7 th Grade Intensive Math. McGraw-Hill Supplemental Resources

Transcription

1 7 th Grade Intensive Math McGraw-Hill Supplemental Resources Student Edition October 2014 January 2015

2 NAME DATE PERIOD Lesson 4 Reteach Multiply Integers The product of two integers with different signs is negative. The product of two integers with the same sign is positive. Example 1 Find 5(-2). 5(-2) = -10 The integers have different signs. The product is negative. Example 2 Find -3(7). -3(7) = -21 The integers have different signs. The product is negative. Example 3 Find -6(-9). -6(-9) = 54 The integers have the same sign. The product is positive. Example 4 Find (-7) 2. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. (-7) 2 = (-7)(-7) There are 2 factors of -7. = 49 The product is positive. Example 5 Find -2(-3)(4). -2(-3)(4) = 6(4) Multiply -2 and -3. = 24 Multiply 6 and 4. Exercises Multiply (8) 2. -3(-7) 3. 10(-8) 4. -8(3) (-12) 6. (-8) (7) 8. 3(-2) 9. -6(-3) 10. 5(-4)(5) (-4) 12. 2(-3)(5) (-3) 14. 9(-4) 15. (-3)(-4) (-3)(5) (5) (-3)(-4)(5) Course 2 Chapter 3 Integers 47

3 NAME DATE PERIOD Lesson 4 Extra Practice Multiply Integers Multiply. 1. 5( 2) ( 4) (21) ( 5) (5) (0) ( 5) (8) ( 6) ( 2) ( 4) ( 5) Evaluate each expression if a = 5, b = 2, c = 3, and d = d a ab d b a cd ab c 9 Course 2 Chapter 3 Integers

4 NAME DATE PERIOD Lesson 5 Reteach Divide Integers The quotient of two integers with different signs is negative. The quotient of two integers with the same sign is positive. Example 1 Find 30 (-5). 30 (-5) The integers have different signs. 30 (-5) = -6 The quotient is negative. Example 2 Find -100 (-5) (-5) The integers have the same sign (-5) = 20 The quotient is positive. Exercises Divide (-7) (-3) Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use (-25) (-3) ALGEBRA Evaluate each expression if d = -24, e = -4, and f = e f 13. d d e 15. f e 16. e 2 f 17. -d e 19. f ef d - e 5 Course 2 Chapter 3 Integers 49

5 NAME DATE PERIOD Lesson 5 Extra Practice Divide Integers Divide ( 2) ( 8) ( 2) ( 3) ( 8) ( 1) ( 8) ( 7) ( 11) 11 Evaluate each expression if a = 2, b = 7, x = 8, and y = x y x a ax y bx y y x ay y a x 2 y ab xy a 16 Course 2 Chapter 3 Integers

6 NAME DATE PERIOD Lesson 1 Reteach Terminating and Repeating Decimals To write a fraction as a decimal, divide the numerator by the denominator. Division ends when the remainder is zero. You can use bar notation to indicate that a number pattern repeats indefi nitely. A bar is written over the digits that repeat. Example 1 Write 3 as a decimal Divide 3 by The remainder is 0. So, 3 20 = Example 3 Write as a fraction in simplest form. Example 2 Write 5 as a decimal The remainder after each step is You can use bar notation 0. 5 to indicate that 5 repeats forever. So, 5 9 = Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use = = Exercises The 2 is in the hundredths place. Simplify. Write each fraction or mixed number as a decimal. Use bar notation if the decimal is a repeating decimal Write each decimal as a fraction in simplest form Course 2 Chapter 4 Rational Numbers 51

7 NAME DATE PERIOD Lesson 1 Extra Practice Terminating and Repeating Decimals Write each fraction or mixed number as a decimal. Use bar notation if needed Write each decimal as a fraction or mixed number in simplest form Course 2 Chapter 4 Rational Numbers

8 NAME DATE PERIOD Lesson 6 Reteach Multiply Fractions To multiply fractions, multiply the numerators and multiply the denominators = = = 1 2 To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply the fractions = = = Example 1 Find Write in simplest form = 2 4 Multiply the numerators. 3 5 Multiply the denominators = 8 15 Simplify. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 2 Find Write in simplest form = Exercises = = 5 6 Rename as an improper fraction, 5 2. Multiply. Simplify. Multiply. Write in simplest form ( ) ( ) Course 2 Chapter 4 Rational Numbers 63

9 NAME DATE PERIOD Lesson 6 Extra Practice Multiply Fractions Multiply. Write in simplest form _ Course 2 Chapter 4 Rational Numbers

10 NAME DATE PERIOD Lesson 8 Reteach Divide Fractions To divide by a fraction, multiply by its multiplicative inverse or reciprocal. To divide by a mixed number, rename the mixed number as an improper fraction. Example Find Write in simplest form = Rename = = 10 / 3 9 / / / = 15 Multiply. as an improper fraction. Multiply by the reciprocal of 2 9, which is 9 2. Divide out common factors. Exercises Divide. Write in simplest form Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use ( ) (-10) ( ) Course 2 Chapter 4 Rational Numbers 67

11 NAME DATE PERIOD Lesson 8 Extra Practice Divide Fractions Divide. Write in simplest form _ Course 2 Chapter 4 Rational Numbers

12 Name: Period: Date: Multiplying and Dividing Decimals Find each product x (-4.87) x (-2.1) x (-1.6) x (-3.1) x (-7.2) x (-11.6) x (-7.1) x x (-8.3) x x x (-5.2) Find each quotient (-0.3) (-0.009) (-8.02) Solve each problem. Check your solution. 25. Mrs. Johnson harvested 107 pounds of tomatoes from her garden. She sold them for $0.85 a pound. How much did she receive from selling all the tomatoes? 26. Emma bought 2.5 yards of cording for the trim around the edge of a square pillow. How much will she use for each side of the pillow? 27. Sean has a loan of $ including interest. He makes payments of $ each month on the simple interest loan. How many months will it take Sean to repay his loan? 28. Travis painted for 6.25 hours. He received $27 an hour for his work. How much was Travis paid for doing this painting job?

13 NAME DATE PERIOD Lesson 1 Reteach Algebraic Expressions To evaluate an algebraic expression you replace each variable with its numerical value, then use the order of operations to simplify. Example 1 Evaluate 6x - 7 if x = 8. 6x - 7 = 6(8) - 7 Replace x with 8. = 48-7 Use the order of operations. = 41 Subtract 7 from 48. Example 2 Evaluate 5m - 3n if m = 6 and n = 5. 5m - 3n = 5(6) - 3(5) Replace m with 6 and n with 5. = Use the order of operations. = 15 Subtract 15 from 30. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 3 Evaluate ab if a = 7 and b = 6. 3 ab 3 = (7)(6) Replace a with 7 and b with 6. 3 = 42 The fraction bar is like a grouping symbol. 3 = 14 Divide. Example 4 Evaluate x if x = 3. x = Replace x with 3. = Use the order of operations. = 31 Add 27 and 4. Exercises Evaluate each expression if a = 4, b = 2, and c = ac 2. 5b 3 3. abc c 5. ab 8 7. b a - 3b 8. c - a bc 10. 2bc 11. ac - 3b 12. 6a c 14. 6a - b 15. ab - c Course 2 Chapter 5 Expressions 69

14 NAME DATE PERIOD Lesson 1 Extra Practice Algebraic Expressions Evaluate each expression if a = 3, b = 4, c = 12, and d = a + b 7 2. c d a + b + c b a 1 5. c ab 0 6. a + 2d 5 7. b + 2c ab a + 3b a + c c d abc (a + b) c b abc ab b a c a b b 3 + c a2 d a + 2d d2 b a a d (2c + b) b (b2 + 2d) a (2c + ab) c (3.5c + 2) 11 4 Course 2 Chapter 5 Expressions

15 NAME DATE PERIOD Lesson 2 Reteach Sequences An arithmetic sequence is a list in which each term is found by adding the same number to the previous term. 1, 3, 5, 7, 9, Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 1 Describe the relationship between terms in the arithmetic sequence 17, 23, 29, 35, Then write the next three terms in the sequence. 17, 23, 29, 35,. Each term is found by adding 6 to the previous term = = = 53 The next three terms are 41, 47, and 53. Example 2 MONEY Brian s parents have decided to start giving him a monthly allowance for one year. Each month they will increase his allowance by $10. Suppose this pattern continues. What algebraic expression can be used to find Brian s allowance after any given number of months? How much money will Brian receive for allowance for the 10th month? Make a table to display the sequence. Position Operation Value of Term n n 10 10n Each term is 20 times its position number. So, the expression is 10n. How much money will Brian receive after 10 months? 10n Write the expression. 10(10) = 100 Replace n with 10 So, Brian will receive $100 after 10 months. Exercises Describe the relationship between terms in the arithmetic sequences. Write the next three terms in the sequence. 1. 2, 4, 6, 8, 2. 4, 7, 10, 13, , 0.6, 0.9, 1.2, , 212, 224, 236, , 2.0, 2.5, 3.0, 6. 12, 19, 26, 33, 7. SALES Mama s bakery just opened and is currently selling only two types of pastry. Each month, Mama s bakery will add two more types of pastry to their menu. Suppose this pattern continues. What algebraic expression can be used to find the number of pastries offered after any given number of months? How many pastries will be offered in one year? Course 2 Chapter 5 Expressions 71

16 NAME DATE PERIOD Lesson 2 Extra Practice Sequences Describe the relationship between the terms in each arithmetic sequence. Then write the next three terms in each sequence. 1. 5, 9, 13, 17, 2. 3, 5, 7, 9, 3. 10, 15, 20, 25, 4 is added to the previous 2 is added to the previous 5 is added to the previous term; 21, 25, 29 term; 11, 13, 15 term; 30, 35, , 93, 96, 99, 5. 8, 14, 20, 26, , 5.4, 6.3, 7.2, 3 is added to the previous 6 is added to the previous 0.9 is added to the previous term; 102, 105, 108 term; 32, 38, 44 term; 8.1, 9.0, , 0.4, 0.5, , 3.4, 4.5, 5.6, , 9.1, 9.3, 9.5, 0.1 is added to the previous 1.1 is added to the previous 0.2 is added to the previous term; 0.6, 0.7, 0.8 term; 6.7, 7.8, 8.9 term; 9.7, 9.9, , 11, 19, 27, , 375, 400, 425, , 635, 650, 665, 8 is added to the previous 25 is added to the previous 15 is added to the previous term; 35, 43, 51 term; 450, 475, 500 term; 680, 695, , 7, 12, 17, , 17, 24, 31, 15. 9, 90, 171, 252, 5 is added to the previous 7 is added to the previous 81 is added to the previous term; 22, 27, 32 term; 38, 45, 52 term; 333, 414, , 2.8, 3.0, 3.2, , 4.6, 5.1, 5.6, , 7.7, 8.8, 9.9, 0.2 is added to the previous 0.5 is added to the previous 1.1 is added to the previous term; 3.4, 3.6, 3.8 term; 6.1, 6.6, 7.1 term; 11.0, 12.1, , 21, 22.5, 24, , 14.8, 15.1, 15.4, , 0.4, 0.7, 1.0, 1.5 is added to the previous 0.3 is added to the previous 0.3 is added to the previous term; 25.5, 27, 28.5 term; 15.7, 16.0, 16.3 term; 1.3, 1.6, 1.9 Course 2 Chapter 5 Expressions

17 NAME DATE PERIOD Lesson 3 Reteach Properties of Operations Example 1 Name the property shown by the statement u + v = v + u. The order in which the variables are being added changed. This is the Commutative Property of Addition. Example 2 State whether the following conjecture is true or false. If false, provide a counterexample. Subtraction of integers is commutative. Write two subtraction expressions using the Commutative Property State the conjecture. 8-8 Subtract. We found a counterexample. That is, So, subtraction is not commutative. The conjecture is false. Example 3 Simplify the expression. Justify each step. 9 + (3x + 4) 9 + (3x + 4) = 9 + (4 + 3x) Commutative Property of Addition Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. = (9 + 4) + 3x Associative Property of Addition = x Simplify. Exercises Name the property shown by each statement = (3y + 2) = (4 + 3y) + 2 State whether the following conjectures are true or false. If false, provide a counterexample. 3. The product of two even numbers is odd. 4. The difference of two odd numbers is even. 5. Simplify 4 + (5x + 2). Justify each step. Course 2 Chapter 5 Expressions 73

18 NAME DATE PERIOD Lesson 3 Extra Practice Properties of Operations Name the property shown by each statement = 4 Identity ( ) (b + 2) = (6 + b) + 2 Associative (+) 3. 9(6n) = (9 6)n Associative ( ) 4. 8t 0 = 0 8t Commutative ( ) 5. 0(13n) = 0 Multiplicative (0) t = t + 7 Commutative (+) Simplify each expression. Justify each step. 7. (12 + x) (15 + c) = (x + 12) + 9 Commutative (+) = ( ) + c Associative (+) = x + (12 + 9) Associative (+) = 46 + c Simplify. = x + 21 Simplify. 9. (8 + d) (6 m) = (d + 8) + 19 Commutative (+) = (2 6) m Associative ( ) = d + (8 + 19) Associative (+) = 12c Simplify. = d + 27 Simplify. 11. (5 p) (4f) = (p 5) 3 Commutative ( ) = (9 4) f Associative ( ) = p (5 3) Associative ( ) = 36f Simplify. = 15p Simplify. Course 2 Chapter 5 Expressions

19 NAME DATE PERIOD Lesson 4 Reteach The Distributive Property Distributive Property Words To multiply a sum or difference by a number, multiply each term inside the parentheses by the number outside the parentheses. Symbols a (b + c) = ab + ac a (b - c) = ab - ac Examples 3 (2 + 5) = (8-3) = Examples Use the Distributive Property to evaluate each expression. 1 5 (x + 3) 5 (x + 3) = 5 x Expand using the Distributive Property = 5x + 15 Simplify. 2 (4x - y)9 (4x - y) 9 = [4x + (-y)]9 Rewrite 4x - y as 4x + (-y). = (4x)9 + (-y)9 Expand using the Distributive Property. = 36x + (-9y) Simplify. = 36x - 9y Defi nition of subtraction. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 3 MOVIES Alwyn is taking three of his friends to the movies. Tickets cost $8.90 per person. Find Alwyn s total cost. You can use the Distributive Property to find the total cost mentally. 4 ($9 - $0.10) = 4 ($9) - 4 ($0.10) Distributive Property = $36 - $0.40 Multiply. = $35.60 Subtract. Alwyn will pay $35.60 for himself and three friends to go to the movies. Exercises Use the Distributive Property to evaluate or rewrite each expression (w + 4) 2. (x - 5) (-2) 3. 7 (6x - 2y) (4 + 2m) 5. 8 (2n + 7) 6. (3v + 6w) 2 7. BOOKS Mariah bought 7 books costing $11.20 each. Find the total cost of the 7 books. Justify your answer by using the Distributive Property. Course 2 Chapter 5 Expressions 75

20 NAME DATE PERIOD Lesson 4 Extra Practice The Distributive Property Use the Distributive Property to evaluate each expression. 1. 2(4 + 5) (5 + 3) (7 6) 3 4. (2 + 5) (10 4) (1 + 3) 24 Use the Distributive Property to rewrite each expression. 7. 3(m + 4) 3m (y + 7)5 5y (x + 3) 6x (p 4)5 5p (s 9) 3s (x + y) 5x + 5y 13. b(c + 3d) bc + 3bd 14. (a b)( 5) 5a + 5b 15. 6(v 3w) 6v + 18w 16. 5(x + 12) 5x (m 6)(4) 4m (a b) 2a + 2b 19. (8 m)( 3) m 20. 8(p 3q) 8p 24q 21. (2x + 3y)(4) 8x + 12y 22. 2(x + 3) 2x (a + 7) 3a (g 6) 3g (a + 3) 2a (x 6) x (a 5) 4a 20 Course 2 Chapter 5 Expressions

21 NAME DATE PERIOD Lesson 5 Reteach Simplify Algebraic Expressions When a plus or minus sign separates an algebraic expression into parts, each part is called a term. The numerical factor of a term that contains a variable is called the coefficient of the variable. A term without a variable is called a constant. Like terms contain the same variables to the same powers, such as 3x 2 and 2x 2. Example 1 Identify the terms, like terms, coefficients, and constants in the expression 7x x - 3x. 7x x - 3x = 7x + (-5) + x + (-3x) Defi nition of subtraction = 7x + (-5) + 1x + (-3x) Identity Property; x = 1x The terms are 7x, -5, x, and -3x. The like terms are 7x, x, and -3x. The coefficients are 7, 1, and -3. The constant is -5. An algebraic expression is in simplest form if it has no like terms and no parentheses. Examples Write each expression in simplest form. 2 5x + 3x 5x + 3x = (5 + 3) x or 8x Distributive Property; simplify. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. 3-2m m - 3-2m and 6m are like terms. 5 and -3 are also like terms. -2m m - 3 = -2m m + (-3) Defi nition of subtraction = -2m + 6m (-3) Commutative Property = (-2 + 6) m (-3) Distributive Property = 4m + 2 Simplify. Exercises Identify the terms, like terms, coefficients, and constants in each expression y y 2. -5g g - g a a Write each expression in simplest form. 4. 3d + 6d s z + 3-9z - 8 Course 2 Chapter 5 Expressions 79

22 NAME DATE PERIOD Lesson 5 Extra Practice Simplify Algebraic Expressions Identify the terms, like terms, coefficients, and constants in each expression. 1. 8b + 7b 4 6b z 3 + 5z 3. 11q 5 + 2q 7 terms: 8b, 7b, 4, 6b terms: 9, 8z, 3, 5z terms: 11q, 5, 2q, 7 like terms: 8b, 7b, 6b like terms: 8z and 5z, 9 and 3 like terms: 11q and 2q, coefficients: 8, 7, 6 coefficients: 8, 5 5 and 7 constant: 4 constants: 9, 3 coefficients: 11, 2 constants: 5, 7 4. a a + 8a c 3c j 6 + 8j 5 terms: a, 1, 2a, 8a terms: 1, 2c, 3c, 100 terms: 14j, 6, 8j, 5 like terms: a, 2a, 8a like terms: 1 and 100, like terms: 14j and 8j, coefficients: 1, 2, 8 2c and 3c 6 and 5 constant: 1 coefficients: 2, 3 coefficients: 14, 8 constants: 1, 100 constants: 6, 5 Write each expression in simplest form. 7. 3x + 2x 5x 8. 6x 3x 3x 9. 2a 5a 3a 10. 5x 6x x 11. 8a 3a 5a 12. a 4a 3a 13. 3a + 2a 6 5a x + 2x 3 8x a 3 + 2a 7a x + 7 5x 2x x 3 + 5x 6x x 3x 2 3x a 2a + 5 a x 2 + 7x 13x a 7a + 2 2a a + 2 7a 5 3a a 2 + 5a 7 8a x 3x x 3 Course 2 Chapter 5 Expressions

23 NAME DATE PERIOD Lesson 6 Reteach Add Linear Expressions You can use models to add linear expressions. Example 1 Add (3x + 5) + (2x + 3). Step 1 Model each expression. Example 2 Add (x - 2) + (- 2x + 4). Step 1 Model each expression. x x x x x x + 5 2x x -x -x x - 2 (-2x) + 4 Step 2 Combine like tiles and write an expression for the combined tiles. Step 2 Combine like tiles and write an expression for the combined tiles. x x 3x x x x x x -x -x x + (-2x) + (-2) + 4 So, (3x + 5) + (2x + 3) = 5x + 8. Step 3 Remove all zero pairs and write an expression for the remaining tiles. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Exercises Add. Use models if needed. x -x -x (-x) So, (x - 2) + (- 2x + 4) = - x (5x + 2) + (3x + 1) 2. (- 8x + 1) + (- 2x + 6) 3. (- 7x + 4) + (x - 5) 4. (- 6x + 1) + (4x - 1) Course 2 Chapter 5 Expressions 81

24 NAME DATE PERIOD Lesson 6 Extra Practice Add Linear Expressions Add. Use models if needed. 1. (3x + 5) + (4x 1) 7x (5x 3) + ( 2x + 1) 3x 2 3. ( 7x + 4) + ( 5x 12) 12x 8 4. (4x 10) + ( 5x 2) x (7x 9) + ( x 6) 6x ( 3x + 9) + (14x 2) 11x (6x 7) + (3x 5) 9x ( 7x 5) + (9x + 6) 2x (4x + 2) + (3x 1) 7x (3x + 5) + ( 2x 2) x (2x + 4) + (4x 2) 10x ( 3x + 1) + (6x + 3) 6x (4x 5) + (6x 4) 14x ( 7x + 12) + ( 4)(2x + 3) 15x 15. (2x + 6) + 8(3x 7) 26x ( 3x + 6) + (10x 15) 2x + 9 Course 2 Chapter 5 Expressions

25 NAME DATE PERIOD Lesson 7 Reteach Subtract Linear Expressions When subtracting expressions, subtract like terms. You can use models or the additive inverse. Example 1 Find (- 3x - 2) - (4x). Step 1 Model the expression - 3x x -x -x -1-1 (-3x) + (-2) Step 2 Since there are no positive x-tiles to remove, add four zero pairs of x-tiles. Remove four positive x-tiles. -x -x -x x x x x -x -x -x -x -1-1 Zero pairs So, (- 3x - 2) - (4x) = - 7x - 2. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 2 Subtract (4x + 6) - (-7x + 1). The additive inverse of - 7x + 1 is 7x x + 6 Arrange like terms in columns. + 7x - 1 Add. 11x + 5 So, (4x + 6) - (- 7x + 1) = 11x + 5. Exercises Subtract. Use models if needed. 1. (9x + 10) - (2x + 4) 2. (3x + 4) - (2x - 5) 3. (6x + 3) - (- x - 2) 4. (4x - 1) - (x + 3) 5. (3x - 1) - (2x - 6) Course 2 Chapter 5 Expressions 83

26 NAME DATE PERIOD Lesson 7 Extra Practice Subtract Linear Expressions Subtract. Use models if needed. 1. (6x + 2) (9x + 3) 3x 1 2. ( 4x + 7) ( 7x 8) 3x (6x 7) (2x + 5) 4x (6x 8) (4x 7) 2x 1 5. (4x 8) ( 3x + 10) 7x (9x 11) (x 5) 8x (3x + 4) (x + 1) 2x (2x + 4) (x + 2) x (6x + 3) (4x 4) 2x (x + 4) ( 2x + 6) 3x (3x 2) (x 2) 2x 12. (x 9) (2x 1) x 8 Course 2 Chapter 5 Expressions

27 NAME DATE PERIOD Lesson 8 Reteach Factor Linear Expressions A linear expression is in factored form when it is expressed as the product of its factors. Example 1 Factor 5x Use the GCF to factor the linear expression. 5x = 5 x Write the prime factorization of 5x and = 5 2 Circle the common factors. The GCF of 5x and 10 is 5. Write each term as a product of the GCF and its remaining factors. 5x + 10 = 5(x) + 5(2) = 5(x + 2) Distributive Property So, 5x + 10 = 5(x + 2). Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 2 Factor 3x x = 3 x 8 = There are no common factors, so 3x + 8 cannot be factored. Exercises Factor each expression. If the expression cannot be factored, write cannot be factored x x x x x x x x x x + 45 Course 2 Chapter 5 Expressions 85

28 NAME DATE PERIOD Lesson 8 Extra Practice Factor Linear Expressions Find the GCF of each pair of monomials , 42x a, m, 56n x, 56y c, 28cd 4c 6. 5ab, 6b b 7. 7x, 14xy 7x 8. 14b, 56bc 14b 9. 21a, 63ab 21a Factor each expression. If the expression cannot be factored, write cannot be factored. Use algebra tiles if needed x + 3 3(4x + 1) 11. x 2 cannot be factored 12. 4x 3 cannot be factored 13. 3x + 9 3(x + 3) 14. 6x 12 6(x 2) 15. 2x 7 cannot be factored 16. 7x (x + 2) x 10 2(6x 5) 18. 3x (x + 12) 19. 4x (x + 5) x + 7 cannot be factored x (5x + 9) Course 2 Chapter 5 Expressions

29 NAME DATE PERIOD Lesson 3 Reteach Solve Equations with Rational Coefficents Multiplicative inverses, or reciprocals, are two numbers whose product is 1. To solve an equation in which the coefficient is a fraction, multiply each side of the equation by the reciprocal of the coefficient. Example 1 Solve 15 = 0.5n. Check the solution. 15 = 0.5n Write the equation = 0.5n 0.5 Division Property of Equality 30 = n Simplify. Example 2 Solve 4 x = 8. Check your solution. 5 4 x = 8 Write the equation. 5 ( 5 4 ) 4 5 x = ( 5 4 ) 8 Multiply each side by the reciprocal of 4 5, 5 4. x = 10 The solution is 10. Simplify. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Exercises Solve each equation. Check your solution = 0.7m = - 6 h = 4b x = = 10 a 6. 9 = 0.3n y = = 0.75a = b Course 2 Chapter 6 Equations and Inequalities 91

30 NAME DATE PERIOD Lesson 3 Extra Practice Solve Equations with Rational Coefficients Solve each equation. Check your solution m = = 0.6x y = = 6.2z t = x = a = or x = r = t = = 1 4 h m = n = = 1 10 b x = = 1 5 y m = z = c = f = x = Course 2 Chapter 6 Equations and Inequalities

31 NAME DATE PERIOD Lesson 4 Reteach Solve Two-Step Equations To solve a two-step equation, undo the addition or subtraction fi rst. Then undo the multiplication or division. Example 1 Solve 7v - 3 = 25. Check your solution. 7v - 3 = 25 Write the equation. +3 = +3 Undo the subtraction by adding 3 to each side. 7v = 28 Simplify. 7v 7 = 28 7 Undo the multiplication by dividing each side by 7. v = 4 Simplify. Check 7v - 3 = 25 Write the original equation. 7(4) Replace v with Multiply. 25 = 25 The solution checks. The solution is 4. Example 2 Solve -10 = 8 + 3x. Check your solution. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. -10 = 8 + 3x Write the equation. -8 = -8 Undo the addition by subtracting 8 from each side. -18 = 3x Simplify = 3x 3 Undo the multiplication by dividing each side by = x Simplify. Check -10 = 8 + 3x Write the original equation (-6) Replace x with (-18) Multiply. -10 = -10 The solution checks. The solution is -6. Exercises Solve each equation. Check your solution. 1. 4y + 1 = x + 2 = = 5k n + 4 = = -3c p + 3 = = -5t r + 12 = n = = 7 + 4b p = = y = 4t x - 10 = = 12z g = 7 Course 2 Chapter 6 Equations and Inequalities 93

32 NAME DATE PERIOD Lesson 4 Extra Practice Solve Two-Step Equations Solve each equation. Check your solution. 1. 3x + 6 = r 7 = d = b + 4 = w 12 = t 4 = q 6 = g 3 = = 6y s 4 = f = p = e + 14 = b = m 9 = c = t 14 = x + 24 = w 4 = d 3 = g 16 = k + 13 = = 5 2x z + 15 = 1 2 Course 2 Chapter 6 Equations and Inequalities

33 NAME DATE PERIOD Lesson 5 Reteach More Two-Step Equations An equation in the form p(x + q) = r contains two factors, p and (x + q) and is considered a two-step equation. Example 1 Solve 6(x + 2) = 42. Check your solution. 6(x + 2) = 42 Write the equation. 6(x + 2) = 42 Division Property of Equality 6 6 Simplify. x + 2 = 7-2 = -2 Subtraction Property of Equality x = 5 Simplify. Check 6(x + 2) = 42 Write the original equation. 6(5 + 2) 42 Replace x with 5. 6(7) 42 Add. Multiply. 42 = 42 The solution checks. The solution is 5. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 2 Solve 4 (x - 5) = 4. Check your solution. 5 4 (x - 5) = 4 5 Write the equation (x - 5) = Multiplication Property of Equality (x - 5) = = 1; write 4 as 4 1. x - 5 = 5 Simplify. + 5 = +5 Addition Property of Equality x = 10 Simplify. 4 Check (x - 5) = 4 Write the original equation. 5 Replace x with (10-5) = 4 Subtract then multiply. 5 4 (5) = 4 5 The solution checks. The solution is 10. Exercises Solve each equation. 1. 7(x + 4) = (x - 8) = (x + 3) = (x - 3) = (x - 12) = (x + 4) = (x + 5) = (x - 15) = 4 8 Course 2 Chapter 6 Equations and Inequalities 95

34 NAME DATE PERIOD Lesson 5 Extra Practice More Two-Step Equations Solve each equation. Check your solution. 1. 2(x + 4) = (x 4) = ( 10 + x)2 = (x 12)4 = (x 4) = (x + 7) = (x 4) = (x + 12) = = 6(x + 2) (x 15) = (18 + x)2 = (11 + x) 3 4 = (x + 12) = (x 4) = (x 9) = (x 7) = (x + 12) = (x + 24) = Course 2 Chapter 6 Equations and Inequalities

35 NAME DATE PERIOD Lesson 6 Reteach Solve Inequalities by Addition or Subtraction Solving an inequality means fi nding values for the variable that make the inequality true. You can use the Addition and Subtraction Properties of Inequality to help solve an inequality. When you add or subtract the same number from each side of an inequality, the inequality remains true. Examples Solve each inequality. 1 x + 4 > 9 Write the inequality. x > 9-4 Subtract 4 from each side. x > 5 Simplify. Any number greater than 5 will make the statement true. Therefore, the solution is x > n - 9 Write the inequality n Add 9 to each side. -3 n Simplify. The solution is -3 n or n -3. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. 3 Solve a + 1 < 1. Graph the solution set on a number line. 3 a + 1 < 1 Write the inequality. 3 a < a < 2 3 Exercises Solve each inequality. Subtract 1 from each side. 3 Simplify. 1. t - 6 > 3 2. b < r < p + 4 Solve each inequality. Graph the solution set on a number line. 5. s + 8 < d Course 2 Chapter 6 Equations and Inequalities 99

36 NAME DATE PERIOD Lesson 6 Extra Practice Solve Inequalities by Addition or Subtraction Solve each inequality. Graph the solution set on a number line. 1. y + 3 > 7 y > 4 2. c 9 < 5 c < x x 5 4. y 3 < 15 y < t 13 5 t x + 3 < 10 x < 7 7. y 6 2 y 8 8. x 3 6 x 3 9. a a c 2 11 c a a y y y 6 17 y > a a 16 Course 2 Chapter 6 Equations and Inequalities

37 NAME DATE PERIOD Lesson 7 Reteach Solve Inequalities by Multiplication or Division When you multiply or divide each side of an inequality by a positive number, the inequality remains true. However, when you multiply or divide each side of an inequality by a negative number, the direction of the inequality must be reversed for the inequality to remain true. Example 1 Solve t -4. Then graph the solution set on a number line. -6 t -4 Write the inequality. -6 t (-6) -4(-6) Multiply each side by -6 and reverse the inequality symbol. -6 t 24 Simplify. To graph the solution, place a closed circle at 24 and draw a line and arrow to the right Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 2 Solve 4 5 x - 5 < x - 5 < 23 Write the inequality. 5 4 x < Add 5 to each side. 5 4 x < 28 Simplify. 5 ( 5 4) 4 5 x < ( 5 28 Multiply each side by 4) 5 4. x < 35 Simplify. Exercises Solve each inequality. Then graph the solution on a number line. 1. 3a > r Solve each inequality. Check your solution c > y h 5-6 < a Course 2 Chapter 6 Equations and Inequalities 101

38 NAME DATE PERIOD Lesson 7 Extra Practice Solve Inequalities by Multiplication or Division Solve each inequality. Graph the solution set on a number line. 1. 5p 25 p x < 12 x < m m 5 4. d 3 > 15 d > < r 7 r > g < 27 g < p 24 p > k 3 k > z 5 > 2 z < x 9 x x > 35 x < a 6 < 1 a > 6 Course 2 Chapter 6 Equations and Inequalities

39 NAME DATE PERIOD Lesson 8 Reteach Solve Two-Step Inequalities A two-step inequality is an inequality that contains two operations. To solve a two-step inequality, use inverse operations to undo each operation in reverse order of the order of operations. Example 1 Solve 4x Graph the solution set on a number line. 4x Write the inequality Addition Property of Inequality 4x 20 Simplify. 4x 20 Division Property of Inequality 4 4 x 5 Simplify. The solution is x 5. Graph the solution set Draw a closed dot at 5 with an arrow to the left. Check 4x Write the inequality. 4(3) Replace x with a number less than or equal to 5. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use This statement is true. Exercises Solve each inequality. Graph the solution set on a number line. 1. 3x - 4 < x < 2x x > x x Course 2 Chapter 6 Equations and Inequalities 103

40 NAME DATE PERIOD Lesson 8 Extra Practice Solve Two-Step Inequalities Solve each inequality. Graph the solution set on a number line. 1. 2x 3 > 11 x > x x x 6 x < 4x + 1 x > x x x 6 > 19 x < 5 7. x > 1 x > < 3x + 6 x < x 3 x x 5 x x 5 > 7 x < x x 10 Course 2 Chapter 6 Equations and Inequalities

41 NAME DATE PERIOD Lesson 1 Reteach Classify Angles An angle is formed by two rays that share a common endpoint called the vertex. An angle can be named in several ways. The symbol for angle is. Angles are classified according to their measures. Two angles that have the same measure are said to be congruent. Two angles are vertical if they are opposite angles formed by the intersection of two lines. Vertical angles are congruent. Two angles are adjacent if they share a common vertex, a common side, and do not overlap. Acute Angle Right Angle Obtuse Angle Straight Angle less than 90 exactly 90 between 90 and 180 Example Name each angle below. Then classify the angle as acute, right, obtuse, or straight. exactly 180 Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use Use the vertex as the middle letter and a point from each side, ABC, CBA, or use the vertex or the number only, B or 1. The angle is 90, so it is a right angle Use the vertex or the number only, D or 2. The angle is less than 90, so it is an acute angle. 3. What is the value of x in the figure at the right? The angle labeled 5x and the angle labeled 55 are vertical angles. Since vertical angles are congruent, the value of x is 11. Exercises Name each angle. Then classify the angle as acute, right, obtuse, or straight x 4. Find the value of x in the figure at the right. (3x - 4) Course 2 Chapter 7 Geometric Figures 105

42 NAME DATE PERIOD Lesson 1 Extra Practice Classify Angles Name each angle in four ways. Then classify each angle as acute, right, obtuse, or straight , ABC, CBA, B; right 2, DEF, FED, E; obtuse , LMN, NML, M; straight 4, XYZ, ZYA, Y; acute , KLM, MLK, L; acute 6, RST, TSR, S; obtuse Refer to the diagram at the right. Identify each angle pair as adjacent, vertical, or neither and 2 adjacent 8. 2 and 5 neither 9. 1 and 3 vertical and 4 adjacent and 5 neither and 4 neither Course 2 Chapter 7 Geometric Figures

43 NAME DATE PERIOD Lesson 2 Reteach Complementary and Supplementary Angles Two angles are complementary if the sum of their measures is 90. Two angles are supplementary if the sum of their measures is 180. Examples Identify each pair of angles as complementary, supplementary, or neither = = 90 The angles are supplementary. The angles are complementary. Example 3 ALGEBRA Find the value of x. Since the two angles form a straight line, they are supplementary. The sum of their measures is x 5x + 35 = 180 Write the equation. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use = -35 Subtract 35 from each side. 5x = Divide each side by 5 x = 29 Simplify. Exercises Identify each pair of angles as complementary, supplementary, or neither ALGEBRA Find the value of x in each figure x 4x x Course 2 Chapter 7 Geometric Figures 107

44 NAME DATE PERIOD Lesson 2 Extra Practice Complementary and Supplementary Angles Identify each pair of angles as complementary, supplementary, or neither complementary supplementary complementary neither Find the measure of x in each figure Course 2 Chapter 7 Geometric Figures

45 NAME DATE PERIOD Lesson 3 Reteach Triangles Every triangle has at least two acute angles. One way you can classify a triangle is by using the third angle. Another way to classify triangles is by their sides. Sides with the same length are congruent segments. Classify Triangles Using Angles all acute angles 1 right angle 1 obtuse angle acute triangle right triangle obtuse triangle Classify Triangles Using Sides no congruent sides at least 2 congruent sides 3 congruent sides scalene triangle isosceles triangle equilateral triangle Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example The figure shows a triangular pennant tied to a pole. Classify the marked triangle by its angles and by its sides. The triangle has three acute angles and two sides the same length. So, it is an acute, isosceles triangle. Exercises Draw a triangle that satisfies each set of conditions. Then classify each triangle. 1. a triangle with three acute angles and three congruent sides 2. a triangle with one right angle and no congruent sides 20 Course 2 Chapter 7 Geometric Figures 109

46 NAME DATE PERIOD Lesson 3 Extra Practice Triangles Classify each triangle by its angles and by its sides acute scalene right scalene acute equilateral right scalene Find the value of x Course 2 Chapter 7 Geometric Figures

47 NAME DATE PERIOD Lesson 4 Reteach Scale Drawings A scale drawing represents something that is too large or too small to be drawn or built at actual size. Similarly, a scale model can be used to represent something that is too large or built too small for an actual-size model. The scale gives the relationship between the drawing/model measure and the actual measure. Example On this map, each grid unit represents 50 yards. Find the horizontal distance from Patrick s Point to Agate Beach. map actual Patrick s Point Scale to Agate Beach 1 unit 50 yards = 8 units map x yards actual 1 x = 50 8 Cross products N Patrick's Point Agate Beach x = 400 Simplify. It is 400 yards from Patrick s Point to Agate Beach. Exercises Find the actual distance between each pair of cities. Round to the nearest tenth if necessary. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Cities 1. Los Angeles and San Diego, CA 2. Lexington and Louisville, KY 3. Des Moines and Cedar Rapids, IA 4. Miami and Jacksonville, FL Map Distance Scale 6.35 cm 1 cm = 20 mi 15.6 cm 1 cm = 5 mi cm 2 cm = 15 mi cm 1 cm = 20 mi 2 Find the length of each object on the scale drawing with the given scale. Then find the scale factor. 5. an automobile 16 feet long; 1 inch:6 inches 6. a pond 85 feet across; 1 inch = 4 feet 7. a parking lot 200 meters wide; 1 centimeter:25 meters Actual Distance 8. a flag 5 feet wide; 2 inches = 1 foot Course 2 Chapter 7 Geometric Figures 113

48 NAME DATE PERIOD Lesson 4 Extra Practice Scale Drawings Use the scale drawing to find the actual length and width of each room. Then find the actual area of each room. Master Bedroom Master Bath Bedroom 2 Kitchen and Dining Area Living Room Half Bath Key 1 cm = 3 ft 1. master bedroom 2. bedroom 2 15 ft by 12 ft; 180 ft 2 12 ft by 12 ft; 144 ft 2 3. kitchen and dining area 4. half bath 18 ft by 12 ft; 216 ft 2 6 ft by 9 ft; 54 ft 2 On a map, the scale is 1 inches = 50 miles. For each map distance, find the actual distance inches 250 mi inches 600 mi inches mi inch 40 mi inches mi inches mi Course 2 Chapter 7 Geometric Figures