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MATH 7 Workbook

Unit 3: Number Relationships

Unit 4: Equations and Inequalities

Name:_______________________________

Block:_____

Teacher:__________________________

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Unit 3 Number Relationships

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Unit 3 - Table of Contents

Day

Pages Topic

1 5 Unit 3 Pretest

1&2 6-14 Properties of Operations

3 15-21 Translate Expressions and Equations

4 22-28 Evaluate algebraic expressions

5&6 29-35 Arithmetic and Geometric Sequences

7-9 36-56 Relations and Functions

10 57-61 Study Guide

11 Unit 3 TEST

Unit 3 SOL Objectives

7.16: Apply the following properties of operations with real numbers:

a. commutative and associative properties for add. and mult.

b. distributive property

c. additive and multiplicative identity properties

d. additive and multiplicative inverse properties

e. multiplicative property of zero

7.13a: Write verbal expressions as algebraic expressions and sentences

as equations and vice versa.

7.13b: Evaluate algebraic expressions for given replacement values of

the variables.

7.2: Describe and represent arithmetic and geometric sequences,

using variable expressions.

7.12: Represent relationships with tables, graphs, rules and words.

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Unit 3 Pretest

Translate. 1) Three less than twice a number, x 2) The quotient of 4 and a number, x

Sequences. 3) -5, -3, -1, 1, 3, … What is the variable

expression for the rule? 4) 28, -14, 7,

What is the common

ratio?

Properties. Which property is shown? 5) 5 + 9 + (-2) = 5 + (-2) + 9 6) 4 + 5 + (-5) = 4 + 0

Distributive Property. 7) ( )

8) ( )

Evaluate. ) ( )

Evaluate. )

) ( ) ) ( )

13) Determine whether the relation is a function and state the domain and range.

{(-3,9), (4,8), (-7,4), (0,4), (-3,8)} Is this relation a function?______

Domain:_____________ Range:______________

14) Which is the rule of this function?

y = 2x + 3

y = 2x – 3

y = -3x

y = 3x

x y

-2 -6

2 6

5 15

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Warm Up

1. -9 – 2 = ______ 2. 8 – 12 = ______ 3. -4 + -5 = ____

4. -12 + 5 = _______ 5. -1 – (-2) = ______ 6. -8 + 4 = _____

7. 10 + (-3) = ______ 8. -7 – (-2) = ______ 9. 4 – (-7) = ____

Notes: Properties of Operations with Real Numbers

SOL Objective 7.16: Apply the following properties of operations with real numbers:

a. commutative and associative properties for addition and multiplication

b. distributive property

c. additive and multiplicative identity properties

d. additive and multiplicative inverse properties

e. multiplicative property of zero.

Distributive Property

To multiply a number by a sum or difference, multiply each number inside the

parentheses by the number outside the parentheses.

a(b + c) = ab + ac OR a(b – c) = ab – ac

Let’s Practice:

1. -3(x – 9) __________ 4. 5(x + 10) ____________

2. (y + 7) 5 ____________ 5. 4(2x + 3y) _____________

3. b (c + 3d) ____________ 6. -2(y + 4) ____________

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Commutative Property

The order in which two numbers are added or multiplied does not change their sum or

product.

3 + 5 = 5 + 3 Commutative Property of Addition

10 • 15 = 15 • 10 Commutative Property of Multiplication

Let’s Practice:

1. 5 + 4 = ____ + 5 2. 8 • 9 = 9 • ____

3. (9 + 1) + 3 = 3 + ____________ 4. 2 • 5x = 5x • ____

Associative Property

The way in which three numbers are grouped when they are added or multiplied does

not change their sum or product.

(6 + 3) + 7 = 6 + (3 + 7) Associative Property of Addition

(4 • 3) • 5 = 4 • (3 • 5) Associative Property of Multiplication

Let’s Practice:

1. 7 + (8 + 1) = ______________________ 2. (5 • 8) • 9 = ________________

3._______________________ = x + (y + 3)

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Identity Property

The sum of an addend and zero is the addend.

7 + 0 = 7 Identity Property of Addition

The product of a factor and one is the factor.

5 • 1 = 5 Identity Property of Multiplication

Let’s Practice:

1. 9 + ____= 9 2. 7 • ____ = 7

Multiplicative Property of Zero

When any number is multiplied by 0, the product is 0.

9 • 0 = 0

Let’s Practice:

1. 8 • ___ = 0 2. 2 • 0 = ____

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Inverse Property

The sum of a number and its opposite (additive inverse) is 0.

-6 + 6 = 0 Inverse Property of Addition

The product of a number and its reciprocal (multiplicative inverse) is 1.

•

= 1 Inverse Property of Multiplication

Let’s Practice:

1. 9 + _____ = 0 2. 7 • _____ = 1

3. What is the multiplicative inverse of 4? ________

4. What is the additive inverse of 5?________

Name the property shown by each statement.

1. 1 • 4 = 4 _______________________________________________

2. 6 + (b + 2) = (6 + b) + 2 ________________________________________________

3. 9 •

= 1 ________________________________________________

4. 8t • 2 = 2 • 8t ________________________________________________

5. 0 (13n) = 0 ________________________________________________

6. 7 (x + 7) = 7x + 49 ________________________________________________

Simplify each expression.

1. (12 + x) + 9 ____________________

2. 2 • (6x) ____________________

3. (5m) • 3 ____________________

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PRACTICE: Properties of Addition and Multiplication

Use the word bank to name the property shown by each statement.

Commutative Property of Addition Commutative Property of Multiplication

Associative Property of Addition Associative Property of Multiplication

Inverse Property of Addition Inverse Property of Multiplication

Identity Property of Addition Identity Property of Multiplication

Distributive Property Multiplicative Property of Zero

1. 2(x + 3) = 2x + 6 _______________________________

2. 9 ● 0 = 0 _______________________________

3. 7 + 10 = 10 + 7 _______________________________

4. 3 + (2 + 9) = (3 + 2) + 9 _______________________________

5. 10 ● 1 = 10 _______________________________

6. -6 + 6 = 0 _______________________________

7. r ● s = s ● r _______________________________

8. 7 + 0 = 7 _______________________________

9. 2 ● (4 ● 11) = (2 ● 4) ● 11 _______________________________

10.

•

= 1 _______________________________

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1. Apply the commutative property to the equations below.

3 + 5 = __________________ 2 • 4 = __________________

2. Apply the associative property to the equations below.

(1 + 3) + 5 = ________________ (2 • 6) • 8 = _____________

3. Apply the distributive property to the equations below.

3(x + 5) = _________________ 5(y – 7) = ________________

4. Apply the identity property of addition to the equations below.

13 + ___ = _________ 37 + ___ = ________

5. Apply the identity property of multiplication to the equations below.

8 • ____ = _______ 62 • ____ = ________

6. Apply the inverse property of addition to the equations below.

13 + _____ = ________ 37 + _____ = _________

7. Apply the inverse property of multiplication to the equations below.

½ ● _____ = ________ ______ ● ¾ = _______

8. Apply the multiplicative property of zero to the equations below.

8 ● ____ = _______ f ● ____ = _______

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Practice – The Distributive Property

1. 4(d + 2) = ________________ 16. 4(h – 16) = ___________________

2. 3(u – 3) = ________________ 17. -3(w – 10) = __________________

3. -6(f + 5) = ________________ 18. -10(c + 9) = __________________

4. -2(x – 3) = _______________ 19. 2(11 – q) = ___________________

5. 3(x – 7) = ________________ 20. -4(12 – f) = ___________________

6. 8(-b + 4) = _________________ 21. 12(n + 2) = ___________________

7. (9 – h)5 = __________________ 22. -3(-x - 1) = ___________________

8. (c + 1)(-4) = __________________ 23. -8(9 + b) = ___________________

9. -1(2 – y) = __________________ 24. -5(z – 4) = ____________________

10. -7(a + 1) = __________________ 25. 6(r – 20) = ____________________

11. 11(k – 20) = _________________ 26. 7(2 – j) = _____________________

12. -9(r – 1) = __________________ 27. -1(m + 1) = ___________________

13. 5(1 – b) = ___________________ 28. -2(v – 8) = ____________________

14. 8(x + 12) = __________________ 29. 5(q – 16) = ___________________

15. -6(p + 15) =___________________ 30. -10(c – 7) = ___________________

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Properties Practice – Multiple Choice

1. Which shows the commutative 2. Which shows the 3. Which shows the

property? associative property? distributive property?

A. (3 + 5) + 8 = 3 + (5 + 8) F. 7 + 4 = 4 + 7 A. 8(2 + 5) = 16 + 40

B. 3(5 + 8) = 15 + 24 G. 2 + (4 + 7) = (2 + 4) + 7 B. 8 + 2 + 5 = 2 + 8 + 5

C. 3 + 5 = 5 + 3 H. 2(7 + 4) = 14 + 8 C. 8 ● 2 = 2 ● 8

D. 5 + (-5) = 0 J. 4 + 0 = 4 D. 8 ● 1 = 8

4. Which is the additive inverse 5. Which is the multiplicative 6. Which shows the

of 13? inverse of 8? mult property of 0?

A. 1 A. -8 F. 7 ● 0 = 0

B. 1/13 B. -1/8 G. 7 ● 0 = 7

C. -1/13 C. 1/8 H. 7 + 0 = 7

D. -13 D. 0 J. 7 + (-7) = 0

7. Which statement is NOT true? 8. Which statement is NOT true? 9. Which statement

A. 3(2 + 4) = 6 + 4 F. 4 ÷ 5 = 5 ÷ 4 is NOT true?

B. 3 + 2 + 4 = 2 + 4 + 3 G. 4 ● 5 = 5 ● 4 A. ¾ ● 4/3 = 1

C. 3 ● 2 ● 4 = 2 ● 3 ● 4 H. 4(5 + 2) = 20 + 8 B. 3(-4 + 8) = -12 + 24

D. 3 ● 1/3 = 1 J. 4 ● 1 = 4 C. -3/4 ● 4/3 = 1

D. 3 ● 4 ● 0 = 0

10. Which shows the commutative 11. Which is the multiplicative 12. Which is not a true

property? inverse of ¾? statement?

A. 7(2 ● 8) = (7 ● 2) 8 F. 3 A. 4 ● 0 ● 7 = 0

B. 7(2 – 8) = 14 – 56 G. 4 B. 4 + 0 + (-4) = 8

C. 7 + (2 + 8) = (7 + 2) + 8 H. 4/3 C. 4(0 – 4) = -16

D. 7 ● 2 ● 8 = 2 ● 7 ● 8 J. -3/4 D. 4(2 + 4) = 8 + 16

13. Anne’s utility bills for three months were 14. Which statement is false?

$59, $67, and $33. To add the utility bills monthly, A. -7/1 = -1

Anne thought: (59 + 67) + 33 = 59 + (67 + 33) B. 7 + 0 = 7

C. 7 + (-7) = 0

What property did Anne use? D. 7 ● -1 = -7

A. Inverse property of addition

B. Identity property of addition

C. Commutative property of addition

D. Associative property of addition

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15. Which property is shown in the following number sentence?

(¾ x + 9) + 0 = ( ¾ x + 9)

A. Multiplicative identity property

B. Additive identity property

C. Multiplicative inverse property

D. Additive inverse property

16. 1/7 ● y = 1/7 17. Which number sentence illustrates the

commutative property of multiplication?

If the number sentence is true, then y is the -- A. 14 + (13 ● 7) = 14 + (7 ● 13)

B. 14 + (13 ● 7) = 13 + (14 ● 7)

F. additive identity C. 14 + (13 ● 7) = 14 ● 13 + 14 ● 7

G. additive inverse D. 14 + (13 ● 7) = (14 + 13) ● 7

H. multiplicative identity

J. multiplicative inverse

18. 1/7 + y = 1/7 19. Which number sentence illustrates the

distributive property?

If the number sentence is true, then y is the – F. 14 + (13 ● 7) = 14 + (7 ● 13)

G. 14 + (13 ● 7) = 13 + (14 ● 7)

F. additive identity H. 14 + (13 ● 7) = (14 + 13) ● 7

G. additive inverse J. 14(13 + 7) = 14 ● 13 + 14 ● 7

H. multiplicative identity

J. multiplicative inverse

Use the Distributive Property to simplify each expression.

1. -2 (x – 3) = ______________________ 2. 4 (y + 5) = ______________________

3. -3(x + 7) = _______________________ 4. 8 (y – 2) = _______________________

5. 10(x + 4) = _______________________

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Warm Up

1. -4(x – 9) = ___________ 2. 3(y + 2) = __________

3. 3(x – 7) = __________ 4. -4(2 – x) = _________

5. (y + 3)(-2) = ____________ 6. (1 – y) (-5) = _________

Notes: Translate Expressions and Equations

SOL Objective 7.13a: Write verbal expressions as algebraic expressions and

sentences as equations and vice versa.

Numerical Expression:_______________________________________________

__________________________________________________________________

Example:_______________________

Algebraic Expression:_________________________________________________

__________________________________________________________________

Example:_______________________

Equation:__________________________________________________________

__________________________________________________________________

Example:_______________________

Variable:__________________________________________________________

__________________________________________________________________

Example:_______________________

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PRACTICE – Translate verbal phrases and sentences into expressions and equations

Translate each verbal phrase into a numerical expression.

1. The product of eight and seven. ______________________

2. The difference of nine and three. ______________________

3. The sum of seven, four, and eighteen. ______________________

4. The quotient of eighty-one and three. ______________________

Addition Phrases Expression

8 more than a number The sum of 8 and a number x increased by 8 the total of x and 8

x + 8

Multiplication Phrases Expression

Twice a number The product of 2 and n 2 multiplied by a number 2 times a number

2n

Subtraction Phrases Expression

The difference of r and 6 r decreased by 6 6 less than a number 6 subtracted from a number

r - 6

Division Phrases Expression

The quotient of z and 3 A number divided by 3 The ratio of z and 3

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Translate each verbal phrase into an algebraic expression.

5. Twelve points more than the Dolphins scored. ____________________

6. Four times a number decreased by 6. ____________________

7. The quotient of thirty and ten times a number. ____________________

8. Five times the sum of three and some number. ____________________

9. Six times a number, minus seven _____________________

10. The difference of sixty and a number _____________________

11. Three times the number of tickets sold _____________________

12. Twelve more than four times a number _____________________

13. Half the distance to school _____________________

14. Four times a number decreased by 6 _____________________

15. The quotient of thirty and ten times a number _____________________

16. Five times the sum of three and some number _____________________

Translate each verbal sentence into an algebraic equation.

17. 5 more than a number is 6. ____________________

18. The product of 7 and b is equal to 63. _____________________

19. The sum of r and 45 is 79. _____________________

20. The quotient of x and 7 is equal to 13. _____________________

Translate:

1. k + 12 __________________________________________________________________________

2. 17(y + 11) _______________________________________________________________________

3. 3b – 8 ___________________________________________________________________________

4. 2c – 5 = 2___________________________________________________________________

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Translate each phrase or sentence into an expression or equation.

1. six minutes less than Bob’s time ________________

2. five less than twice a number is 7 ________________

3. four points more than the Bearcubs scored ________________

4. Joan’s temperature increased by two degrees ________________

5. the cost decreased by ten dollars ________________

6. Ten more than the quotient of a number and 3 is 12 ________________

7. the sum of four feet and seven feet ________________

8. the difference of 150 lb and 8 lb ________________

9. five more than x ________________

10. fifteen less than c ________________

11. one less than the product of four and a number is 11 _______________

12. three less than a number ________________

13. the product of a certain number and nine ________________

14. the product of 2 and the sum of 5 and t is 8 ________________

15. a number increased by six ________________

16. seven times a certain number ________________

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17. twice a number, decreased by four ________________

18. the quotient of ten and five ________________

19. eight decreased by y ________________

20. twice the sum of two and y ________________

21. ten more than the quotient of a number and 3 is 12 ________________

22. p more than twenty-nine ________________

23. fifty minus k ________________

24. sixteen less than m ________________

25. the quotient of x and 2 ________________

26. the sum of 9 and the quotient of x and 7 is 11 ________________

27. eleven times a number, decreased by three ________________

28. four times the sum of a number and eight ________________

29. five increased by seven times a number ________________

30. nine more than the quotient of b and 4 ________________

31. three times the difference between x and 5 ______________

32. ten less than the quotient of a number and −2 is three ______________

33. the product of six and a number, increased by six ______________

34. two-fifths of a number, minus seven ______________

35. a number increased by four times the number ______________

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PRACTICE: Translating Phrases to Algebraic Expressions

Translate the sentences to algebraic equations.

1. The sum of a number and 16 is equal to 45. ___________________________

2. The product of 6 and m is 216. ___________________________

3. The difference of 100 and x is 57. ___________________________

4. The quotient of z and 10 is 32. ___________________________

5. $18 less than the original price is $48. ___________________________

6. 17 more than some number is equal to 85. ___________________________

7. The number of members divided by 6 is 15. ___________________________

8. The total of Joshua’s savings and $350 is $925. ___________________________

Translate the phrases to algebraic expressions.

1. 5 more than 2 times a number ________________________

2. 7 less than 5 times a number ________________________

3. 5 times a number, decreased by 2 ________________________

4. 2 diminished by 7 times a number ________________________

5. 2 increased by 5 times a number ________________________

6. Twice a number, decreased by 7 ________________________

7. 2 more than 7 times a number ________________________

8. 9 less than a number ________________________

9. 4 times a number, plus 9 ________________________

10. 9 decreased by 4 times a number ________________________

11. One-fourth of a number ________________________

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12. 9 times a number, decreased by 4 ________________________

13. 4 more than 9 times a number ________________________

14. 9 times a number, increased by 4 times the number ________________________

15. 2 times a number, increased by 8 ________________________

16. 2 times the sum of a number and 8 ________________________

17. 3 more than 8 times a number ________________________

18. 8 times the sum of a number and 3 ________________________

19. 3 times the sum of a number and 8 ________________________

20. Two-thirds of a number ________________________

21. 3 times the sum of twice a number and 8 ________________________

22. 10 meters higher than height x ________________________

Mixed Review

1. Write 10-3 as a decimal and a fraction:________ _________

2. Write 56,700,000 in scientific notation. ____________________

3. Write 4.2 x 10-2 in standard form. ____________________

4. Put in ascending order: 56% .5 5.2 x 103

__________ _________ _________ __________

5. Write 68% as a decimal and a simplified fraction: ________ ________

6. Distribute: 8(9x – 2) = _____________________

7. A. -2 – (-7) = _______ B. 9 ● -7 = _______

C. -18 ÷ -3 = _______ D. -4 + -3 = _______

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Warm Up

Translate:

1. four more than the difference of x and 7. ______________

2. ten decreased by the product of a number and 2. __________

3. the quotient of a number and 3, decreased by 5. ___________

4. the sum of four and a number is 13. ________________

5. the difference of 8 and a number is equal to -12. ______________

6. Two less than the sum of 4 and a number is 20. _________________

Notes: Evaluate Algebraic Expressions

SOL Objective 7.3b: Evaluate algebraic expressions for given replacement

values of the variables.

To evaluate an algebraic expression…

replace the variable or variables with known values

and then use the order of operations.

Example: x + y – 9 if x = 15 and y = 26

15 + 26 – 9

41 – 9

32

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Let’s practice evaluating expressions!

1. 6m – 3k if m = 7 and k = 2 2.

if m = 7 and n = 4

3. n + (k + 5m) if m = 7, k = 2, n = 4 4.

if a = 5 and c = 8

When substituting negative numbers,

put (parentheses) around them

so you get the calculations right.

If a = -2, then….

-a = _____________ = _________

a2 = _____________ = _________

-a2 = ______________ = _________

Let’s practice substituting negative numbers…. Let x = -3

3 – x x2 + 4 9 – x2

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Practice: Evaluating Algebraic Expressions

Evaluate for a = -5, b = 2, c = -6.

1. 8a

2. a + b + c

3. 12b

4. 50 - c

5. ab

6. 7(a + c)

7. 4bc

8.

Evaluate for w = 9, x = 10, y = 3.

9. 5(x + 2)

10. (4w) ÷ y

11. 8(x + y)

12.

13.

14. 100 – (x + y) 15. x2 16.

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Evaluate for x = -4.

17. 9x

20.

18. 2x + 7

21. 3x2

19. x2

22. (3x)2

Evaluate for a = 7 and b = 2.

23. 6ab 24. 8a – 5b 25. ab2

26. a2 + b2 27. (a + b)2 28. (a – b)3

29.

30. b3(a – 2b) 31.

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Use the Order of Operations to simplify each expression.

1. 18 ÷ 2 · 3 + 22 2. 12 ÷ (-3) + [52 – 7 · 3] 3. -2(-4) + 8[6(9) – 72]

4. 2(64 – 24) + 12 ÷ 6 5.

√

MORE PRACTICE– Evaluating Algebraic Expressions

Evaluate for x = 4.

1. 9x 2. 2x + 7 3. x2

4.

5. 3x2 6. (3x)2

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Evaluate for a = 7 and b = 2.

7. 6ab 8. 8a – 5b 9. ab2

10. a2 + b2 11. (a + b)2 12. (a – b)3

13.

14. b3 (a – 2b) 15.

Review

Solve each problem using the Order of Operations. Show all of your steps.

1. 12 ÷ 4 + 2 2. 25 – 15 ÷ 5 3. 52 – 4 • 6 ÷ 3

4. 43 ÷ (16 – 12) • 3 5. 18 + 1(12) ÷ 6 6. 16 – 6 + 5 • 23

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7. 52 – 2 • 4 + (7 – 2) 8. 2 [18 – (5 + 32) ÷ 7] 9. 67 + 84 -12 • 4 ÷ 16

Translate the phrases to algebraic expressions.

1. 5 more than 2 times a number_______________ 2. 7 less than 5 times a number _____________

3. 5 times a number, decreased by 2______________ 4. 2 diminished by 7 times a number________

5. 2 increased by 5 times a number________________ 6. Twice a number, decreased by 7__________

7. 2 more than 7 times a number__________________ 8. One-fourth of a number_______________

Use the Distributive Property to simplify each expression.

1. 3(m + 4) = _____________ 2. -5(9 – z) = _____________ 3. (y + 7)5 = ________________

4. 21(k – 3) = ______________ 5. -6(x+ 3)= ______________ 4. (7 – 2h)(-3) = ________________

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Warm Up

Evaluate each expression if a = 2, b = 4, and c = -3.

1. c4 2. ac3 3. 3a + b3 4. c2 + a2

Notes: Arithmetic and Geometric Sequences

SOL Objective 7.2: Describe and represent arithmetic and geometric sequences, using variable expressions.

A sequence is _______________________________________________________.

Each number is called a __________________.

Example: 1, 2, 4, 7, 11…

Terms

Examples: 3, 5, 7, 9, 11….. 2, 6, 18, 54, 162….. 2, 4, 12, 48, 240….

Can you write the next term in each sequence? How did you figure it out?

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Arithmetic Sequence

A sequence in which ________________________________________________

___________________________________________________________________

Example: 3, 5, 7, 9, 11…. The COMMON DIFFERENCE is 2. The RULE is x + 2.

Geometric Sequence

A sequence in which_________________________________________________

___________________________________________________________________

Example: 2, 6, 18, 54, 162…. The COMMON RATIO is 3. The RULE is 3x.

What about a pattern like: 2, 4, 12, 48, 240….. ??

Is there a common difference? Is there a common ratio?

_____________________________________________________________

The difference between any two consecutive terms in an arithmetic sequence is

called the ______________ ______________________.

Sometimes the arithmetic sequence looks like the term is subtracted, but it’s really adding a negative.

State whether the sequence –3, –5, –7, –9, –11, … is arithmetic. If it is, state the

common difference and write the next three terms.

–3, –5, –7, –9, –11, _____, _____, _____, …

Arithmetic? ____ Common difference is ______

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Write the rule as an algebraic expression: “The previous term n, increased by 2” ____

State whether the sequence is arithmetic. If it is, find the values of the following:

17, 12, 7, 2, –3, … 4, 1, –2, –5, … –20, –16, –12, …

Common difference: ______ Common difference: _______ Common diff: _____

Next 3 terms: ____, ____, ____ Next 3 terms: ____, ____, ____ Next 3 terms: ____, ____, ____

Ninth term: __________ Eighth term: __________ Seventh term: _________

Rule: __________ Rule: __________ Rule: __________

If the numbers are growing, then the common difference is ___________________.

If the numbers are getting smaller, then the common difference is __________________.

Practice - Sequences

Complete the table……type of sequence, CD or CR, rule and next 3 numbers.

Sequence Type of Sequence

Common Difference

or Common Ratio

Rule Next 3 numbers

17, 12, 7, 2, -3…. 1, 10, 100, 1000, 10000…

1, 1, 2, 6, 24….

125, 25, 5, 1,

….

–2, –6, –18, –54 … 2, 6, 24, 120…

–8, –6, –4, –2…

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Sequence Type of Sequence

Common Difference

or Common Ratio

Rule Next 3 numbers

-20, -16, -12….

5, 5, 10, 30, 120…

-32, -8, -2….

0, 3, 6, 9, 12, . . .

48, 24, 12, 6, 3, . . .

6, 11, 16, 21, 26, . . .

0, 1, 3, 6, 10, . . .

, 1, 3, 9, . . .

30, 26, 22, 18, 14, . . .

–3, –6, –12, –24, . . .

–4, 4, –4, 4, –4, . . .

–5, 10, –20, 40, . . .

1, 2, 1, 2, 1, . . .

448, 224, 112, 56, . . .

35, 28, 21, 14…

1, 3, 9, 27…

2, –4, 8, –16…

121, 1221, 12221 …

, 1, –2, 4, . . .

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State whether the sequence is arithmetic. If it is, find the values of the following:

1) , , , , ... 2) 17, –, –, –, ... 3) –, –, –, –, ...

Common difference: _______ Common difference: _______ Common difference:___

Sixth term: __________ Ninth term: __________ Seventh term: ______

Rule: __________ Rule: __________ Rule: __________ 4) –, –, –, ... 5) , , , ... 6) 5, 3, 1, –1

Common difference: _______ Common difference: _______ Common difference:_____

Next 3 terms: ___, ___, ___ Next 3 terms: ___, ___, ___ Next 3 terms: ___, ___, ___

Fifth term: __________ Eighth term: __________ Seventh term: ______

Rule: __________ Rule: __________ Rule: __________

7. Write in the missing terms and the rule for each arithmetic sequence. Use n to represent the previous term. a. 2, 7, 12, 17, ___, ___, … b. 5, –3, –11 , ___, ___, … c. –11, –15, ___, –23, ___ , … Rule: __________ Rule: __________ Rule: __________

If a new candle is lit and burns at a steady rate, its height in centimeters at the end of each successive hour might produce the following arithmetic sequence: 24, 21, 18, 15, . . . 8. How many centimeters of the candle burn each hour? _____ 9. What is the common difference of this sequence? _____ 10. The first term, 24, represents the height of the candle after it had burned for 1 hour.

How tall was the candle before it was lit? ______ 11. Copy and continue the sequence until you get to the term 0. ______________________ 12. How many hours did the candle last? _____

34

Practice – Cumulative Review

1. Use the Order of Operations to evaluate each expression. Circle your answers.

a. 9 + [5(7 - 8)]2 b. 5(4

2 - 8) ÷ (-10) c. 9 – 12 + 8 ∙ 2

2. Write a numerical or algebraic expression for each phrase.

a. the difference of fifty and two ________________________

b. the quotient of a number and five ________________________

c. the product of a number and ten, decreased by four ________________________

d. four times the sum of a number and ten ________________________

e. twelve decreased by twice a number ________________________

3. Evaluate each expression if x= 15, y= 8, and z= -2.

a. y + (2z -1) b. yz – 2x c. 60 - xz - y d. (x – y) + z2

4. Apply the Distributive Property.

a. -9(x – 7) = ___________________ b. 8(y + 2) = _________________

c. (h + 2)(-3) = ___________________ d. (x – 3)7 = __________________

e. 5(-2 - y) = ____________________ f. -7(-x + 4) = _________________

35

5. Evaluate.

a. -6 + 7 b. -5 (-6) c. 36 ÷ -9 d. -5 – 4

e. 8 (-7) f. -32 ÷ -8 g. -4 + (-5) h. -4 – (-2)

i. (-5)(-9) j. 15 + (-16) k. 9 – (-12) l. -42 ÷ 6

6. Write a variable expression for each sequence.

a. -5, 2, 9, 16, 23, … _____________ b. 15, 7, -1, -9, … ______________

c. 2, -6, 18, -54,… _____________ d. 48, 24, 12, 6, 3, 3/2, … ____________

7.

36

Warm Up

State if the sequence is arithmetic, geometric or neither. Then, state the CD or CR.

1. 11, 14, 19, 26…… 2. 13, 8, 3, -2…. 3. 48, 12, 3,

…. 4. 6, 8, 12, 18….

Notes: Relations and Functions

SOL Objective 7.12: Represent relationships with tables, graphs, rules and words.

A ________________ is any set of ordered pairs such as {(1,2), (2,4), (3,0), (4,5)}.

A Function is ____________________________________________________________

________________________________________________________________________

Examples Non-Examples

{(1,2), (4,2), (6,7)} {(2,3), (2,4), (1,9)}

{(5,7), (-5,4),(0,1)} {(-2,5), (5,4), (-2,-3)}

Relations Some relations are functions. All functions are relations. Functions

37

Determine whether each relation is a function. Explain.

1. {(-3,-3), (-1,-1), (0,0), (-1,1), (3,3)} ____________________________________

2. {(-1,6), (4,2), (2,36), (1,6)} ___________________________________

3. {(5,-4), (-2,3), (5,-1), (2,3)} ___________________________________

4. {(-10,34), (0,-22), (10, -9), (20, 3)} ______________________________________

5. {(-10,-34), (-10,-22), (10,-9), (20,3)} ___________________________________

How can you represent relations?

38

39

40

41

42

43

Practice – Represent Relations and Functions in Tables and Graphs

Graph the function y = x + 4 using any domain values

Which relation represents the equation y = x ?

a) {(0, 6), (1, 7), (2, 8), (3, 9)}

b) {(6, 0), (7, 1), (8, 2), (9, 3)}

c) {(1, 6), (2, 12), (3, 18), (4, 24)}

Which relation is not a function?

a. {(–1, 4), (–2, 5), (–3, 0)}

b. {(–2, 1), (6, 1), (5, 2)}

c. {10, 8), (2, 0), (10, 0)}

d. {(1, 3), (2, 2), (1, 3)}

(-5,-8), (-3,-7), (-1,-6), (0,-5), (1,-5) Function? ______________ Function? ________

Domain ________________ Domain: ____________ Range _________________ Range: _____________

x y

x y

0 -1

2 -5

4 4

2 5

44

A function usually connects input, x, with an output, y, by a rule. Example: Suppose you can buy DVDs for $15 each.

1. Complete the function table

2. If 6 DVDs are purchased, the total cost is $ ______

3. To find the total cost of 6 DVDs

__________________________

4. To find the total cost of 9 DVDs __________________________

Complete each table and graph each function.

1. y = 2x 2. y = –3x 3. y = x – 4

DVDs Cost($)

1 15

2 30

3

4

5

x y

x y

x y

45

Practice: Relations and Functions

46

47

Warm Up

Notes: The RULE (equation) of a function

48

49

50

51

52

53

Multiple Choice. Choose the best answer.

54

55

56

57

Unit 3 STUDY GUIDE

1. What is the sixth term in this arithmetic sequence?

25, 41, 57, 73, …

a) 89 b) 105

c) 121 d) 137

2. What is the common ratio in this geometric sequence?

100, 20, 4,

, …

a) 5 b) 4

c)

d)

3. Daniel read b number of books this winter. Marion read 3 more than twice the number of books Daniel read. Which can be used to represent the number of books Marion read?

a) b)

c) d)

4. If n represents a number in the sequence, -6, -1, 4, 9, … which variable expression could be used to determine the next term in the sequence?

a) b)

c) d)

58

5. Christopher is evaluating the expression x3, for a value for x. If the expression, x3 = 27, which is the replacement value for x?

a) 3 b) 9

c) 24 d) 30

6. Which shows “the sum of twice a number and 4”?

a) b) ( )

c) d) ( )( )

7. Which phrase best represents the following?

a. Seven added to the quotient of a number and five

c. Seven added to the quotient of five and a number

b. The sum of seven and five divided by a number

d. Five divided by a number added to seven

8. If n represents the number in the sequence,

1, -3, 9, -27, 81, …,

Which variable expression could be used to determine the next term in the sequence? a. n – 3 b. n + 3 c. 3n d. -3n

59

9. Which is equivalent to ( ) when ?

a) 2 b) 3

c) 8 d) 9

Word Bank:

Distributive Property Associative Property of Multiplication Commutative Property of Addition Inverse Property of Addition Multiplicative Property of Zero Identity Property of Multiplication

Use the word bank above to answer the questions 10-12.

10. What property makes 3 + 7 + (-7) = 3 + 0 true?

11. What property makes 22 + (5 · 0) · (-5) = 22 + 0 · (-5) true?

12. What property makes (3 x 13) x 19 = 3 x (13 x 19) true?

13. Rewrite ( ) using the distributive property.

14. Rewrite ( ) using the commutative property.

15. Using the variable expression n – 3 and a starting number of 10. Give the sequence for the first six values.

10, ______, ______, ______, ______, ______

60

Translate each of the verbal phrases into variable expressions for questions 16-17.

16. Twice the difference of a number, c, and 5 _______________________________

17. Three less than twice Julia’s age _________________

18. What is the value of the following expression? ( )

19. What is the value of ( ) if a= -3, b=2, and c=4

20. What is the value of the expression, ( )

?

21. Jim is given an expression. ( )

Which expression shows Jim correctly applying the distributive property?

A. ( ) ( ) B. ( ) ( ) C. ( ) D. ( )

22. Write a rule that describes the function below.___________________________

23. Complete the

table.

x 4 5 6 7 8 9

y -24 -30 -36

61

24. Which of these tables represents the rule (equation) y = x + 5 ?

25. Create a table of values for y = –3x + 1.

Graph the function.

x y

-1 -4

0 -5

1 -6

2 -7

x y

-2 3

-1 4

0 5

1 6

x y

0 -5

1 2

2 1

3 4

x y

-2 -3

1 -2

0 -1

1 0

x y

-2

-1

0

1

62

63

Unit 4

Equations & Inequalities

64

Unit 4 - Table of Contents

Day

Pages Topic

1 65 Unit 4 Pretest

1 66-72 Solve One-Step Equations

2 73-78 Solve Two-Step Equations

3 79-83 Solve Equations with Distributive Property

4 84-88 Solve Word Problems Using Equations

5 89-92 Solve One-Step Inequalities

6 93-95 Solve Two-Step Inequalities

7 96-103 Graph Solutions to inequalities

8 104-109 Study Guide

9 Unit 4 TEST

Unit 4 SOL Objectives

7.14a: Solve one and two step linear equations in one variable.

7.14b: Solve practical problems requiring the solution of one and two

step linear equations.

7.15a: Solve one step inequalities in one variable.

7.15b: Graph solutions to inequalities on the number line.

65

Unit 4 Pretest

Solve:

1. 2.

3.

4. x + 8 = 10 5. Model and solve the following equation: –

=

6. At a concert you purchase three t-shirts and a $15 concert program and paid a total of $90. How much was each t-shirt? Write an equation and solve.

7. Graph the solution: x ≤ -2

8. Graph the solution: 8 > x

66

Notes: Solve One Step Equations

SOL Objective 7.14a: Solve one and two step linear equations in one variable.

.

Term Definition Action

Equation

Expression

Inequality

Directions: equations expressions inequalities

( )

67

To solve an equation, you find the value or values that make the statement TRUE.

Directions: In the following problems, you are GIVEN a set of values. Substitute each

possible solution to find the correct value of the variable x.

EQUATION POSSIBLE SOLUTIONS

Substitute each possible solution

Solution (x = ?)

2x + 4 = 16

{-2, 0, 6, 8}

3(x – 4) = 15

{3, 6, 9, 12}

-4x + 15 = -5x

{-15, 5, 10, 15}

4x = 28

{-2, 0, 7}

10 = x - 3

{3, 0, 13}

68

To solve an equation or inequality when we are not given possible solutions, we

must use ________________ operations to ______________ the variable.

In order to keep an equation or inequality balanced, every operation must be

performed to _____________ sides of the equation or inequality.

Addition

Subtraction

Example 1:

Example 2:

Example 3:

Example 4:

Multiplication Division

Example 1:

Example 2:

Example 3:

Example 4:

69

What would we do if the equation looked like:

?

To isolate the variable, multiply both sides of the equation by the _____________ of

.

Write the reciprocal of each fraction:

FRACTION

RECIPROCAL

-5

4

Example 1:

Example 2:

Example 3:

Example 4:

70

Now let’s model equations with pictures.

What is the equation?________________

What is the equation?________________

What is the equation?________________

Draw pictures to represent each equation.

71

Practice: Solve One Step Equations

Show your work on a separate sheet of paper.

72

Practice: Solve One Step Equations

0

73

WARM-UP

Solve each equation. Show your steps.

1. x – 5 = 8 2. -5 =

3.

x = 4 4. - 4 = x + 2

5. Explain in your own words how to solve an equation._________________________

______________________________________________________________________

______________________________________________________________________

NOTES: Solve Two Step Equations

SOL Objective 7.14a: Solve one and two step linear equations in one variable.

Remember: To solve an equation, ___________________ the variable by using

___________________ operations.

Problem x + 5 = 8 7 = y – 6 4x = -12

= -3

Show Your Work

Answer

What Inverse Operation did you use?

74

Now let’s look at TWO STEP Equations.

Example: 2x + 5 = 15

The variable x is on the LEFT side of this equation.

What is the inverse of multiplying by 2?__________________________

What is the inverse of adding 5?________________________________

Which inverse operation MUST we perform FIRST? Discuss with your group.

To solve two step equations, we ____________________________________________

______________________________________________________________________

______________________________________________________________________

Let’s practice solving two step equations. Show all of your steps.

1. 2w + 3 = 9 2. 5p – 8 = 22 3.

+ 15 = 20

4.

= -2 5. -44 = 4x – 8 6. 15 + 2x = 75

75

7.

– 14 = 10 8. -10x + 90 = -50 9. -80 = 10d – 20

10. -81 = 3r – 6 11. -15 =

+ 6 12. 4 – 2x = 20

Write the equation for each model. Use the key:

Equation:____________________

Equation:____________________

Equation:____________________

76

More Practice: Solving Two Step Equations

Draw a model for each equation. Use the same key.

Solve each equation. Show all steps.

1.

2.

3.

4.

77

5. 6. 7.

8.

9. 10. 11.

12.

Directions: Model and solve the following equations.

13. 2x + 7 = 15 14. 4x – 1 = 19

78

15. Which of the following is a solution to:

4y – 11 = -19 A y = 2

B y = -2

C y = 7.5

D y = 2 and y = -2

16. Which of the following is a solution to:

7

x + 23 = 20

A x = 21

B x = -21

C x = 6

D x = 6 and x = 21

17. Which of the following is a solution to:

-10 = -3x – 82 A x = 24 and 12

B x = -216

C x = -12

D x = -24

18. Which of the following equations models the statement: “8 less than 3 times x is 24.”? A

B ( )

C

D

Review: Represent the following in decimal, fraction, and expanded form.

19. 10-2

20. 103 21. 10-3

22. 100 23. 10-5 24. 10-1

79

WARM-UP

1. Model and solve: 2x + 1 = 9 2. Solve:

3. Solve: 4x – 20 = 16 4. Which of the following is a solution to:

A. -36 B. 36 C. 18 D. -18

NOTES: Solve Equations with Distributive Property

SOL Objective 7.14a: Solve one and two step linear equations in one variable.

Review: Solve the two step equations. Show all of your steps.

1. -7x + 12 = -2 2.

+ 12 = 15 3. 8x – 19 = 21

80

4.

+ -3 = -4 5. 3x + -25 = -19 6.

+ -10 = -17

Now, let’s review the Distributive Property.

The Distributive Property: To multiply a number by a sum or difference, multiply each

number inside the parentheses by the number outside the parentheses.

a(b + c) = ab + ac OR a(b – c) = ab – ac

Let’s Practice: 1. -4(x – 9) = ____________ 2. 35(x + 10) = _____________

3. (y + 7) 5 = ____________ 4. 4(2x + 3y) = _____________

5. c (a+ 3d) = ____________ 6. -2(y + 4) = _____________

Now, solve equations with the Distributive Property.

1. 4(x – 3) = 4 2. 3(a – 5) = 18 3. 32 = 4(x + 9)

4. 3(g – 3) = 6 5. 4(k + 1) = 16 6. 2(n – 5) = 8

81

Practice: Solve Equations with the Distributive Property

1. 2(a + 3) = -12 2. 4(2r + 8) = 88 3. 3(p + 2) = 18

4. 2(3a + 2) = -8 5. 5 = 5(y – 2) 6. –(3x – 12) = 48

7. -2(x -3) = 30 8. 4 = 4(x + 3) 9.

(4x – 6) = 11

For Problem #3…. Check your work. Describe the steps you used to solve it.

_______________________________________

_______________________________________

_______________________________________

82

Practice: Solve Two Step Equations

83

84

WARM UP: Solve and Check your answers.

2. -3(x + 1) = -12

3. -x + 7 = 9 4.

NOTES: Solve Word Problems Using Equations

SOL Objective 7.14b: Solve practical problems requiring the solution of one and two step linear equations.

85

1. Ken is thinking of a number. Nine more than the product of 4 and the number is 73. Find Ken’s number.

2. Barbie is thinking of a number. Twenty less than one third of the number is 72. Find Barbie’s number.

3. The length of a rectangular field is 75 yards. This is 3 yards more than twice the width. How wide is the field?

4. Grandpa Gump is 63 years old. His age is 2 years less than 5 times the age of Billy Gump. How old is Billy?

5. Zoe weighs 92 pounds. Her weight is 6 pounds more than half of her father’s weight. How much does her father weigh?

6. A banana has 85 calories. This is 10 calories less than one eighth of the calories in a banana split. How many calories are in a banana split?

86

7. The Space Club is having some posters printed. The printer charges $250 plus $2 per poster. How many posters can be printed for $1000?

8. Pizzazz Publications is having some books printed. The printer charges $800 plus $5 per book. How many books can be printed for $4000?

9. Mr. Lock’s car broke down on the turnpike. Acme Towing charged $30 plus $3 per mile to tow the car. If Mr. Lock paid $117, how far was the car towed?

10. Rolex worked 40 hours last week. He had $74 deducted from his earnings for taxes. If he had $286 left after the deduction, how much does Rolex earn per hour?

11. A table and 8 chairs together weigh 97 pounds. If the table weighs 25 pounds, how much does each chair weigh?

12. Three desks and a bookcase together weigh 157 pounds. The bookcase weighs 34 pounds. How much does each desk weigh?

87

Practice: Solving Word Problems with Equations

1. Angelica sets aside $1000 in her annual budget to pay for her gym fees. If

her annual membership fee is $720 and personal trainers are available for

$35 an hour, how many hours can she spend with a personal trainer?

2. A telephone company advertises long distance service for $.07 per minute

plus a monthly fee of $3.95. If your bill one month was $12.63, find the

number of minutes you used making long distance calls.

3. You return a book that is 5 days overdue. Including a previous unpaid

overdue balance of $1.30, your new balance is $2.05. Find the fine for a

book that is one day overdue.

4. A furniture rental store charges a down-payment of $100 and $75 per

month for a table. Steve paid $550 to rent the table. For how many

months did he rent the table?

88

5. At work, Jack must stuff 1000 envelopes with advertisements. He can stuff

12 envelopes in one minute, and he has 112 envelopes already finished.

How many minutes will it take Jack to complete the task?

6. For Jillian’s cough, her doctor says that she should take eight tablets the

first day and then four tablets each day until her prescription runs out.

There are 36 tablets. Find the number of days she will take four tablets.

7. Twice a number is 60 more than five times the number. What is the number?

8. The fastest speed recorded for a cheetah is 70 mph. This is 11 mph less than 3 times the fastest running speed for a man. What is the fastest running speed for a man?

89

WARM-UP

1. The sum of seven times a number and twelve is 62.Find the number.

2. How old am I if 400 reduced by 2 times my age is 244?

3. The Empire State Building is 1250 feet tall. This is 140 feet more than twice the height of the Washington Monument. How tall is the Washington Monument?

4. Jack said, “Five times my age in 2 years is 100.” How old is Jack now?

NOTES: Solve One Step Inequalities

SOL Objective 7.15a: Solve one step inequalities in one variable.

Suppose we have the inequality 10 > 2

Add 2 to both sides. Still true?____________________________________

Subtract 2 on both sides. Still true?_________________________________

Multiply by 2 on both sides. Still true?_______________________________

Multiply by -2 on both sides. Still true?_______________________________

Divide by 2 on both sides. Still true?_________________________________

Divide by -2 on both sides. Still true?________________________________

Talk in your group about the results above.

90

Example #1: Divide by a negative Example #2: Multiply by a negative

-7x > 14

< 8

<

Reverse the symbol! -3 ∙

> 8 ∙ -3 Reverse the symbol!

x < -2 The answer! x > -24 The answer!

Let’s solve these inequalities!

-5x ≥ 125

> 24

t < 4

14 > -2x -4 ≤

> 2

91

Should you reverse the inequality symbol in these next problems? Explain.

> -5 -18 < 3x

Circle the numbers that satisfy each inequality.

1. x < 8 6 7 8 9 10 2. x + 1 ≥ 5 2 3 4 5 6

State whether the inequality is true or false. Show your work.

3. y – 7 < 20 (y = 28) 4. 12 ≤ 2y – 6 (y = 9)

Evaluate the expression if x = 3, y = 5 and z = 2. Then write <, > or = in the box.

5. 10 – xz y 6. yz 2x 7. 3x – y 2z

92

Practice: Solve One Step Inequalities

1. a + 7 < 21 2. c + 10 < 9 3. 5 + x ≤ 18

4. 10 + n ≥ -2 5. -4 < k + 6 6. 3 < y + 8

7. r – 9 ≤ 7 8. g – 4 ≥ 13 9. -2 < b – 6

10. 5x < 15 11. 9n ≤ 45 12. 14k ≥ -84

13. -12 > 3x 14. -100 ≤ 50y 15. 2y < -22

16. -4w ≥ 20 17. -3r > 9 18. -72 < -12h

93

WARM-UP

1. 2x > -6 2. 10 ≤ 5y 3.

≤ 10

4. -3 ≥

5. -7x < 14 6.

< -5

NOTES: Solve Two Step Inequalities

Let’s practice solving two step inequalities.

94

1. 3y – 1 ≤ 5 2. 3 + 4c > -13 3. 22 ≥ -3x - 2

4. 4 – 3x ≤ 19 5.

– 5 < 6 6.

+ 3 ≥ -11

7. 3y + 2 < -7 8.

– 6 ≤ 3 9. 7 +

< 4

95

PRACTICE: Solving Two Step Inequalities

1. 3x – 17 < 19 2.

– 6 ≥ -2 3. 4x – 15 ≤ 17

4. -12y + 10 ≤ -14 5. 4x – 5 ≥ 27 6. 4y + 7 < -5

7. 2x – 3 > 19 8.

+ 15 < 21 9. 6 ≥

+ 1

96

Warm Up

Solve:

1. -4x + 2 < 10 2. 8 ≤ 4 – 2y 3.

+ 3 > -4

NOTES: Graph Inequalities

SOL Objective 7.15b: Graph solutions to inequalities on the number line.

Fill in each blank. Follow the example.

Example: x < -8 is the same as -8 > x

1. x < 10 is the same as __________. 5. z ≥ 9 is the same as __________.

2. y ≥ 2 is the same as ___________. 6. x < -7 is the same as __________.

3. -4 > y is the same as ___________. 7. 12 > x is the same as _________.

4. 10 ≤ x is the same as ___________. 8. -17 ≤ t is the same as _________.

97

Inequality

Read the inequality

starting with the variable

Graph the inequality

Write three solutions

to the inequality

b ≥ 4

“b is greater than or equal to 4”

4

4, 5, 6

-2 ≥ x

“x is less than or equal to -2”

-2

-2, -3, -4

m ≤ 0

10 > n

x ≥ -16

100 < y

x ≥ -12

98

Inequality

Read the inequality

starting with the variable

Graph the inequality

Write three solutions

to the inequality

-4

“x is greater than 2”

10

“x is less than or equal to -8”

Complete the chart.

Inequality Read the inequality starting with the

variable

Graph the inequality Write three solutions to the inequality

x ≤ -8

x is greater than 2

7

99

Practice: Graphing Inequalities

Fill in each blank. Follow the example.

Example: x > -6 is the same as -6 < x

1. y < 8 is the same as ___________. 2. -4 ≥ x is the same as __________.

3. x ≤ 0 is the same as ___________. 4. 10 > y is the same as __________.

Write an inequality for each sentence.

5. y is greater than or equal to -4. ________________________

6. The sum of y and 8 is less than 12. ________________________

7. At least 15 people went to the restaurant. ________________________

8. At most 10 people went to the movies. ________________________

Circle the numbers that satisfy each inequality.

9. x < -2 -4 -3 -2 -1 0

10. 8 ≥ y 6 7 8 9 10

11. 12 ≤ 2x 4 5 6 7 8

State whether each inequality is true or false for the given value. Show your

work.

12. 15 + n ≥ 15 (n = 6) 13. 2 ≥ x – 3 (x = 3)

100

14. 4y – 4 < 20 (y = 7) 15. 29 < 24 + a (a = 6)

Graph each inequality on the number line.

16. x < -2 17. y ≥ 12

Write the inequality for each graph.

18. 19.

6 -5

Answer:___________________ Answer:__________________

Evaluate the expression if a = 2, b = 4, and c = 6. Then write <, > or = in the box.

20. ac ab 21. 5 – a bc 22. 10 – c ab - 4

101

Practice: Inequalities

Write an inequality for each sentence.

1. x is less than 10. ______________

2. At least 295 students attend Smith School. ______________

3. 20 is greater than or equal to y. ______________

4. A bill increased by $15 is more than $80. ______________

5. 14 is greater than a. ______________

6. The product of 8 and a number is less than 15. ______________

7. b is less than or equal to 8. ______________

8. Citizens who are 18 or older can vote. ______________

9. 6 is less than the product of f and 20. _____________

10. 80 runners at most showed up for the race. ______________

11. The sum of t and 9 is greater than or equal to 36. ______________

12. More than 3400 people attended the concert. ______________

13. Her earnings at $11 per hour were no more than $121. ______________

14. A savings account increased by $70 is now more than $400. ______________

102

Circle the numbers that satisfy each inequality.

1. r > 10 5 10 15 20

2. t ≥ 10 5 10 15 20

3. 2 + n < 5 0 1 2 3 4

4. 6 + m ≤ 10 3 4 5 6 7

5. 30 ≥ 5d 4 5 6 7 8

6. 30 ≤ 5d 4 5 6 7 8

State whether each inequality is true or false for the given value. Show your

work.

7. b + 10 < 12 (b = 4) 8. 3 < x – 8 (x = 12)

9. 6m + 3 ≤ 8 (m = 1) 10. 12 ≤ 2p – 6 (p = 9)

11. k – 12 < 18 (k = 31) 12. 13 > 4 + c (c = 9)

103

Graph each inequality on a number line.

13. a < -2 14. x > -6

15. d ≥ 7 16. x ≤ -5

Write the inequality for each graph

17. 18.

12 -3

Evaluate the expression if a = 2, b = 4, and c = 6. Then write <, > or = in

the box.

19. bc ac 20. c + 6 3a + 2c 21. 5b – 2a 4b

104

Unit 4 Equations and Inequalities STUDY GUIDE

1. Solve each equation. Show your work. Circle your answers.

a.

= 8 b.

- 2 = 11 c. -28 = -4w

d. -5x + 18 = 28 e. 6 +

= 0 f. 17 = - 4 – 7x

g. 11x – 31 = 24 h. 3(x – 6) = 18 i. x – 7 = -12 2. Determine whether the given x-value is a solution to the equation. Explain. a. 6x + 2 = 14 (x = 2) b. x – 8 = -13 (x = 5)

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3. Use the key to model the following equations. Key: x = 1 = + -1 = –

a. 2x + 4 = 9 b. 2x – 5 = 3

= =

c. y + 4 = 6 d. y + (-2) = -1

= =

4. Write and solve an equation for each word problem. Show your steps. Circle your

answers. a. Five more than twice a number is 27. b. The product of a number and three, What is the number? increased by 5, is 32. What is the number? c. Mandy bought a DVD player. The sales clerk says that if she pays $80 now, her monthly payments will be $32. The total cost will be $400. How many months will she make payments? d. Tom and his two friends want to evenly share the bill at their favorite restaurant. If each person paid $14, what was the total bill?

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e. Shari had $30 last week. She babysat last night and earned some money. She now has $76. How much did Shari earn from babysitting?

5. Write an inequality for each situation.

a. A number increased by 7 is more than 19. _____________

b. Twice a number is at least 12. _____________

c. Sharon’s earnings at $12 per hour were less than $150. _____________

d. If 8 times an integer is decreased by 12, the result is less than 44._____________

e. The class must raise at least $125. _____________

f. The bus holds no more than 65 students. _____________

g. John’s earnings at $11 per hour were less than $80. _____________

h. It takes no less than three years to complete law school. _____________

i. Fewer than 60 points were scored. _____________

j. A number is greater than or equal to 20. _____________

6. Write two equivalent inequalities for each phrase. The first one is done for you.

a. A number less than 10. x < 10 10 > x

b. A number greater than -7. ______ ______

c. A number greater than or equal to 20. ______ ______

d. A number less than or equal to 5. ______ ______

7. For the given value, state whether the inequality is true or false. Show your work.

a. 5p + 7 ≥ 25 (p = 5) b. 18 – x > 4 (x = 12) c. b + 10 < 12 (b = 4)

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8. Evaluate each expression if a =2, b=4, and c=6. Then write <, >, or = in the

blank.

a. bc _____ ac b. c + 6 ____ 3a + 2c c. 5b – 2a ____ 4b

9. Complete the chart.

Inequality Graph Three solutions to the inequality

n ≥ 3

4

-1 < y

-3

x ≤ 5

0

9 > x

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10. Solve each inequality and graph on a number line. Show your steps, and circle your answers.

a. p + 5 < -6 b. -35 > 7y c.

≤ -3

d.

≥ 5 e. 2a – 8 < -24 f. x – 12 ≤ -5

g. 3 < y + 8 h. -4w ≥ 20 i.

+ 8 < 1

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11. Write an inequality for each word problem, and then solve the inequality.

a. Chris is saving money for a ski trip. He has $62.50, but his goal is to save at least

$100. What is the least amount Chris needs to have to reach his goal?

b. The difference between a number and eleven is less than or equal to 8.

c. Julia delivers pizzas on weekends. Her average tip is $1.50 for each pizza that she

delivers. How many pizzas must she deliver to earn at least $20 in tips?

d. The quotient of a number and −6 is at most five.

e. Six less than four times a number is greater than two times the same number

plus 8.

f. Robert makes $3.50 per hour working at a convenience store. If he gets a bonus

of $25 this week, how many hours must he work to make at least $165?