9, 27 Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 36: 1, 2, 3,, 6, 9, 12, 18, 36 Factors of 5: 1, 3, 5, 9, 15, 5 The common factors are 1, 3, and 9. The GCF is 9.

62 Method 2 Write the prime factorization of each number. The GCF is the product of the prime factors p 3 p 3 2 p 3 p 3 2 p 2 p 3 p 3 3 p 3 p 5 The common prime factors are 3 and 3. The GCF is 3 p 3 9. Answer: The greatest number of groups that can be formed is 9. Each group will have freshman, sophomores, 36 9 juniors, and seniors. Checkpoint Find the greatest common factor of the numbers. 1. 5, , 8, 66 Example Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime. a. 28, 63 b. 2, 55 a. List the factors of each number. Identify the greatest number that the lists have in common. Factors of 28: Identifying Relatively Prime Numbers 1, 2,, 7, 1, 28 Factors of 63: 1, 3, 7, 9, 21, 63 The GCF is 7. So, the numbers are not relatively prime. b. Write the prime factorization of each number. 2 2 p 3 p p 11 The GCF is 1. So, the numbers are relatively prime.

63 Checkpoint Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime , 9. 52, 78 1; relatively prime 26; not relatively prime Example 3 Finding the GCF of Monomials Find the greatest common factor of 16x 2 y and 26x 2 y 3. Factor the monomials. The GCF is the product of the common factors. 16x 2 y 26x 2 y 3 2 p 2 p 2 p 2 p x p x p y 2 p 13 p x p x p y p y p y Answer: The GCF is 2x 2 y. Checkpoint Find the greatest common factor of the monomials x 3,18x xy 3,2xy 6x 2 8xy

64 .3 Equivalent Fractions Goal: Write equivalent fractions. Vocabulary Equivalent fractions: Simplest form: Two fractions that represent the same number are called equivalent fractions. A fraction is in simplest form when its numerator and its denominator are relatively prime. Equivalent Fractions Words To write equivalent fractions, multiply or divide the numerator and the denominator by the same nonzero number. Algebra For all numbers a, b, and c, where b 0 and c 0, a b a p c and a b p c b a c. b c Numbers p p Example 1 Writing Equivalent Fractions 6 Write two fractions that are equivalent to 1 8. Multiply or divide the numerator and the denominator by the same nonzero number. 6 6 p Multiply numerator and denominator by p Divide numerator and denominator by Answer: The fractions and 2 are equivalent to

65 Checkpoint Write two fractions that are equivalent to the given fraction , , , Example 2 Writing a Fraction in Simplest Form 8 Write in simplest form. 3 6 Write the prime factorizations of the numerator and denominator p 3 2 The GCF of 8 and 36 is Divide numerator and denominator by GCF Simplify. Checkpoint Write the fraction in simplest form

66 Example 3 Simplifying a Variable Expression 2 Write 1 x y 3 in simplest form. 35x 2 1 x y 3 35x 2 p 7 p x p x p y 5 p 7 p x p x p x p 7 p x p x p y 5 p 7 p x p x p x Factor numerator and denominator. Divide out common factors y 5x Simplify. Checkpoint Write the variable expression in simplest form. 9a a m st 28n 2 3s t n m n 5 a 7 1 3t s

67 . Least Common Multiple Goal: Find the least common multiple of two numbers. Vocabulary Multiple: Common multiple: A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple. Least common multiple (LCM): The least common multiple of two or more numbers Least common denominator (LCD): The least common multiple of the denominators of two or more fractions Example 1 Find the least common multiple of 6 and 1. You can use one of two methods to find the LCM. Method 1 List the multiples of each number. Identify the least number that is on both lists. Multiples of 6: Multiples of 1: Finding the Least Common Multiple 6, 12, 18, 2, 30, 36, 2, 8 1, 28, 2, 56 Method 2 Find the common factors of the numbers. The LCM of 6 and 1 is p p 7 The common factor is 2. Multiply all of the factors, using each common factor only once. LCM 2 p 3 p 7 2 Answer: Both methods get the same result. The LCM is 2.

68 Example 2 Finding the Least Common Multiple of Monomials Find the least common multiple 6xy and 16x 2. 6xy 2 p 3 p x p y 16x 2 2 p 2 p 2 p 2 p x p x LCM 2 p 2 p 2 p 2 p 3 p x p x p y 8x 2 y Answer: The least common multiple of 6xy and 16x 2 is 8x 2 y. Checkpoint Find the least common multiple of the numbers or the monomials. 1. 8, 18 2., 5, x, 18x 2. xy,10xz 2 Example 3 36x 2 20xyz 2 Comparing Fractions Using the LCD Summer Sports Last year, a summer resort had 165,000 visitors, including,000 water skiers. This year, the resort had 180,000 visitors, including 63,000 water skiers. In which year was the fraction of water skiers greater? 1. Write the fractions and simplify. Number of water skiers,000 Last year: 1 Total number of visitors 65, Number of water skiers 63,000 This year: 1 Total number of visitors 80, Find the LCD of and. The LCM of 15 and 20 is So, the LCD of the fractions is 60.

69 3. Write equivalent fractions using the LCD. Last year: p p 3 This year: p 1 5 p Compare the numerators: <, so < Answer: The fraction of water skiers was greater this year. Example Ordering Fractions and Mixed Numbers 5 Order the numbers 1 2, 9 2, and 3 3 from least to greatest Write the mixed number as an improper fraction p Find the LCD of, , and 3 3. The LCM of 12, 2, 8 and 8 is 2. So, the LCD is Write equivalent fractions using the LCD p p 9 12 p p p 8 p Compare the numerators: < and <, so < and < 9. 2 Answer: From least to greatest, the numbers are ,, and

70 .5 Rules of Exponents Goal: Multiply and divide powers. Product of Powers Property Words To multiply powers with the same base, add their exponents. Algebra a m p a n a m n Numbers 3 p Example 1 Using the Product of Powers Property a. 7 p 11 Product of powers property b. 2x 2 p 7x 6 2 p 7 p x 2 p x 6 2 p 7 p x 2 p 7 p x Add exponents. Commutative property of multiplication Product of powers property Add exponents. 1x 8 Multiply. Checkpoint exponents. Find the product. Write your answer using p p 5 2 p x 6 p x 13. b 2 p b p b x 19 b 7

71 Quotient of Powers Property Words To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. m Algebra a a n a mn, where a Numbers Example 2 Using the Quotient of Powers Property 8 a Quotient of powers property Subtract exponents. 7 3x b x 7 3 Quotient of powers property 2x 12 3x 12 Subtract exponents. Divide numerator and denominator by 3. x Checkpoint exponents. Find the quotient. Write your answer using x 7. 2 x x 6x11 x 7 x 5 6 3

72 Example 3 Using Both Properties of Powers Simplify m p. 2 12m 3 m m 3 p m 2 m 3 12m 12m 2 Product of powers property 7 m 12m 2 Add exponents. m 7 2 Quotient of powers property 12 m m 3 Subtract exponents. Divide numerator and denominator by. Checkpoint Simplify m p m 3 15m 10. n2 p 10n 6 3 5n 3 2m 2n 5 5

73 .6 Negative and Zero Exponents Goal: Work with negative and zero exponents. Negative and Zero Exponents For any nonzero number a, a 0 1. For any nonzero number a and any integer n, a n a 1 n. Example 1 Powers with Negative and Zero Exponents Write the expression using only positive exponents. a. 3 Definition of negative exponent 13 b. m 5 n 0 m 5 p 1 Definition of zero exponent m 15 Definition of negative exponent 1 3x 8 y c. 13xy 8 Definition of negative exponent Checkpoint Write the expression using only positive exponents , z 2. 3x y z 2 2 x 3y Example 2 Rewriting Fractions Write the expression without using a fraction bar. 1 a. 15 Definition of negative exponent b. a c 5 3 a 3 c 5 Definition of negative exponent

74 Checkpoint Write the expression without using a fraction bar c 2 8. x 5 y c 2 x 5 y 7 Example 3 Find the product or quotient. Write your answer using only positive exponents. a p 6 b. 7n n Using Powers Properties with Negative Exponents a p 6 6 Product of powers property 6 12 () 8 Add exponents. b. 7n 7n n 7n 1 5 Quotient of powers property Subtract exponents. n 75 Definition of negative exponent Checkpoint Find the product or quotient. Write your answer using only positive exponents p d d2 3 3 d 76