Download Patterns and Expressions

1-1
Patterns and Expressions
Vocabulary
Review
Write the correct word to continue each pattern.
Thursday
1. Monday, Wednesday, Friday, Sunday, Tuesday, c
April
2. January, April, July, October, January, c
red
3. red, blue, red, yellow, red, blue, c
triangle
4. circle, square, triangle, circle, square, c
Write the next number in each pattern.
6. 6, 3, 0, 23, 26
5. 2, 4, 6, 8, 10
7. 2, 8, 32, 128, 512
variable (noun)
VERH
x and y
are often used as
variables
ee uh bul
Related Words: vary
Main Idea: A variable is usually a letter that can change or vary.
Definition: A variable is a symbol that can represent one or more numbers.
Math Usage: A variable represents an unknown number in equations
and inequalities.
Use Your Vocabulary
8. Write N if the expression is a numerical expression. Write A if the expression is an
algebraic expression.
5x A
5
312 N
92z?5 A
6
17
r
A
9. Circle the variables in each algebraic expression below.
32x
Chapter 1
4w 1 d
8?v
k 1 2q 2 7
2
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Vocabulary Builder
Problem 1 Identifying a Pattern
Got It? Look at the figures from left to right. What is the pattern?
Draw the next figure in the pattern.
10. There are 4 yellow squares in the first figure.
11. There are 4 more yellow squares in the second figure.
12. Describe the pattern. Answers may vary. Sample:
Each figure increases by 4 yellow squares. One square is added to
_______________________________________________________________________
each end of the shape formed by the existing yellow squares.
_______________________________________________________________________
13. Draw the next figure in the pattern.
Problem 2 Expressing a Pattern with Algebra
Use a table of values with a process column.
14. Complete the table of values.
Number of Tiles
(Output)
1
1 (2 ∙ 1 1)
4
2
1 +2 ∙ 2 1,
6
3
1 +2 ∙ 3 1,
8
4
1 +2 ∙ 4 1,
10
…
Process Column
…
Figure Number
(Input)
…
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Got It? How many tiles are in the 25th figure in this pattern?
n
1 2n 1
2n 2
15. There are 25 Q 2 R 1 2, or 52 , tiles in the 25th figure of this pattern.
3
Lesson 1-1
16. Explain how the table of values helps you find the number of tiles in the
25th figure.
Answers may vary. Sample: It helps me see a pattern to write an
_______________________________________________________________________
expression for finding the number of tiles in any figure in the pattern.
_______________________________________________________________________
_______________________________________________________________________
Problem 3
Using a Graph
Got It? The graph shows the total cost of platys at the aquarium shop.
Cost of Platys
17. Circle the description of the input value. Underline the description of the
output value.
number of platys
process
total cost
18. Complete the table of values.
4
2
0
Number of Platys
Total Cost
(Input)
Process Column
1
1∙2
2
2
2 ∙2
4
0
2
4
6
Number of Platys
(Output)
6
6 ∙ 2
12
…
6
…
3∙ 2
…
3
n
n ∙ 2
2 n
Underline a number or expression to complete each sentence.
19. In the table, when the input value is 6, the output value is 2 / 6 / 12 .
20. The expression n / 2n / 3n describes the cost of 1 / 2 / n platys.
21. At the aquarium shop, six platys cost $ 12 .
Chapter 1
6
4
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cost of platys
Total Cost ($)
Use a table to find the cost of six platys.
Lesson Check • Do you know HOW?
Make a table to represent each pattern. Use a process column.
Process
Column
Output
Input
Process
Column
Output
1
2(1)
2
1
3(1)
3
2
2(2)
4
2
3(2)
6
3
2(3)
6
3
3(3)
9
4
2(4)
8
4
3(4)
12
…
…
…
2(n)
2n
n
3(n)
3n
…
Input
…
23.
…
22. 2, 4, 6, 8
n
Lesson Check • Do you UNDERSTAND?
Error Analysis Your friend looks for a pattern in the
table at the right and claims that the output equals
the input divided by 2. Is your friend correct? Explain.
Input
3
6.8
8
10
25
Output
2
3.4
4
5
12.5
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24. Circle the rule that applies to most of the input
and output values.
Output 5 Input 4 3
Output 5 Input ? 2
Output 5 Input 4 2
25. Does the input value of 3 follow this rule?
Yes / No
26. Does the table represent a pattern? Explain. Answers may vary. Sample:
No. The input value of 3 corresponds to the output value of 2, which
_______________________________________________________________________
does not follow the same rule as the other input and output values.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
variable
algebraic expression
Rate how well you can determine patterns.
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Lesson 1-1
Properties of
Real Numbers
1-2
Vocabulary
Review
1. Circle the pairs of numbers that are opposites.
7 and 17
24 and 4
5 and 25
1.2 and 2.1
4 and u 4 u
2. Write the opposite of each number.
u 23 u
1
23
21
217
17
1
2
!16
0
212
24
0
Vocabulary Builder
Real Numbers
real numbers (noun) reel NUM burz
1
, 8, 0,
3 6
1.55, and 4
Definition: The set of real numbers includes all rational and
irrational numbers.
Related Words: natural numbers, whole numbers, integers, rational numbers,
irrational numbers
Use Your Vocabulary
3. Circle the real numbers.
23
!5
!21
0
7
8
!21
0
7
8
4. Circle the real numbers that are integers.
23
!5
5. How does the set of real numbers differ from the set of integers?
Answers may vary. Sample: The set of real numbers includes
_______________________________________________________________________
rational and irrational numbers.
_______________________________________________________________________
Chapter 1
6
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5,
Problem 1 Classifying a Variable
Got It? Your school is sponsoring a charity race. If each participant made a
donation d of $15.50 to a local charity, which subset of real numbers best describes
the amount of money raised?
6. Circle the values below that could represent the number of participants making
a donation.
!5
3
8
0
2.7
10
150
7. Circle the set of numbers that best describes the numbers you chose in Exercise 6.
counting numbers
rational numbers
real numbers
whole numbers
Problem 2 Graphing Numbers on the Number Line
Got It? What is the graph of the numbers !3, 1.4, and 13 ?
8. Reasoning Why should your first step be to write all of the numbers as decimals?
Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect.
✓
It is easier to compare numbers when they are all in the same form.
✗
There are fewer decimals than radicals or fractions.
✓
Radicals cannot be written as fractions.
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9. Write each number as a decimal. Round to the nearest hundredth.
!3 <
1.4 <
1.73
1
3<
1.44
0.33
10. Graph the numbers !3, 1.4, and 13 on the number line.
63
3
2
1
0 1
3
1
1.4
2
3
Problem 3 Ordering Real Numbers
Got It? How do !26 and 6.25 compare? Use S or R.
11. Find a perfect square close to 26.
5 2 5 25
12. Find a square root of a perfect square close to 6.25.
" 36
5
6
7
Lesson 1-2
13. Use , or . twice to compare your answers to Exercises 11 and 12.
"
25
"
R
36
R 6.25
14. Now write . or , to compare !26 and 6.25.
!26 R 6.25
Key Concept Properties of Real Numbers
Let a, b, and c represent real numbers.
Property
Addition
Multiplication
Closure
a b is a real number.
ab is a real number.
Commutative
abba
ab ba
Associative
(a b) c a (b c)
(ab)c a(bc)
Identity
a 0 a, 0 a a
0 is the additive identity
a ∙ 1 1, 1 ∙ a a
1 is the multiplicative identity
Inverse
a (a) 0
1 1, a 0
a ∙
a
a(b c) ab bc
Distributive
Column A
Column B
2(3x 1 7) 5 (3x 1 7)(2)
Inverse Property of Addition
(x 1 y) 1 5 5 x 1 (y 1 5)
Commutative Property of Multiplication
x105x
Associative Property of Addition
x?15x
Distributive Property
8 1 (28) 5 0
Identity Property of Multiplication
5(2x 1 1) 5 5 ? 2x 1 5 ? 1
Identity Property of Addition
Problem 4 Identifying Properties of Real Numbers
Got It? Which property does the equation 3(g 1 h) 1 2g 5 (3g 1 3h) 1 2g
illustrate?
16. Circle the operations used in the equation.
division
addition
multiplication
subtraction
17. Circle the algebraic equation that models 3(g 1 h) 5 3g 1 3h.
hg 5 gh
h1g5g1h
a(g 1 h) 5 ag 1 ah
a 1 (g 2 h) 5 (g 2 h) 1 a
18. The equation 3(g 2 h) 1 2g 5 3g 2 3h 1 2g illustrates the 9 Property.
Chapter 1
8
Distributive
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15. Draw a line from the expression in Column A to the property that it illustrates
in Column B.
Lesson Check • Do you UNDERSTAND?
Reasoning There are grouping symbols in the equation (5 1 w) 1 8 5
(w 1 5) 1 8, but it does not illustrate the Associative Property of Addition. Explain.
19. Circle the equation below that illustrates the Associative Property of Addition.
a1b1c5b1a1c
a 1 (b 1 c) 5 ab 1 ac
(a 1 b) 1 c 5 a 1 (b 1 c)
20. The Associative Property of Addition is a rule about the order / grouping of the
terms of an addition statement.
21. Write the terms grouped on each side of the equation.
Left side: 5 and w
Right side: 5 and w
22. Now explain why (5 1 w) 1 8 5 (w 1 5) 1 8 does NOT illustrate the Associative
Property of Addition.
Answers may vary. Sample: Although there are grouping symbols in
_______________________________________________________________________
the equation, the terms being grouped do not change as required by
_______________________________________________________________________
the Associative Property.
_______________________________________________________________________
_______________________________________________________________________
23. Circle the property that the equation illustrates.
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Closure Property of Addition
Commutative Property of Addition
Inverse Property of Addition
Distributive Property
Math Success
Check off the vocabulary words that you understand.
additive inverse
multiplicative inverse
opposite
reciprocal
squares
Rate how well you understand the properties of real numbers.
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Lesson 1-2
1-3
Algebraic Expressions
Vocabulary
Review
Simplify each numerical expression.
1. 6(5 2 2) 1 7 5 25
2.
7 1 5 ? (23)
5 22
4
3. (25)2 2 (4)2 5
9
Vocabulary Builder
evaluate (noun) ee VAL you ayt
Definition: To evaluate an expression means to substitute a number for each
variable and then simplify to get a value.
Use Your Vocabulary
Evaluate each expression for the given values of the variables.
b
4. a 1 4 for a 5 22 and b 5 8
5. x 2 x2y for x 5 3 and y 5 24
0
39
Key Concept Properties for Simplifying Algebraic Expressions
6. Draw a line from each property in Column A to an algebraic example of the
property in Column B. Let a, b, and c represent real numbers.
Column A
Column B
Distributive Property for Subtraction
2(a 1 b) 5 2a 1 (2b) 5 2a 2 b
Multiplication by 0
2(2a) 5 a
Multiplication by 21
21 ? a 5 2a
Opposite of a Sum
a(b 2 c) 5 ab 2 ac
Opposite of a Difference
0?a50
Opposite of an Opposite
2(a 2 b) 5 2a 1 b 5 b 2 a
Chapter 1
10
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Example: To evaluate xy for x 5 2 and y 5 3, substitute 2 for x and 3 for y;
xy 5 (2)(3) 5 6.
Problem 1 Modeling Words With Algebraic Expressions
Got It? Which algebraic expression models the word phrase two times the sum of
a and b?
7. The word “times” means you should use multiplication / addition / division .
8. The word “sum” means you should use multiplication / addition / division .
9. Now write the expression.
Answers may vary. Sample: 2(a 1 b) or 2 ? (a 1 b)
Complete each numerical or algebraic expression by writing a letter, number or
operation sign in each box.
10. The difference of 7 and 4.
7 2 4
11. The product of 3 and x.
3 ? x
12. The number y increased by 2.
y 1
13. The quotient of 48 and 3.
48 4 3
2
14. The number t is doubled then decreased by 1.
2 t 2 1
15. Five taken away from q.
q
2
5
Problem 2 Modeling a Situation
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Got It? You had $150, but you are spending $2 each day. What algebraic expression
models this situation?
16. Define the variable. Let d 5
the number of days
.
17. Complete the model to write the algebraic expression.
Relate
Write
Starting
the amount
amount spent per day
150
2
∙
the number
of days
∙
d
18. Now write the expression.
150 2 2d
Problem 3
Evaluating Algebraic Expressions
Got It? What is the value of the expression
2(x2 2 y2)
3
for x 5 6 and y 5 23?
19. Substitute the given values for each variable into the expression.
2Q 6
2(x 2 2 y 2)
5
3
2
2
2 23 R
3
11
Lesson 1-3
20. Now simplify the numerical expression.
Answers may vary. Sample:
Problem 4
2(36 2 9)
3
5
2(27)
3
5 2(9) 5 18
Writing and Evaluating an Expression
Got It? In basketball, teams can score by making two-point shots, three-point
shots, and one-point free throws. What algebraic expression models the total number
of points that a basketball team scores in a game? If a team makes 10 two-point shots,
5 three-point shots, and 7 free throws, how many points does it score in all?
21. Define the variables.
Let t 5 the number of two-point shots,
Let h 5
the number of three-point shots
, and
Let f 5
the number of free throws
.
22. Complete the expression for the total number of points a team can score in
one game.
2 t
13 h
1
f
2t 1 3h 1 f 5 2(10) 1 5(3) 1 7
5 20 1 15 1 7
5 42
24. The team scored 42 points.
The expression 5ax 1 6y 2 7 has three terms: 5ax , 6y , and 27.
The coefficient is the numerical factor of a term: 5, 6
The constant term is the term with no variables: 27.
Identify the coefficients and the constant term in each expression.
25. 2x2 2 3x 1 5
26. 24yx 1 8x 2 3
Coefficients: 2 and 23
Coefficients: 24 and 8
Constant: 5
Constant: 23
Chapter 1
12
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23. Evaluate the expression for t 5 10, h 5 5, and f 5 7.
Problem 5 Simplifying Algebraic Expressions
Got It? Combine like terms. What is a simpler form of the expression
24j 2 2 7k 1 5j 1 j 2 ?
At the right is one student’s solution.
Rose’s Solution
–4j2 – 7k + 5j + j2 = –3j2 – 7k + 5j
= –3j2 – 2kj
27. What error did Rose make?
Explanations may vary. Sample: The terms 27k and 5j have different
_______________________________________________________________________
variables, so they cannot be combined.
_______________________________________________________________________
28. Simplify the expression correctly.
24j2 2 7k 1 5j 1 j2 5 (24j2 1 j2) 2 7k 1 5j
5 3j2 2 7k 1 5j
Lesson Check • Do you UNDERSTAND?
Compare and Contrast How are algebraic expressions and numerical expressions
alike? How are they different? Include examples to justify your reasoning.
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29. How is an algebraic expression different from a numerical expression?
Answers may vary. Sample: An algebraic expression contains one or
_______________________________________________________________________
more variables, while a numeric expression contains only numbers.
_______________________________________________________________________
30. Put an N next to each numerical expression. Put an A next to each algebraic expression.
3x 1 2 A
6?4 N
a27 A
4 ? 1 1 10 N
5g
A
3 1h
Math Success
Check off the vocabulary words that you understand.
term
evaluate
coefficient
constant
terms
like terms
Rate how well you can write and evaluate algebraic expressions.
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Lesson 1-3
Solving Equations
1-4
Vocabulary
Review
Circle the equations.
2(3x 1 7) 5 10
3x 1 14
3x 1 2y 5 6
4x 1 (y 1 5)
Vocabulary Builder
x22x
inverse operations (noun)
IN
vurs ahp uh RAY shunz
inverse operations
Related Words: opposite, reverse
Main Idea: Inverse operations undo each other.
Use Your Vocabulary
1. Write each inverse operation.
Add 7.
Subtract 4.
Subtract 7.
Divide by 5.
Add 4.
Multiply by 5.
Key Concepts Properties of Equality
Assume a, b, and c represent real numbers.
Property
Definition
Example
Reflexive
aa
55
Symmetric
If a b, then b a.
(4)(2) 8, so 8 (4)(2)
If a b and b c, then a c.
6 (2)(3) and (2)(3) (3)(2),
so 6 (3)(2)
If a b, then you can replace
a with b and vice versa.
If a b and 9 a 15,
then 9 b 15
Transitive
Substitution
Chapter 1
14
Multiply by 3.
Divide by 3.
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Example: Addition and subtraction are inverse operations.
Draw a line from each example in Column A to the property that it illustrates
in Column B.
Column A
Column B
2. a 5 4 and a 1 b 5 5, so 4 1 b 5 5
Reflexive Property
3. 2 1 (x 1 8) 5 2 1 (x 1 8)
Symmetric Property
4. x 1 y 5 z and z 5 4, so x 1 y 5 4
Transitive Property
5. y 1 12 5 9, so 9 5 y 1 12
Substitution Property
Key Concepts Properties of Equality, Continued
Assume a, b, and c represent real numbers.
Property
Definition
Example
Addition
If a b, then a c b c.
If x 12, then x 3 12 3.
Subtraction
If a b, then a c b c.
If x 12, then x 3 12 3.
Multiplication
If a b, then a ∙ c b ∙ c.
If x 12, then x ∙ 3 12 ∙ 3.
Division
If a b, then a c b c
(with c 0).
If x 12, then x 3 12 3.
Write the Property of Equality that justifies each statement.
Subtraction
Property of Equality
Division
Property of Equality
8. If 220 5 5 then 220 ? 5 5 5 ? 5.
Multiplication
Property of Equality
9. If y 2 3 5 212 then y 2 3 1 3 5 212 1 3.
Addition
Property of Equality
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6. If x 1 2 5 5, then x 1 2 2 2 5 5 2 2.
4x
7. If 4x 5 12, then 4 5 12
4.
x
x
Problem 1 Solving a One-Step Equation
Got It? What is the solution of 12b 5 18?
10. Circle the multiplicative inverse of 12.
1
12
212
1
212
1
11. Use the multiplicative inverse to solve the equation and check your solution.
12 Q 32 R 0 18
12b 5 18
1
1
12b ? 12
5 18 ? 12
18 5 18
3
b 5 18
12 5 2
15
Lesson 1-4
Problem 2
Solving a Multi-Step Equation
Got It? What is the solution of 3(2x 2 1) 2 2(3x 1 4) 5 11x?
Division Property of Equality
12. The equation has been solved below. Use one of the reasons
in the box to justify each step.
Combine like terms.
Simplify.
Distributive Property
3(2x 2 1) 2 2(3x 1 4) 5 11x
6x 2 3 2 6x 2 8 5 11x
211 5 11x
11x
211
11 5 11
21 5 x
Problem 3
Distributive Property
Combine like terms.
Division Property of Equality.
Simplify.
Using Properties of Equations to Solve Problems
Got It? The carpet at the right has perimeter 320 feet. What are the dimensions
x
of the carpet?
13. Let x 5 the width of the carpet. Then 3x 5 length of the carpet.
14. Use the formula for perimeter of a rectangle, P 5 2w 1 2/. Write an equation for
the perimeter of the carpet.
3x
15. Solve for x.
2x 1 2(3x) 5 320
8x 5 320
8x
8
5 320
8
x 5 40
16. The width of the carpet is 40 ft, and the length of the carpet is 120 ft.
Problem 4
Equations With No Solutions and Identities
Got It? Is the equation 7x 1 6 2 4x 5 12 1 3x 2 8 always, sometimes, or never true?
17. Combine like terms on each side of the equation.
3x 1 6 5 3x 1 4
18. Now solve the equation.
3x 1 6 5 3x 1 4
0 5 22 or 2 5 0
19. The equation is always / sometimes / never true.
Chapter 1
16
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2x 1 2(3x) 5 320
Problem 5 Solving a Literal Equation
Got It? The equation S 5 3F 2 24 relates shoe size S and length of a foot in inches F.
What is F in terms of S?
20. What two operations will you undo?
subtraction and multiplication
_______________________________________________________________________
21. Now solve the equation for F.
S 5 3F 2 24
S 1 24 5 3F 2 24 1 24
S 1 24 5 3F
S 1 24
3
5F
Lesson Check • Do you UNDERSTAND?
Reasoning Suppose you solve an equation and find that your school needs
4.3 buses for a class trip. Explain how to interpret this solution.
22. Why does the solution 4.3 not make sense as a solution to this problem?
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Answers may vary. Sample: There can only be a whole number of
_______________________________________________________________________
buses. There cannot be a fraction number of buses.
_______________________________________________________________________
23. What is the minimum number of buses that are needed for the trip? Explain.
Answers may vary. Sample: A minimum of 5 buses are needed. 4.3 is
_______________________________________________________________________
greater than 4, so 4 buses are not enough.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
equation
solution
inverse operations
Rate how well you can solve equations.
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Lesson 1-4
1-5
Solving Inequalities
Vocabulary
Review
1. Write I if the math sentence is an inequality. Write E if it is an equation.
I 15 . 32x
I
45 , 46
I
E 17x 5 34
227 2 3x
Complete each statement with ,, ., or 5.
9
2. 2 S 4
3. 5(2 1 3) R 27
4. 0.6 5 23
5. 26 R 4
Vocabulary Builder
compound inequality
compound (noun)
KAHM
pound
x 7 or x 5
Definition: A compound is made up of separate parts.
Example: A compound inequality is made up of two or more simple inequalities
joined by and or or.
Use Your Vocabulary
6. A student uses the word compound in three different sentences. Place a ✓ next to
the sentences that use the word correctly. Place an ✗ next to those that do not.
✓
In chemistry class we learn about various chemical compounds.
✓
Compound inequalities contain more than one inequality symbol.
✗
To simplify an expression, you can compound like terms.
Write an inequality for each verbal description.
7. x is at most 25
x K 25
Chapter 1
8. x is not equal to 25
9. x is greater than 25
x u 25
x S 25
18
10. x is less than 25
x R 25
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Related Words: compound inequality, less than,
greater than
Problem 1 Writing an Inequality From a Sentence
Got It? What inequality represents the sentence The quotient of a number and
3 is no more than 15?
11. Circle the expression that represents the phrase “the quotient of a number and 3.”
n23
n
3
3
n
3n
12. The symbol , > . > $ > # represents the phrase “no more than”.
13. Now write the inequality.
n
3
Problem 2
K 15
Solving and Graphing an Inequality
Got It? What is the solution of 22(x 1 9) 1 5 L 3? Graph the solution.
14. Solve the inequality for x.
22(x 1 9) 1 5 L 3
22x 2 18 L 22
22x L 16
x K 28
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15. Underline the correct word or number to complete each sentence.
The graph includes all numbers less / greater than 28 / 22 / 3 / 5 / 9 / 10 .
The graph will include a closed / an open circle at 28 / 22 / 3 / 5 / 9 / 10 .
16. Graph the solution.
0
10 9 8 7 6 5 4 3 2 1
Problem 3
1
2
3
4
5
6
7
8
9
10
Using an Inequality
Got It? A digital music service offers two subscription plans. The first has a $9
membership fee and charges $1 per download. The second has a $25 membership
fee and charges $.50 per download. How many songs must you download for the
second plan to cost less than the first plan?
17. Define the variables. Answers may vary. Sample:
Let r
5 the number of songs downloaded.
Let C1 5 cost of the first plan.
Let C2 5 cost of the second plan
.
19
Lesson 1-5
Write an algebraic expression to describe the cost of the first plan and the cost of the
second plan.
9 11 r
18. C1 5
19. C2 5 25 1 0.5 r
20. Write an inequality to represent the situation. Then solve.
25 1 0.5r R 9 1 r
16 R 0.5r
32.6 R r
21. You must download 33 songs for the second plan to cost less than the first plan.
Problem 4
No Solution or All Real Numbers as Solution
Got It? Is 4(2x 2 3) R 8(x 1 1) always, sometimes, or never true?
22. Complete the solution.
4(2x 2 3) , 8(x 1 1)
Write the original inequality.
8x 2 12 R 8x 1 8
Distributive Property
8x 2 20 R 8x
Subtract 8 from each side.
220 R 0
Subtract 8x from each side.
23. The inequality is always / sometimes / never true.
The set of all real numbers.
_______________________________________________________________________
Problem 5
Solving an And Inequality
Got It? What is the solution of 8 K 3x 2 1and 2x R 12? Graph the solution.
25. Solve each inequality.
8 K 3x 2 1
9 K 3x
3Kx
2x R 12
xR6
26. Graph each inequality.
1
2
3
4
5
6
7
1
2
3
27. Graph the solution set. Then write the solution as a compound inequality.
1
Chapter 1
2
3
4
5
6
7
23 K x R 6
20
4
5
6
7
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24. Describe the solution set of the inequality.
Problem 6 Solving an Or Inequality
Got It? What is the solution of 7w 1 3 S 11 or 4w 2 1 R 213? Graph the solution.
28. Cross out the inequality that is NOT a solution to either 7w 1 3 . 11or 4w 2 1 , 213.
w . 1 17
w , 23
w.1
29. Graph the solution.
5
4
3
2
0
1
1
2
3
4
5
30. The solution to 7w 1 3 . 11 or 4w 2 1 , 213 is all real numbers
less than 23 or greater than 1 17 .
Lesson Check • Do you UNDERSTAND?
Reasoning Make up an example to help explain why you must reverse the inequality
symbol when you multiply or divide by a negative number.
Write T for true or F for false.
T
31. 5 . 24
F
32. (21)5 . (21)(24)
T
33. 21 , 24
21
5
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34. Explain what to do to multiply or divide an inequality by a negative number.
Explanations may vary. Sample: In order for the inequality still to be
_______________________________________________________________________
true after multiplying both sides by 21, I must reverse the inequality
_______________________________________________________________________
symbol.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
compound inequality
“and” inequality
“or” inequality
Rate how well you can solve and graph inequalities.
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Lesson 1-5
Absolute Value Equations
and Inequalities
1-6
Vocabulary
Review
1. Circle the solution of 3x 1 8 5 24.
x 5 29
x 5 24
x 5 43
x54
Vocabulary Builder
absolute value
absolute value (noun)
AB
suh loot VAL yoo
2
Definition: The absolute value of a real number x is its
distance from zero on the number line.
2 1
Main Idea: If x is positive, then u x u 5 x . If x is negative,
then u x u 5 2x .
2
0
1
2
2 and 2 are
2 units from 0
Use Your Vocabulary
Graph each absolute value on the number line.
2. u 3 u
3. u 22 u
4. 2u 1 u
U1U
6
5
4
3
2
1
U2U
0
1
2
5. 2u 22 u
U3U 2U2U
3
4
5
Complete each graph.
6. u x u 5 4
4
Chapter 1
7. u x u 5 5
0
4
4
22
0
4
6
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Examples: u 5 u 5 5, u 25 u 5 5
Problem 1 Solving an Absolute Value Equation
Got It? What is the solution of »3x 1 2… 5 4? Graph the solution.
8. Write as two equations.
3x 1 2 5 4
3x 1 2 5 24
or
9. Complete the steps to find x.
3x 1 2 5 4
Write the two equations from Exercise 8.
3x 1 2 5 24
3x 5 2
Subtract 2 from each side.
3x 5 26
x 5 23
Divide each side by 3.
x 5 22
10. Graph the solution.
6
5
4
3
2
1
0
1
2
3
4
5
6
Problem 2 Solving a Multi-Step Absolute Value Equation
Got It? What is the solution of 2»x 1 9… 1 3 5 7? Graph the solution.
11. Complete the steps to find x.
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2u x 1 9 u 1 3 5 7
Write the original equation.
2u x 1 9 u 5
4
Subtract 3 from each side.
ux 1 9u 5
2
Divide each side by 2.
12. Write as two equations.
x1952
x 1 9 5 22
or
13. Solve for x.
x1952
Write the two equations from Exercise 12.
x 1 9 5 22
x 5 27
Subtract 9 from each side.
x 5 211
14. Graph the solution.
12
10
8
6
4
2
0
2
23
4
6
8
10
12
Lesson 1-6
Problem 3
Solving for Extraneous Solutions
Got It? What is the solution of »5x 2 2… 5 7x 1 14? Check for extraneous solutions.
15. Solving the equation yields x 5 21 and x 5 28. Substitute each value into the
original equation.
u 5 ? 28 2 2 u 5 7 ? 28 1 14
u
242
u 5
u 5 ? 21 2 2 u 5 7 ? 21 1 14
u 27 u 5
242
7
16. Which, if any, solutions are extraneous? Explain.
28 ; Explanations may vary. Sample: Because »42… 5 42, not 242.
_______________________________________________________________________
Problem 4
Solving the Absolute Value Inequality »A… R b
Got It? What is the solution of »3x 2 4… K 8? Graph the solution.
17. The inequality is solved below. Use the justifications to complete each step.
28 # 3x 2 4 # 8
Write the compound inequality.
24
#
3x
#
12
Add 4 to each part.
243
#
x
#
4
Divide each part by 3.
4
18. The endpoints of the graph are 23 and 4 .
19. Now graph the solution.
10
Problem 5
8
6
4
2
0
2
4
6
8
10
Solving the Absolute Value Inequality »A… L b
Got It? What is the solution of »5x 1 10… S 15? Graph the solution.
20. Write the absolute value inequality as a compound inequality.
5x 1 10 , 215
5x 1 10 S 15
or
21. Solve each inequality for x.
5x 1 10 R 215
5x R 225
x R 25
5x 1 10 S 15
5x S 5
xS1
22. Now graph the solution.
10
Chapter 1
8
6
4
2
0
2
24
4
6
8
10
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The endpoints are indicated on the graph with open / closed dots.
Problem 6 Using an Absolute Value Inequality
Got It? Suppose the smallest allowable height of a racecar is 52 in. and the
desirable height is 52.5 in. What absolute value inequality describes heights
of a racecar within an indicated tolerance?
23. Find the difference between the desirable height and the smallest allowable height.
52.5 2
5
52.0
0.5
24. Circle the expression that models the difference between the desirable height and
the actual height h.
52.0 2 h
52.5 2 h
25. Write a compound inequality.
20.5 # 52.5 2 h # 0.5
26. Rewrite the compound inequality as an absolute value inequality.
u
52.5 2 h
u#
0.5
Lesson Check • Do you UNDERSTAND?
Reasoning When is the absolute value of a number equal to itself?
0
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
27. The absolute value of 0 is equal to 9 .
28. When a number is negative, its absolute value is equal to 9 .
its opposite
29. When a number is positive, its absolute value is equal to 9 .
itself
30. Now answer the question.
Answers may vary. Sample: The absolute value of a number is equal to
_______________________________________________________________________
itself when it is 0 or positive.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
absolute value
graph
inequality
equation
Rate how well you can use absolute value to solve equations and inequalities.
Need to
review
0
2
4
6
8
Now I
get it!
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Lesson 1-6