math 7 workbook...f. additive identity h. 14 + (13 7) = (14 + 13) 7 g. additive inverse j. 14(13 +...
TRANSCRIPT
1
MATH 7 Workbook
Unit 3: Number Relationships
Unit 4: Equations and Inequalities
Name:_______________________________
Block:_____
Teacher:__________________________
2
4
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.
5
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
6
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) ____________
7
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)
8
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 = ____
9
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 ____________________
10
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 _______________________________
11
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) = ___________________
13
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
14
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) = _______________________
15
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:_______________________
16
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
17
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___________________________________________________________________
18
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 ________________
19
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 ______________
20
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 = _______
22
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
23
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
24
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.
25
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.
26
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
27
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
28
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) = ________________
29
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?
30
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 ______
31
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…
32
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, . . .
33
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?