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Name
1-1
Class
Date
Additional Vocabulary Support
Patterns and Expressions
Choose the word from the list that best matches each sentence.
algebraic expression
consecutive
diagram
pattern
variable
consecutive
1. Following one another in order.
2. A symbol, usually a letter, that represents one or more numbers.
variable
3. A drawing that is included with a math problem.
diagram
algebraic expression
4. An expression that contains one or more variables.
pattern
5. The same type of change between any two shapes or numbers.
Use a word from the list above to complete each sentence.
6. The numbers 21 and 22 are
consecutive
numbers.
7. In the algebraic expression 5x2 2 2x 1 9, x is called the
8. The picture included with a problem is called a
variable
diagram
.
.
9. 2x 2 10 is called an algebraic expression .
10. The
pattern
in a list of numbers is that each one is 5 more than the last one.
Describe the pattern for each group of figures. What would the next consecutive
figure in the pattern look like?
11.
Answers may vary. Sample: a black square in the
corner of a larger square moves clockwise from one
corner to the next corner; a square with a black
square in the lower right corner
12.
Answers may vary. Sample: equilateral triangles
alternating pointing up, then down; an equilateral
triangle pointing up
13.
Answers may vary. Sample: alternate a circle inside a
square with a square inside a circle; a circle inside a
square
14.
Answers may vary. Sample: a stack of small squares,
first one square, then two squares, then 3 squares; a
stack of 4 small squares
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1-1
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Date
Think About a Plan
Patterns and Expressions
Use the graph shown.
y
15
a. Identify a pattern of the graph by making a table of the
Output
inputs and outputs.
b. What are the outputs for inputs 6, 7, and 8?
1. What are the ordered pairs of the points in the graph?
(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)
12
9
6
3
O
x
1
2
3
4
Input
2. Complete the table of the input and output values shown in the ordered pairs.
Input
Output
1
3
2
6
3
9
4
12
5
15
3. Complete the process column with the process that takes each input value
and gives the corresponding output value.
4. output 5
Input
Process
Column
Output
1
1( 3 )
3
2
2( 3 )
6
3
3( 3 )
9
4
4( 3 )
12
5
5( 3 )
15
input ∙ 3
5. Complete the process column for inputs 6, 7, and 8. Then find the outputs for
inputs 6, 7, and 8.
Input
Process
Column
Output
6
6( 3 )
18
7
7( 3 )
21
8
8( 3 )
24
18, 21, 24
6. The outputs for inputs 6, 7, and 8 are
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5
Name
Class
1-1
Date
Practice
Form G
Patterns and Expressions
Describe each pattern using words. Draw the next figure in each pattern.
1.
Answers may vary. Sample: Rotate
the previous figure 90° clockwise.
2.
Answers may vary. Sample: Each
figure is obtained by moving the
previous corner block up one row
and right one column, shading this
block and all blocks to the left of
and below this new corner block.
Answers may vary. Sample:
With the center of the middle
of the three consecutive blocks
as pivot, each figure is a 90°
counterclockwise rotation of the
previous figure.
3.
Copy and complete each table. Include a process column.
4.
Input
Output
P. C.
1
4
2
5.
Input
Output
P. C.
5(1) ⫺ 1
1
22
9
5(2) ⫺ 1
2
3
14
5(3) ⫺ 1
4
19
5
6.
Input
Output
P. C.
⫺2(1)
1
0.5
0.5(1)
24
⫺2(2)
2
1.0
0.5(2)
3
26
⫺2(3)
3
1.5
0.5(3)
5(4) ⫺ 1
4
28
⫺2(4)
4
2.0
0.5(4)
24
5(5) ⫺ 1
5
210
⫺2(5)
5
2.5
0.5(5)
6
29
5(6) ⫺ 1
6
212
⫺2(6)
6
3.0
0.5(6)
n
5n 2 1
5n ⫺ 1
n
22n
⫺2n
n
0.5n
0.5n
7. Describe the pattern using words.
Answers may vary. Sample: Draw a square. For each subsequent figure in the
pattern, make a new square having vertices at the midpoints of the sides of
the previous innermost square, shading all but the new innermost square.
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Practice (continued)
Form G
Patterns and Expressions
A gardener plants a flower garden between his house and
a brick pathway parallel to the house. The table at the right
shows the area of the garden, in square feet, depending on
the width of the garden, in feet.
Width
Area
1
3.5
2
7
3
10.5
4
14
8. What is the area of the garden if the width is 8 feet?
28 ft²
9. What is the area of the garden if the width is 15 feet? 52.5 ft²
Identify a pattern and find the next three numbers in the pattern.
10. 25, 210, 220, 240, c
Each term is double the previous term;
280, 2160, 2320
11. 5, 8, 11, 14, c
Each term is 3 more than the previous
term; 17, 20, 23
12. 3, 1, 21, 23, c
Each term is 2 less than the previous
term; 25, 27, 29
2 3 4 5
14. 3, 4, 5, 6, c
The numerator and denominator of
each term are each one more than the
previous term; 67, 78, 89
13. 1, 3, 6, 10, 15, c
Each term is n 1 1 more than the previous
term where n is the difference between
the previous two terms; 21, 28, 36
15. 10, 9, 6, 1, 26, c
Each term is n 1 2 less than the previous
term where n is the difference between
the previous two terms; 215, 226, 239
The graph shows the cost depending on the
number of DVDs that you purchase.
$80
$64
Cost
16. What is the cost of purchasing 5 DVDs? $80
17. What is the cost of purchasing 10 DVDs? $160
$48
$32
$16
18. What is the cost of purchasing n DVDs? $16n
0
1
2
3
4
Number of DVDs
Keesha earns $320 a week working in a clothing store. As a bonus, her employer
pays her $15 more than she earned the previous week, so that at the end of the
second week she earns $335, and after 3 weeks, she earns $350.
19. How much will Keesha earn at the end of the fifth week? $380
20. How much will Keesha earn at the end of the tenth week? $455
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Name
1-1
Class
Date
Practice
Form K
Patterns and Expressions
Describe each pattern using words. Draw the next figure in each pattern.
1.
Answers may vary. Sample: Start with a row of two squares. Add a row of two squares
below the previous figure for each new figure.
2.
Answers may vary. Sample: Start with one square. Add a row of squares below the
previous figure that has one more square than the row above it for each new figure.
3.
Answers may vary. Sample: Start with an isosceles triangle that points upward. Rotate
the triangle 90° clockwise for each new figure.
Make a table with a process column to represent the pattern. Write an
expression for the number of circles in the nth figure. The table has been started
for you.
4.
The expression for the
number of circles in
the nth figure is 2n.
Figure Number
(Input)
Process
Column
Number of
Circles (Output)
1
1(2)
2
2
2(2)
4
3
3(2)
6
4
4(2)
8
■
■
■
n
n(2)
2n
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Name
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Practice (continued)
1-1
Form K
Patterns and Expressions
Number of cups of flour
The graph shows the number of cups of flour needed for
baking cookies.
5. How many cups of flour are needed for baking 4 batches
of cookies?
8 cups
6. How many cups of flour are needed for baking 30 batches
of cookies?
10
Baking Cookies
8
6
4
2
0
0 2 4 6 8 10
Number of batches of cookies
60 cups
7. How many cups of flour are needed for baking n batches
of cookies?
2n cups
Identify a pattern by making a table. Include a process column.
(1, 4), (2, 5), (3, 6), (4, 7)
Hint: To start, list the points on the graph. Make a table of input and
output values shown in the ordered pairs. Use the process column to
figure out the pattern.
10
10
9.
8
8
6
6
Output
Output
8.
4
2
4
Input
6
0
8
2
4
Input
6
8
Identify the pattern and find the next three numbers in the pattern.
10. 1, 4, 16, 64, . . .
11. 3, 6, 12, 24, . . .
Each term is 4 times the previous term;
256, 1024, 4096
Each term is 2 times
the previous term;
48, 96, 192
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Process
Column
(Output)
1
113
4
2
213
5
3
313
6
4
413
7
n
n13
n13
(1, 1.5), (2, 3), (3, 4.5), (4, 6)
2
2
0
4
(Input)
(Input)
Process
Column
(Output)
1
1(1.5)
1.5
2
2(1.5)
3
3
3(1.5)
4.5
4
4(1.5)
6
n
n(1.5)
1.5n
Name
Class
1-1
Date
Standardized Test Prep
Patterns and Expressions
Multiple Choice
For Exercises 1–5, choose the correct letter.
1. What is the next figure in the pattern at the right? C
2. Which is the next number in the table? H
14
15
16
20
Input
Output
1
1
2
3
3
6
4
10
5
■
3. How many toothpicks would be in the tenth figure? A
21
20
4. What is the next number in the pattern?
21
23
2, 7, 12, 17, c G
22
5. What is the next number in the pattern?
23
11
23
27
1, 21, 2, 22, 3, c A
0
3
4
Short Response
6. Ramon has 25 books in his library. Each month, he adds 3 new books to his
collection. How many books will Ramon have after 12 months?
61 books
[2] correct number of books
[1] incorrect number of books
[0] no answer given
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Enrichment
Patterns and Expressions
Often identifying and expressing patterns using algebraic expressions is difficult
because the pattern is not immediately obvious. Organizing information into a
table and looking at the common differences provide a clue.
1. Copy and complete the table below to find the first differences between
consecutive terms.
1
4
Figure Number
Number Pattern
2
10
6
6
First Difference
3
16
4
22
5
28
6
6
6
34
6
7
40
6
8
46
6
9
52
6
10
58
6
2. Because the first difference is constant in the pattern above, the algebraic
expression will have degree 1. Degree 1 expressions will not have any
exponents larger than 1. What is the algebraic expression for the nth term in
this pattern? (6n 2 2)
3. Copy and complete the table below by finding the first and second differences
between consecutive terms.
1
4
Figure Number
Number Pattern
2
10
6
First Difference
8
2
Second Difference
3
18
4
28
10
2
5
40
14
12
2
6
54
2
7
70
16
2
8
88
18
2
20
2
4. Since the second difference is constant in the pattern above, the algebraic
expression will have degree 2. In an expression of degree 2, the largest
exponent is 2. Therefore, we know that the expression for this pattern must
include n2. What is the algebraic expression for the nth term in this pattern?
(n2 1 3n)
Write an algebraic expression for the nth term in each pattern.
5. 23, 2, 7, 12, 17, c (5n 2 8)
6. 2, 8, 18, 32, 50, c (2n2)
7. 0, 7, 26, 63, 124, c (n3 2 1)
8. 2, 5, 10, 17, 26, c (n2 1 1)
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9
108
10
130
22
2
Name
1-1
Class
Date
Reteaching
Patterns and Expressions
Identifying patterns is an important skill in algebra. Identify the underlying
structure of a group of numbers, a set of data, or a sequence of figures to express a
rule describing the relationship.
Problem
Look at the figures from left to right. What is the pattern? What would the next
figure in the pattern look like?
First, identify the basic properties of the figure. The number of elements in each
figure increases by one from each figure to the next.
Second, determine whether the figures change in size. Notice that the arrows
decrease in length by about 13 at each step.
Third, observe whether there is displacement or rotation from one figure to the
next. Each arrow is a 908 clockwise rotation from the previous arrow.
The pattern begins with an arrow pointing to the right. Each subsequent figure adds
a new arrow that is shorter than the previous arrow by about 13 and is rotated 908
clockwise from the previous one. The next figure in the pattern is shown at the right.
Exercises
Describe the pattern in words and draw the next figure in the pattern.
1.
Each figure is twice the size of
the one before it.
2.
Each figure adds a
new triangle whose
dimensions are 50%
of the dimensions of
the previous smallest
triangle and is inside
it, rotated 180°. The
color alternates white
to black from biggest
to smallest triangle,
respectively.
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Reteaching
Date
(continued)
Patterns and Expressions
Problem
$15
Charge
You tell your parents that you will pay the text messaging
portion of your cell phone bill. The graph shows the monthly
charge depending on the number of text messages you send and
receive during the month. How much do you owe if you send and
receive a total of 100 text messages during the month?
$12
$9
$6
$3
First, identify several points on the graph. Three points on the
graph are (10, 3), (20, 6), and (50, 15). Make a table of values for the
given inputs.
Input
Process Column
Output
10
10 3 $.30
$3
20
20 3 $.30
$6
50
50 3 $.30
$15
0
10 20 30 40 50
Number of Text Messages
Identify the pattern in the process column. Each output is equal to the product
of the input and $.30. To summarize the result in words, each text message
costs $0.30.
Therefore, if you send and receive 100 text messages during the month, then
you owe 100 3 $.30 5 $30.
Exercises
Use the graph at the right to answer the questions.
3. What is the cost to send 10 text messages? 20 text messages?
50 text messages? $5; $10; $25
Charge
4. What is the cost to send one text message? $.50
$25
$20
$15
$10
$5
5. How much do you owe your parents if you send and
receive 75 text messages during the month? $37.50
0
10 20 30 40 50
Number of Text Messages
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Additional Vocabulary Support
Properties of Real Numbers
Choose the word from the list that best matches each description.
integers irrational numbers natural numbers rational numbers whole numbers
whole numbers
1. the natural numbers and zero
integers
2. the natural numbers, their opposites, and zero
3. the numbers that can be written as a quotient of integers
rational numbers
4. the numbers used for counting
natural numbers
irrational numbers
5. the numbers that cannot be written as quotients of integers
Write all of the numbers from the list that are examples of each subset.
27
6. whole numbers
212
"81
5
7
0
2105
"44
93
27, !81 , 0, 93
7. natural numbers 27, !81 , 93
5
8. rational numbers 27, 212, !81 , 0, 7 , 2105, 93
9. irrational numbers !44
10. integers 27, 212, !81 , 0, 2105, 93
11. Draw a diagram showing the relationship of whole numbers, natural numbers, rational
numbers, integers, irrational numbers, and real numbers.
Real Numbers
Natural numbers
Whole numbers
Irrational
numbers
Integers
Rational numbers
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Think About a Plan
Properties of Real Numbers
Five friends each ordered a sandwich and a drink at a restaurant. Each sandwich
costs the same amount, and each drink costs the same amount. What are two
ways to compute the bill? What property of real numbers is illustrated by the two
methods?
Understanding the Problem
1. There are
5
sandwiches and
5
drinks on the bill.
2. What is the problem asking you to determine?
two ways to represent the cost of 5 sandwiches and 5 drinks, and the property of real
numbers illustrated by the two representations
Planning the Solution
3. How can you represent the cost of five sandwiches?
Answers may vary. Sample: 5s
4. How can you represent the cost of five drinks?
Answers may vary. Sample: 5d
5. How can you represent the cost of the items ordered by one friend?
Answers may vary. Sample: d 1 s
Getting an Answer
6. Write an expression that represents the cost of five drinks and the cost of five
sandwiches.
Answers may vary. Sample: 5d 1 5s
7. Write an expression that represents the cost of the items ordered by five
friends.
Answers may vary. Sample: 5(d 1 s)
8. What property of real numbers tells you that these two expressions are equal?
Explain.
Distributive Property; the Distributive Property states a(b 1 c) 5 ab 1 ac.
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Name
1-2
Class
Date
Practice
Form G
Properties of Real Numbers
Classify each variable according to the set of numbers that best describes
its values.
1. the area of the circle A found by using the formula pr2 irrational number
2. the number n of equal slices in a pizza; the portion p of the pizza in one slice
natural number; rational number
3. the air temperature t in Saint Paul, MN, measured to the nearest degree
Fahrenheit integer
4. the last four digits s of a Social Security number natural number
1
⫺2 2
Graph each number on a number line.
⫺3 ⫺2 ⫺1
5. 21
6. "3
√ 3 2.8
⫺1
0
7. 2.8
1
2
3
1
8. 22 2
Compare the two numbers. Use + or *.
10. 4, !17 *
9. 2!2, 22 +
11. !29, 5 +
12. !50, 6.8 +
13. 11, !130 *
14. 26, 2!30 *
1
15. 72, !67 *
16. 2!10, 2!12 +
Name the property of real numbers illustrated by each equation.
17. 2(3 1 !5) 5 2 ? 3 1 2 ? !5
18. 16 1 (213) 5 213 1 16
Distributive Property
1
19. 27 ? 27 5 1
Inverse Property of Multiplication
Commutative Property of Addition
20. 5(0.2 ? 7) 5 (5 ? 0.2) ? 7
Associative Property of Multiplication
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Name
1-2
Class
Date
Practice (continued)
Form G
Properties of Real Numbers
Estimate the numbers graphed at the labeled points.
A
B
⫺3 ⫺2 ⫺1
C
0
1
D
2
3
21. point A Answers may vary. Sample: 22.7
22. point B Answers may vary. Sample: 21.2
23. point C Answers may vary. Sample: 0.9
24. point D Answers may vary. Sample: 3.0
Geometry To find the length of side b of a rectangular prism with a
square base, use the formula b 5 ÅVh , where V is the volume of the
prism and h is the height. Which set of numbers best describes the
value of b for the given values of V and h?
25. V 5 100, h 5 5
irrational numbers
26. V 5 100, h 5 25
natural numbers
27. V 5 100, h 5 20
irrational numbers
28. V 5 5, h 5 20
rational numbers
h
Write the numbers in increasing order.
4 5
29. 2"2, 5, 24, 0.9, 21
2 54, 21, 45, 0.9, 2"2
5
2
30. 8, 26, 3, 2p, 20.5
26, 2π, 20.5, 58, 23
Justify the equation by stating one of the properties of real numbers.
31. (x 1 37) 1 (237) 5 x 1 (37 1 (237)) Associative Property of Addition
32. x ? 1 5 x Identity Property of Multiplication
33. x 1 (37 1 (237)) 5 x 1 0 Inverse Property of Addition
34. x 1 0 5 x Identity Property of Addition
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b
b
Name
Class
1-2
Date
Practice
Form K
Properties of Real Numbers
Classify each variable according to the set of numbers that best describes
its values.
1. the number of students in your class natural numbers
To start, make a list of some numbers that could describe the number of
students in your class.
2. the area of the circle A found by using the formula A 5 pr2 irrational numbers
To start, make a list of some numbers that could describe the area of a circle.
3. the elevation e of various land points in the United States measured to the nearest foot integers
To start, make a list of some numbers that could describe elevation levels.
Graph each number on a number line.
1
4. 5 2
6. 2.25
5
⫺4
5. 24
1
2
⫺8⫺7⫺6⫺5⫺4⫺3⫺2⫺1 0 1 2
⫺1 0 1 2 3 4 5 6 7 8 9
2.25
1
7. 26 3
⫺1 0 1 2 3 4 5 6 7 8 9
⫺6
1
3
⫺8⫺7⫺6⫺5⫺4⫺3⫺2⫺1 0 1 2
8. !8
√ 8 ⬇ 2.83
To start, use a calculator to approximate the square root.
ⴚ1 0 1 2 3 4 5 6 7 8 9
Compare the two numbers. Use R or S .
9. !50 and 8.8 R
10. 5 and !23 S
11. 6.2 and !40 R
12. 2!3 and 23 S
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Class
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Practice (continued)
Form K
Properties of Real Numbers
Name the property of real numbers illustrated by each equation.
2 3
13. 3 ? 2 5 1
Inverse Property of Multiplication
14. 6(2 1 x) 5 6 ? 2 1 6 ? x
Distributive Property
15. 2 ? 20 5 20 ? 2
Commutative Property of Multiplication
16. 8 1 (28) 5 0
Inverse Property of Addition
17. 2(0.5 ? 4) 5 (2 ? 0.5) ? 4
Associative Property of Multiplication
18. 211 1 5 5 5 1 (211)
Commutative Property of Addition
Estimate the numbers graphed at
the labeled points.
C
A
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
19. point A Answers may vary. Sample: 22
B
0
1
D
2
3
4
5
20. point B Answers may vary. Sample: 1.5
21. point C Answers may vary. Sample: 24.75 22. point D Answers may vary. Sample: 2.25
To find the length of the side b of the square base of a rectangular prism, use the
formula b 5 ÅVh , where V is the volume of the prism and h is the height. Which
set of numbers best describes the value of b for the given values of V and h?
23. V 5 100, h 5 1
24. V 5 100, h 5 10
natural numbers
irrational numbers
Write the numbers in increasing order.
5
5
25. 6, !28, 22, 20.8, 1
2
26. 3, 24, !32, !13, 20.4
24, 20.4, 23, !13, !32
252, 20.8, 56, 1, "28
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Name
Class
1-2
Date
Standardized Test Prep
Properties of Real Numbers
Multiple Choice
For Exercises 1–5, choose the correct letter.
1. Which letter on the graph corresponds to !5? C
2. Which letter on the graph corresponds to 21.5? F
A
⫺1
BC
0
1
F
2
G
⫺3 ⫺2 ⫺1
D
3
H
0
4
5
2
3
I
1
What property of real numbers is illustrated by the equation?
3. 26 1 (6 1 5) 5 (26 1 6) 1 5 D
Identity Property of Addition
Commutative Property of Addition
Inverse Property of Addition
Associative Property of Addition
4. 2(24 1 x) 5 2(24) 1 2 ? x G
Associative Property of Multiplication
Associative Property of Addition
Distributive Property
Closure Property of Multiplication
2
5. Which of the following shows the numbers 13, 1.3, 17, 24, and 2!10 in
order from greatest to least? B
13, 1.3, 127, 24, 2!10
13, 127, 1.3, 2!10, 24
13, 1.3, 127, 2!10, 24
24, 2!10, 127, 1.3, 13
Short Response
Geometry The length c of the hypotenuse of a right triangle with legs having
lengths a and b is found by using the formula c 5 "a2 1 b2 . Which set of
numbers best describes the value of c for the given values of a and b?
6. a 5 3, b 5 4 [2] natural numbers [1] incorrect set of numbers [0] no answer given
1
1
7. a 5 3, b 5 4 [2] rational numbers [1] incorrect set of numbers [0] no answer given
8. a 5 !3, b 5 !4 [2] irrational numbers [1] incorrect set of numbers [0] no answer given
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Properties of Real Numbers
There are four words beginning with the letter “I” that describe certain types of
operations:
a. Identity: an operation that does not change anything. For example, adding
0 to a number is an identity operation, because adding 0 does not change the
original number.
b. Inverse: an operation that can be undone by another operation. For instance,
the operation of adding 2 to a number can be undone by subtracting 2 from
the number. However, the operation of multiplying a number by 0 cannot be
undone.
c. Idempotent: an operation that, when done twice, is the same as doing it once.
For example, multiplying a number by 1 and then multiplying the result by 1
again has exactly the same effect as multiplying the number by 1 only once.
d. Involutory: an operation that, when done twice, leaves a number unchanged.
For instance, multiplying a number by 21 and then multiplying the result by
21 again returns the original number.
For each of the following operations, state which of the “I” words apply. If none
apply, write none.
1. finding the absolute value of a number idempotent
2. dividing a number by 1 identity, inverse, idempotent, involutory
3. multiplying the absolute value of a number by 21 idempotent
4. finding the reciprocal of a nonzero number inverse, involutory
5. dividing a number by 21 inverse, involutory
6. multiplying a number by 0 idempotent
7. adding 0 to the reciprocal of a nonzero number inverse, involutory
8. multiplying the reciprocal of a nonzero number by 2 inverse, involutory
9. adding the absolute value of a nonzero number to the absolute value of its
reciprocal none
10. finding the reciprocal of the absolute value of the reciprocal of a nonzero
number idempotent
11. finding the absolute value of 21 times a nonzero number, then taking the
reciprocal none
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Reteaching
Properties of Real Numbers
The Properties of Real Numbers are relationships that are true for all real numbers
except zero.
The additive identity for real numbers is 0. This gives the Identity Property of
Addition, which states for any real number a:
a 1 0 5 a and 0 1 a 5 a
The additive inverse of a real number a is 2a. By the Inverse Property of Addition:
a 1 (2a) 5 0
There are two similar properties for multiplication. These use the multiplicative
identity 1 and the multiplicative inverse a1 for any nonzero real number a.
Identity Property of Multiplication: a ? 1 5 a and 1 ? a 5 a
1
Inverse Property of Multiplication: a ? a 5 1
Problem
Using the Properties of Real Numbers, what is the missing number in the
equation?
a.
u1055
According to the Identity Property of Addition, the missing number is 5.
b. 7 ?
u51
The Inverse Property of Multiplication shows that the product of a real
number and its multiplicative inverse is 1. The missing number is the
multiplicative inverse of 7, or 17 .
Exercises
Find the missing number in the equation.
1.
0 1 (24) 5 24
u
2. 22 1
u
3. 1 ? 22 5 22
4.
2 50
u
u ? 32 5 1
2
3
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(continued)
Properties of Real Numbers
The Commutative and Associative Properties of Addition and Multiplication are
properties that help you simplify calculations.
The Commutative Property states that the order of addition or multiplication
does not change the sum or product.
a1b5b1a
ab 5 ba
The Associative Property states that the grouping of three or more addends or
factors does not change the sum or product.
(a 1 b) 1 c 5 a 1 (b 1 c)
(ab)c 5 a(bc)
Problem
What property does the equation illustrate?
1
1
5 ? Q 5 ? 85 R 5 Q 5 ? 5 R ? 85
This equation shows that the product of three numbers is the same regardless of
the order of multiplication. Only the grouping of the factors is different. Therefore,
the equation illustrates the Associative Property of Multiplication.
Exercises
Name the property that the equation illustrates.
1
5. Q 5 ? 5 R ? 85 5 1 ? 85
Inverse Property of Multiplication
6. 1 ? 85 5 85
Identity Property of Multiplication
The Distributive Property combines addition and multiplication: a(b 1 c) 5 ab 1 ac.
Problem
What are the missing values in the equation?
4 ? (6 1 3) 5 4 ?
u14?u
By the Distributive Property, the sum of two numbers multiplied by a third
number is equal to the sum of each multiplied by the third number. Because the
third number is 4, the missing numbers are 6 and 3.
Exercises
Name the missing values in each equation.
3 ?2
u1u
2 5u
9 ( y 1 2)
9. 9 ? y 1 9 ? u
22 ? 5 2 2 ? 1
u 1 1b 5 u
8. 22 a 5
7. 3(a 1 2) 5 3 ? a
z
z
1x b
10. 8(23) 1 8 ? x 5 8 a 23
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Additional Vocabulary Support
Algebraic Expressions
Use the chart below to review vocabulary. These vocabulary words will help you
complete this page.
Addition (1)
Subtraction (2)
Multiplication (3)
Division (4)
sum
difference
product
quotient
more than
less than
times
divided by
increased by
fewer than
total
subtracted from
added to
Circle the word or words in each word phrase that tell you what operations
to use. Write the operation symbol word (1, 2, 3, 4) next to the algebraic
expression.
1. the sum of a number m and 212 1
2. the product of b and c 3
3. 14 less than p 2
4. the total of 275 and t 1
5. the quotient of d and 28 4
Match each word phrase in Column A with the matching algebraic expression
in Column B.
Column A
Column B
6. the difference of a number p and 36
A. y 1 9
7. 15 more than the number q
B. 10(r)
8. the product of 10 and a number r
C. q 1 15
9. the total of a number y and 9
D. p 2 36
Match each algebraic expression in Column A with the matching word phrase
in Column B.
Column A
10. m 1 45 C
m
Column B
A. 45 less than a number m
11. 45 D
B. 45 times the sum of a number m and 1
12. m 2 45 A
C. a number m increased by 45
13. 45(m 1 1) B
D. a number m divided by 45
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Think About a Plan
Algebraic Expressions
Write an algebraic expression to model the situation.
The freshman class will be selling carnations as a class project. What is the class’s
income after it pays the florist a flat fee of $200 and sells x carnations for $2 each?
1. What does the variable represent?
the number of carnations sold
2. How will the class’s income change for each carnation sold?
It will increase by $2.
3. Will paying the florist increase or decrease their income? By how much?
decrease; $200
4. Will the expression include both the income for each carnation and the
florist’s fee? Explain.
Yes; the class’s profit is a function of the proceeds from carnation sales
and the florist’s fee.
5. Write the expression in words.
The
income
z 2200
z
is
and
2 z
z times
x z .
z 6. Write the expression using symbols.
income
5
z 2200
z
1 z
z 2 z
z . z
z x z
z 7. Check your expression by substituting 300 for the number of carnations. Does
your answer make sense? Explain.
2200 1 2(300) 5 2200 1 600 5 400; yes; the class has an income of $400 after they
make $600 and pay the florist $200.
z
z
220012x
8. The algebraic expression models the freshman class income.
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Practice
Form G
Algebraic Expressions
Write an algebraic expression that models each word phrase.
1. seven less than the number t t 2 7
2. the sum of 11 and the product of 2 and a number r 11 1 2r
Write an algebraic expression that models each situation.
3. Arin has $520 and is earning $75 each week babysitting. 520 1 75w
4. You have 50 boxes of raisins and are eating 12 boxes each month. 50 2 12m
Evaluate each expression for the given values of the variables.
5. 24v 1 3(w 1 2v) 2 5w; v 5 22 and w 5 4 212
6. c(3 2 a) 2 c2 ; a 5 4 and c 5 21 0
7. 2(3e 2 5f ) 1 3(e2 1 4f ); e 5 3 and f 5 25 35
Surface Area The expression 6s2 represents the surface area of a cube with edges
of length s. What is the surface area of a cube with each edge length?
8. 3 inches 54 in.2
9. 1.5 meters 13.5 m2
The expression 4.95 1 0.07x models a household’s monthly long-distance
charges, where x represents the number of minutes of long-distance calls during
the month. What are the monthly charges for each number of long-distance
minutes?
10. 73 minutes $10.06
11. 29 minutes $6.98
Simplify by combining like terms.
3(a 2 b)
1 49 b 13a 1 19b
9
12. 5x 2 3x2 1 16x2 13x2 1 5x
13.
t2
14. t 1 2 1 t2 1 t 32t2 1 2t
15. 4a 2 5(a 1 1) 2a 2 5
16. 22( j2 2 k) 2 6( j2 1 3k) 28j2 2 16k
17. x(x 2 y) 1 y(y 2 x) x2 2 2xy 1 y2
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Form G
(continued)
Algebraic Expressions
18. In a soccer tournament, teams receive 6 points for winning a game, 3 points
for tying a game, and 1 point for each goal they score. What algebraic
expression models the total number of points that a soccer team receives in a
tournament? Suppose one team wins two games and ties one game, scoring a
total of five goals. How many points does the team receive? 6w 1 3t 1 1g; 20 points
Evaluate each expression for the given value of the variable.
19. 2t2 2 (3t 1 2); t 5 5 242
20. i2 2 5(i3 2 i2); i 5 4 2224
a⫺b
21. Perimeter Write an expression for the perimeter of the figure
c
at the right as the sum of the lengths of its sides. What is the
simplified form of this expression?
a 1 (a 2 b) 1 c 1 b 1 (a 2 2c) 1 b 1 c 1 (a 2 b); 4a
b
a ⫺ 2c
a
b
22. Simplify 2(2x 2 5y) 1 3(4x 1 2y) and justify each step in
your simplification. 22x 1 5y 1 12x 1 6y, Opposite of a
c
Difference and Distributive Property; 10x 1 11y; Combine like terms
a⫺b
using the Distributive Property
23. Error Analysis Alana simplified the expression as shown. Do you agree
2(x
x + 4) - (5x
x - 7)
with her work? Explain. No; Dist. Prop. and Opp. of a Diff.
x + 4 - 5x
x-7
2x
incorrectly applied. Correct simplification: 2x 1 8 2 5x 1 7 5 23x 1 15
-3x
x-3
24. Open-Ended Write an example of an algebraic expression that always
has the same value regardless of the value of the variable. Answers may vary. Sample: any
expression that results in the variable having a coefficient of 0; sample: x 2 x
Match the property name with the appropriate equation.
25. Opposite of a Difference E
A. 2f(2r) 1 2pg 5 2(2r) 2 2p
26. Opposite of a Sum A
B. 16d 2 (3d 1 2)(0) 5 16d 2 0
27. Opposite of an Opposite F
C. 5(2 2 x) 5 10 2 5x
28. Multiplication by 0 B
D. 2(4r 1 3s) 1 t 5 (21)(4r 1 3s) 1 t
29. Multiplication by 21 D
E. 2(8 2 3m) 5 3m 2 8
30. Distributive Property C
F. 2f2(9 2 2w)g 5 9 2 2w
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Form K
Algebraic Expressions
Write an algebraic expression that models each word phrase.
1. six less than the number r r 2 6
To start, relate what you know. “Less than” means subtraction.
Describe what you need to find. Begin with the number r and subtract 6.
2. twelve more than the number b b 1 12
3. five times the sum of 3 and the number m 5(3 1 m)
Write an algebraic expression that models each situation.
4. Alexis has $250 in her savings account and deposits $20 each week for w weeks. 250 1 20w
5. You have 30 gallons of gas and you use 5 gallons per day for d days. 30 2 5d
Evaluate each expression for the given values of the variables.
6. 22a 1 5b 1 6a 2 2b 1 a; a 5 23 and b 5 2 29
To start, substitute the value
for each variable.
22(23) 1 5(2) 1 6(23) 2 2(2) 1 (23)
7. y(3 2 x) 1 x2; x 5 2 and y 5 12 16
8. 3(4e 2 2f ) 1 2(e 1 8f ); e 5 23 and f 5 10 58
The expression 6s2 represents the surface area of a cube with edges of length s.
What is the surface area of a cube with each edge length?
9. 4 centimeters 96 cm2
10. 2.5 feet 37.5 ft2
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Form K
Algebraic Expressions
Write an algebraic expression to model the total score in each situation.
Then evaluate the expression to find the total score.
11. In the first half, there were fifteen two-point shots, ten three-point shots and
5 one-point free throws. T 5 2w 1 3r 1 1f; 65
To start, define your variables. Let w 5 the number of two-point shots,
r 5 the number of three-point shots, and f 5 the number of one-point free throws.
12. In the first quarter, there were two touchdowns and 1 extra point kick. T 5 6t 1 1e; 13
Hint: A touchdown is worth 6 points. An extra point kick is worth 1 point.
Simplify by combining like terms.
13. 10b 2 b 9b
14. 12 1 8s 2 3s 12 1 5s
15. 3a 1 2b 1 6a 9a 1 2b
16. 5m 1 2n 1 6m 1 4n 11m 1 6n
17. 8r 2 (3s 2 5r) 13r 2 3s
18. 2.5y 2 4y 21.5y
The expression 19.95 1 0.05x models a household’s monthly Internet charges,
where x represents the number of online minutes during the month. What are
the monthly charges for each number of online minutes?
19. 65 minutes $23.20
20. 128 minutes $26.35
Evaluate each expression for the given value of the variable.
21. 3a 1 (2a 1 6); a 5 2 16
22. x 2 5(x 1 2); x 5 25 10
23. 2r 1 (3r2 1 1); r 5 4 45
24. x2 2 5(3x 2 12); x 5 10 10
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Standardized Test Prep
Algebraic Expressions
Multiple Choice
For Exercises 1–3, choose the correct letter.
1. The expression 2p(rh 1 r2) represents the total surface area of a cylinder with
height h and radius r. What is the surface area of a cylinder with height
6 centimeters and radius 2 centimeters? C
16p cm2
32p cm2
28p cm2
96p cm2
2. Which expression best represents the simplified form of
3(m 2 3) 1 m(5 2 m) 2 m2 ? F
22m2 1 8m 2 9
22m2 2 2m 2 9
8m 2 9
22m 2 9
3. The price of a discount airline ticket starts at $150 and increases by $30 each
week. Which algebraic expression models this situation? D
30 1 150w
30 2 150w
150 2 30w
150 1 30w
Extended Response
4. Members of a club are selling calendars as a fundraiser. The club pays $100
for a box of wall and desk calendars. They sell wall calendars for $12 and desk
calendars for $8.
a. Write an algebraic expression to model the club’s profit from selling w wall
calendars and d desk calendars. Explain in words or show work for how you
determined the expression
b. What is the club’s profit from selling 9 wall calendars and 7 desk calendars?
Show your work.
a. The income from wall calendars is 12w. The income from desk calendars is 8d. The
total income is 12w 1 8d. The club must pay $100 from the total income, so the profit
is 12w 1 8d 2 100. (OR equivalent explanation)
b. 12w 1 8d 2 100 5 12(9) 1 8(7) 2 100 5 108 1 56 2 100 5 64; $64
[4] appropriate methods and correct expression with no computational errors
[3] appropriate methods and correct expression, but minor computational error
[2] incorrect expression or multiple computational errors
[1] correct expression and profit, without work shown
[0] no answer or no attempt made
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Enrichment
Algebraic Expressions
Math Puzzle
Algebraic expressions can help reveal the secret behind number puzzles that
appear to be magic. Start by trying the puzzle below. Complete the puzzle two
times and record each step in the table.
First Guess
Directions
Second Guess
Think of a number.
Add 1 to your number.
Multiply your answer by 2.
Subtract 2 from your answer.
Finally, subtract your original number.
What did you notice about your final answer each time?
Answers may vary. Sample: The final number is the same as the original number.
By writing and simplifying algebraic expressions, you can explain the puzzle.
Let n represent your number. Write an algebraic expression for each of the steps
in the puzzle above.
1. Add 1 to your number. n 1 1
2. Multiply your answer by 2. 2n 1 2
3. Subtract 2 from your answer. 2n
4. Subtract your original number. n
5. Explain how your algebraic expressions show why the puzzle always ends with
the original number.
Answers may vary. Sample: The final expression is the same as the original expression.
Write and simplify algebraic expressions for the puzzle below.
6. Think of a number. n
7. Multiply your number by 4. 4n
8. Subtract 2. 4n 2 2
9. Divide your number by 2. 2n 2 1
10. Add 1. 2n
11. Subtract your original number. n
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1-3
Algebraic Expressions
You can model words with algebraic expressions. In a word problem, look for
words and word phrases that indicate mathematical operations.
Addition
added to
plus
sum
more than
longer than
increased by
total
in all
Subtraction
subtracted from
minus
difference
less than
shorter than
decreased by
fewer than
Multiplication
multiplied by
product
times
of
Division
divided by
quotient
fraction of
per
Problem
What is an algebraic expression that models the given word phrase?
The quotient of 4 more than the number z and the number y decreased by 3
division
4
addition
z
4
(4
1
z)
(41z)
y subtraction 3
(y
4
2
3)
(y23)
(41z)
(y23)
Exercises
Write an algebraic expression that models each word phrase.
1. nine less than 5 multiplied by the number p 5p 2 9
2. the product of 2 divided by the number h and 8 more than the number k Q h2 R (k 1 8)
3. two decreased by the quotient of the number a and 7 and increased by a
multiplied by 3
2 2 a7 1 3a
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(continued)
Algebraic Expressions
To simplify an algebraic expression, combine like terms using the basic properties
of real numbers. Like terms have the same variables raised to the same powers.
To evaluate an algebraic expression, replace the variables in the expression with
numbers and follow the order of operations.
Problem
What is the value of the algebraic expression 3(4x 1 5y) 2 2(3x 2 7y) when
x 5 3 and y 5 22?
Simplify the algebraic expression using the basic properties of real numbers.
3(4x 1 5y) 2 2(3x 2 7y) 5 12x 1 15y 2 2(3x 2 7y)
Distributive Property for Addition
5 12x 1 15y 2 (6x 2 14y)
Distributive Property for Subtraction
5 12x 1 15y 2 6x 1 14y
Opposite of a Difference
5 12x 2 6x 1 15y 1 14y
Identify like terms.
5 (12 2 6)x 1 (15 1 14)y
Distributive Property
5 6x 1 29y
Combine like terms.
Evaluate the expression, replacing x with 3 and y with 22 in the simplified
expression.
6(3) 1 29(22)
5 18 2 58
5 240
Exercises
Simplify the algebraic expression. Then evaluate the simplified expression for
the given values of the variable.
4. (4x 1 1) 1 2x; x 5 3
5. 7(t 1 3) 2 11; t 5 4
6x 1 1; 19
7t 1 10; 38
6. 3y 1 4z 1 6y 2 9z; y 5 2, z 5 1
7. 2(u 1 v) 2 (u 2 v); u 5 8, v 5 23
9y 2 5z; 13
8. 5a2
u 1 3v; 21
9. 6p2 2 (3p2 1 2q2); p 5 1, q 5 5
3p2 2 2q2; 247
s
1
r
r
11. 2 1 3 2 4 1 5; r 5 21, s 5 0
1 5a 1 a 1 1; a 5 22
5a2 1 6a 1 1; 9
3
1
10. 4(m 1 n) 2 4(m 2 n); m 5 6, n 5 2
1
2m
r
4
1 n; 5
1
1 3s 1 15; 220
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Additional Vocabulary Support
Solving Equations
The column on the left shows the steps used to solve a problem with an equation.
Use the column on the left to answer each question in the column on the right.
Problem
1. Read the title of the Problem. What
Solve by Setting up and
Solving an Equation
Two planes leave San Antonio at the
same time. The northbound plane travels
70 mi/h faster than the southbound plane.
The planes are 1940 mi apart in 2 h. How
fast is the southbound plane flying?
Relate
process are you going to use to solve
the problem?
Answers may vary. Sample: setting up
and solving an equation
2. What is the formula that relates
total
distance the distance the
distance
southbound northbound
1
5 between the
plane travels plane travels
planes after
in 2 h
in 2 h
2h
Define
distance, rate, and time?
d 5 rt
3. Why can you represent the rate
Let x 5 the rate of the southbound plane.
Let x 1 70 5 the rate of the northbound
plane.
of the northbound plane with the
algebraic expression x 1 70?
It travels 70 mi/h faster than the
southbound plane.
Write
4. What does the expression 2x represent?
2x 1 2(x 1 70) 5 1940
2x 1 2x 1 140 5 1940
4x 1 140 5 1940
4x 1 140 2 140 5 1940 2 140
4x 5 1800
It represents the distance the
Dist.
Property
Combine
like terms.
Subtract.
Simplify.
Calculate
southbound plane travels in 2 h.
5. Why do you divide both sides by 4?
to isolate x
6. What does x represent?
4x
1800
4 5 4
the speed of the southbound plane
Divide.
x 5 450
Answer the question asked.
7. How can you find the speed of the
northbound plane?
The southbound plane is flying at
450 mi/h.
Add 70 mi/h to the speed of the
southbound plane.
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Think About a Plan
Solving Equations
Geometry The measure of the supplement of an angle is 208 more
than three times the measure of the original angle. Find the measures
of the angles.
Know
z
z
1808 .
1. The sum of the measures of the two angles is 2. What do you know about the supplemental angle?
It is 208 more than three times the measure of the original angle.
Need
3. To solve the problem, I need to define:
the measure of the original angle 5 x
the measure of the supplemental angle 5 3x 1 20
Plan
4. What equation can you use to find the measure of the original angle?
x 1 (3x 1 20) 5 180
5. Solve the equation.
x 5 40
6. What are the measures of the angles?
original angle: 408; supplemental angle: 1408
7. Are the solutions reasonable? Explain.
Yes; three times the measure of the first angle plus 208 is 1408; the sum of
the measures of the angles is 1808, so they are supplementary.
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Practice
Form G
Solving Equations
Solve each equation.
1. 7.2 1 c 5 19 11.8
2. 8.5 5 5p 1.7
d
3. 4 5 231 2124
4. s 2 31 5 20.6 51.6
Solve each equation. Check your answer.
5. 9(z 2 3) 5 12z 29
6. 7y 1 5 5 6y 1 11 6
7. 5w 1 8 2 12w 5 16 2 15w 1
8. 3(x 1 1) 5 2(x 1 11) 19
Write an equation to solve each problem.
9. Two brothers are saving money to buy tickets to a concert. Their combined
savings is $55. One brother has $15 more than the other. How much has each
saved? Variable may vary. Sample: s 1 s 1 15 5 55
10. Geometry The sides of a triangle are in the ratio 5 : 12 : 13. What is the length
of each side of the triangle if the perimeter of the triangle is 15 inches?
Variable may vary. Sample: 5x 1 12x 1 13x 5 15
11. What three consecutive numbers have a sum of 126?
Variable may vary. Sample: n 1 (n 1 1) 1 (n 1 2) 5 126
Determine whether the equation is always, sometimes, or never true.
12. 6(x 1 1) 5 2(5 1 3x) never
13. 3(y 1 3) 1 5y 5 4(2y 1 1) 1 5 always
Solve each formula for the indicated variable.
14. S 5 L(1 2 r), for r
r 5 1 2 SL
2 lh
15. A 5 lw 1 wh 1 lh, for w w 5 Al 1
h
Solve each equation for y.
4
16. 9 (y 1 3) 5 g y 5 94 g 2 3
18.
c (a 1 b)
17. a(y 1 c) 5 b(y 2 c) y 5 b 2 a , a u b
y13
2
3
t 5 t y 5 t 2 3, t u 0
19. 3y 2 yz 5 2z y 5 3 2z
2 z, z u 3
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Name
1-4
Class
Date
Practice (continued)
Form G
Solving Equations
Solve each equation.
20. 0.5(x 2 3) 1 (1.5 2 x) 5 5x 0
21. 1.2(x 1 5) 5 1.6(2x 1 5) 21
22. 0.5(c 1 2.8) 2 c 5 0.6c 1 0.3 1
u
u
u
23. 5 1 10 2 6 5 1 15
2
Solve each formula for the indicated variable.
p
24. V 5 3 r2h, for h h 5 3V2
πr
25. D 5 kA c
T2 2 T1
DL
d for T1 T1 5 T2 2 kA
L
Write an equation to solve each problem.
26. Two trains left a station at the same time. One traveled north at a certain
speed and the other traveled south at twice that speed. After 4 hours, the
trains were 600 miles apart. How fast was each train traveling?
Variable may vary. Sample: 4r 1 4(2r) 5 600
27. Geometry The sides of one cube are twice as long as the sides of a second
cube. What is the side length of each cube if the total volume of the cubes is
72 cm3? Variable may vary. Sample: s3 1 (2s)3 5 72
28. Error Analysis Brenna solved an equation
for m. Do you agree with her? Explain your
answer.
No; there is an m on both sides of the equation;
v2
mv1 = (m
M)v
mv
m +M
mv2 + M
Mv
v2
m=
v1
Mv
the correct result should be m 5 v1 2 2v2
Solve each problem.
29. You and your friend left a bus terminal at the same time and traveled in
opposite directions. Your bus was in heavy traffic and had to travel 20 miles
per hour slower than your friend’s bus. After 3 hours, the buses were 270 miles
apart. How fast was each bus going? Your bus: 35 mi/h; Your friend’s bus: 55 mi/h
30. Geometry The length of a rectangle is 5 centimeters greater than its width.
The perimeter is 58 centimeters. What are the dimensions of the rectangle? w 5 12 cm, l 5 17 cm
31. What four consecutive odd integers have a sum of 336? 81, 83, 85, 87
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Name
1-4
Class
Date
Practice
Form K
Solving Equations
Solve each equation.
1. 5x 1 4 5 2x 1 10 2
2. 10w 2 3 5 8w 1 5 4
To start, subtract 2x from each side.
To start, subtract 8w from each side.
3. 4(d 2 3) 5 2d 6
4. s 1 2 2 3s 2 16 5 0 27
Solve each equation. Check your answer.
5. 9(z 2 3) 5 12z 29
6. 7y 1 5 5 6y 1 11 6
7. 5w 1 8 2 12w 5 16 2 15w 1
8. 3(x 1 1) 5 2(x 1 11) 19
Write an equation to solve each problem.
9. Lisa and Beth have babysitting jobs. Lisa earns $30 per week and Beth earns
$25 per week. How many weeks will it take for them to earn a total of $275?
To start, record what you know.
Describe what you need to find.
Lisa earns $30 per week.
Beth earns $25 per week.
Total earned: $275
an equation to find the number of
weeks it takes to earn $275 together
Variable may vary. Sample: 30w 1 25w 5 275
10. The angles of a triangle are in the ratio 2 : 12 : 16. The sum of all the angles in a
triangle must equal 180 degrees. What is the degree measure of each angle of
the triangle? Let x 5 the common factor.
2x 1 12x 1 16x 5 180
11. What two consecutive numbers have a sum of 53?
Variable may vary. Sample: n 1 (n 1 1) 5 53
Determine whether the equation is always, sometimes, or never true.
12. 3(2x 2 4) 5 6(x 2 2)
always
13. 4(x 1 3) 5 2(2x 1 1)
never
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Practice (continued)
1-4
Form K
Solving Equations
Solve each formula for the indicated variable.
1
14. A 5 2bh, for b
b 5 2A
h
15. P 5 2w 1 2l, for w
1
16. A 5 2h(b1 1 b2), for h
h 5 b 2A
1 b
17. S 5 2prh 1 2pr2 , for h
1
w 5 P 22 2l
2pr2
h 5 S 22pr
2
Solve each equation for y.
18. ry 2 sy 5 t
y 5 r 2t s, r u s
3
19. 7(y 1 2) 5 g
y 5 73 g 2 2
y
20. m 1 3 5 n
y 5 m(n 2 3), m u 0
21.
3y 2 1
5z
2
2z 1 1
y5 3
Solve each equation.
22. (x 2 3) 2 2 5 6 2 2(x 1 1)
3
23. 4(a 1 2) 2 2a 5 10 1 3(a 2 3)
7
24. 2(2c 1 1) 2 c 5 213
25
25. 8u 1 2(u 2 10) 5 0
2
26. The first half of a play is 35 minutes longer than the second half of the play. If
the entire play is 155 minutes long, how long is the first half of the play? Write
an equation to solve the problem.
Variable may vary. Sample: (m 1 35) 1 m 5 155; m 5 60
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Standardized Test Prep
1-4
Solving Equations
Gridded Response
Solve each exercise and enter your answer in the grid provided.
1. A bookstore owner estimates that her weekly profits p can be described
by the equation p 5 8b 2 560, where b is the number of books sold that
week. Last week the store’s profit was $720. What is the number of books
sold?
2. What is the value of m in the equation 0.6m 2 0.2 5 3.7?
3. Three consecutive even integers have a sum of 168. What is the value of the
largest integer?
4. If 6(x 2 3) 2 2(x 2 2) 5 11, what is the value of x?
5. Your long distance service provider charges you $.06 per minute plus a
monthly access fee of $4.95. For referring a friend, you receive a $10 service
credit this month. If your long-distance bill is $7.85, how many long-distance
minutes did you use?
Answers
1.
–
160
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2.
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
6 . 5
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
3.
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
58
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
4.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
6 . 25
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
5.
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
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–
215
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
Name
1-4
Class
Date
Enrichment
Solving Equations
Equations can be subdivided into three distinct types:
a. conditional equations, or equations that are true for some values of x. For
example, the equation x 1 1 5 0 is true only for x 5 21.
b. identities, for which every possible value of the variable belongs to the
solution set. For example, the equation x 5 x is an identity, as it is true for all
values of x.
c. impossibilities, for which no possible values of the variable belong to the
solution set. For example, the equation x 5 x 1 1 is an impossibility, as it is
never true.
For each of the following equations, find the solution if it is a conditional
equation, or classify the equation as an identity or an impossibility.
1. x 1 (2x 2 4) 5 11 5
2. x 5 x 1 2 impossibility
3. x 1 (2x 2 1) 5 3x 2 1 identity
4. x 5 (2 1 2x) 2 x impossibility
5. (x 2 2) 1 (2x 1 4) 5 x 21
6. x 1 2 5 x 1 3 impossibility
7. 2x 5 3x 0
2
8. 2x 1 8 5 6 2 x 23
1
9. 2x 2 4 1 3x 5 8x 2 5 3
10. 2x 1 5 5 5 1 2x identity
11. 2(x 1 3) 5 5x 2 (3x 2 6) identity
12. x 1 3(x 1 3) 5 3(x 2 3) 218
13. (x 1 3) 1 (x 2 3) 5 3 3
2
14. (x 1 3) 2 (x 2 3) 5 3 impossibility
15. 2(x 1 5) 2 4 5 3(x 1 2) 2 1 1
16. 1 2 (x 2 5) 5 3 2 (x 2 5) impossibility
17. 4(x 2 1) 1 3 5 4x 2 (x 1 1) 0
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Name
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Reteaching
1-4
Solving Equations
To solve an equation that contains a variable, find all of the values of the variable
that make the equation true. Use the equality properties of real numbers and
inverse operations to rewrite the equation until the variable is alone on one side of
the equation. Whatever remains on the other side of the equation is the solution.
Subtraction Property of Equality
Addition Property of Equality
To isolate w on one side of the equation, add
To isolate 5z on one side of the equation,
subtract 3 from each side.
2
7 to each side.
w
2 7 5 11
2
17
5z 1 3 5 13
2 3 23
5z
5 10
17
w
5 18
2
Division Property of Equality
Multiplication Property of Equality
To isolate w on one side of the equation,
multiply each side by 2.
To isolate z on one side of the equation, divide
each side by 5.
w
5 18
2
32
5z
10
5 5 5
32
z52
w 36
Exercises
Solve each equation.
1. y 1 12 5 8 24
2. p 2 9 5 12 21
3. 23 1 r 5 20 23
4. 8 5 15 1 k 27
5. 9q 5 27 3
t
6. 6 5 24 224
d
7. 5 5 2 7 235
8. 49 5 10m 4.9
n
10. 8 1 19 5 3 2128
9. 3g 2 14 5 13 9
c
11. 27 5 25 2 4 2128
12. 12 5 2f 1 9 1.5
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1-4
Date
Reteaching (continued)
Solving Equations
To solve an equation for one of its variables, rewrite the equation as an equivalent
equation with the specified variable on one side of the equation by itself and an
expression not containing that variable on the other side.
Problem
ax 2 b
The equation 2
5 x 1 2b defines a relationship between a, b, and x. What is
x in terms of a and b?
Use the properties of equality and the properties of real numbers to rewrite the
equation as a sequence of equivalent equations.
ax 2 b
5 x 1 2b
2
2Q
ax 2 b
2 R 5 2(x 1 2b)
Multiply each side by 2.
ax 2 b 5 2(x 1 2b)
Simplify.
ax 2 b 5 2x 1 4b
Distributive Property
ax 2 2x 5 4b 1 b
Add and subtract to get terms with x on one side and terms
without x on the other side.
ax 2 2x 5 5b
Simplify.
x(a 2 2) 5 5b
Distributive Property
5b
x5a22
Divide each side by a 2 2.
The final form of the equation has x on the left side by itself and an expression not
containing x on the right side.
Exercises
Solve each equation for the indicated variable.
13. 3m 2 n 5 2m 1 n, for m m 5 2n
14. 2(u 1 3v) 5 w 2 5u, for u u 5 w 27 6v
b
15. ax 1 b 5 cx 1 d, for x x 5 da 2
2c
1 20
16. k (y 1 3z) 5 4(y 2 5), for y y 5 23kz
k24
1
17. 2 r 1 3s 5 1, for r r 5 2 2 6s
5
2
12 2 5g
18. 3 f 1 12 g 5 1 2 f g , for f f 5
8 1 12g
19.
3j 2 4k
x1k
3
5 4 , for x x 5 4
j
20.
a 2 3y
2 ab
1 4 5 a 1 y, for y y 5 a 1b4b
1 3
b
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Additional Vocabulary Support
Solving Inequalities
is greater than is greater than or equal to is less than is less than or equal to
S
L
R
K
To write an inequality from a sentence, first identify the operation and then identify
the inequality.
Example What inequality represents the sentence “6 more than a number is at least 20”?
“more than” means addition
“is at least” means is greater than or equal to
6 more than a number is at least 20
6 1 x $ 20
Underline the word or words that indicate an operation.
1. the product of 12 and a number
2. 8 less than a number
3. the difference between a number and 24
4. the sum of a number and 7
Circle the word phrase that identifies the inequality to use. Then write
the inequality that represents the sentence.
5. The product of 12 and a number is more than 190. 12x S 190
6. 8 less than a number is at least 34. x 2 8 L 34
7. The difference between a number and 24 is no more than 4. x 2 24 K 4
8. The sum of twice a number and 7 is less than 25. 2x 1 7 R 25
Some word phrases are very similar, but have different meanings.
Example Does the sentence indicate an operation or an inequality?
A number is four 5
greater than 15.
x
5 4
Four is 5
greater than a number.
1
15
operation
4
x
.
inequality
Does the sentence indicate an operation or an inequality?
9. 22 is 7 greater than a number.
10. A number is greater than 99.
inequality
operation
11. A number is 8 less than another number.
12. 50 is less than a number.
inequality
operation
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Think About a Plan
1-5
Solving Inequalities
Your math test scores are 68, 78, 90, and 91. What is the lowest score you can earn
on the next test and still achieve an average of at least 85?
Understanding the Problem
1. What information do you need to find an average of scores? How do you find
an average?
The sum of the scores and the total number of scores;
divide the sum of the scores by the total number of scores.
5
2. How many scores should you include in the average? ___________
z
z z
z
greater than or equal
to what score?
3. You want to achieve an average that is 85
Planning the Solution
4. Assign a variable, x.
x 5 the score on the next test
5. Write an expression for the sum of all of the scores, including the next test.
327 1 x
6. Write an expression for the average of all of the scores.
327 1 x
5
7. Write an inequality that can be used to determine the lowest score you can
earn on the next test and still achieve an average of at least 85.
327 1 x
5
L 85
Getting an Answer
8. Solve your inequality to find the lowest score you can earn on the next test and
still achieve an average of at least 85. What score do you need to earn? at least 98
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Date
Practice
Form G
Solving Inequalities
Write the inequality that represents the sentence.
1. Four less than a number is greater than 228. x 2 4 + 228
2. Twice a number is at least 15. 2x # 15
3. A number increased by 7 is less than 5. x 1 7 * 5
4. The quotient of a number and 8 is at most 26. x8 K 26
Solve each inequality. Graph the solution.
5. 3(x 1 1) 1 2 , 11 x * 2
3 2 1
0
1
2
6. 5t 2 2(t 1 2) $ 8 t # 4
1
1
8. 3(7a
3
7. 2f(2y 2 1) 1 yg # 5( y 1 3) y " 17
1
2
5
6
7
2 1) # 2a 1 7 a " 22
10. 22(w 2 7) 1 3 . w 2 1 w * 6
9. 5 2 2(n 1 2) # 4 1 n n # 21
0
4
3
19 20 21 22 23 24 25
14 15 16 17 18 19 20
3 2 1
2
3
3
4
5
6
7
8
9
Solve each problem by writing an inequality.
11. Geometry The length of a rectangular yard is 30 meters. The perimeter is at
most 90 meters. Describe the width of the yard. at most 15 m
12. Geometry A piece of rope 20 feet long is cut from a longer piece that is at least
32 feet long. The remainder is cut into four pieces of equal length. Describe
the length of each of the four pieces. at least 3 ft
13. A school principal estimates that no more than 6% of this year’s senior class
will graduate with honors. If 350 students graduate this year, how many will
graduate with honors? no more than 21 students
14. Two sisters drove 144 miles on a camping trip. They averaged at least 32 miles
per gallon on the trip. Describe the number of gallons of gas they used.
at most 4.5 gal
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Practice (continued)
Form G
Solving Inequalities
Is the inequality always, sometimes, or never true?
15. 3(2x 1 1) . 5x 2 (2 2 x) always true
16. 2(x 2 1) $ x 1 7 sometimes true
17. 7x 1 2 # 2(2x 2 4) 1 3x never true
18. 5(x 2 3) , 2(x 2 9) sometimes true
Solve each compound inequality. Graph the solution.
19. 3x . 26 and 2x , 6 x + 22 and x * 3
3 2 1
0
1
2
3
20. 4x $ 212 and 7x # 7 x # 23 and x " 1
3 2 1
0
1
2
3
21. 5x . 220 and 8x # 32 x + 24 and x " 4
6 4 2
0
2
4
6
22. 6x , 212 or 5x . 5 x * 22 or x + 1
3 2 1
0
1
2
3
23. 6x # 218 or 2x . 18 x " 23 or x + 9
6 3
3
6
9
12
24. 2x . 3 2 x or 2x , x 2 3 x + 1 or x * 23
4 3 2 1
0
1
2
0
Solve each problem by writing and solving a compound inequality.
25. A student believes she can earn between $5200 and $6250 from her summer
job. She knows that she will have to buy four new tires for her car at $90 each.
She estimates her other expenses while she is working at $660. How much can
the student save from her summer wages? between $4180 and $5230
26. Before a chemist can combine a solution with other liquids in a laboratory,
the temperature of the solution must be between 39°C and 52°C. The chemist
places the solution in a warmer that raises the temperature 6.5°C per hour. If
the temperature is originally 0°C, how long will it take to raise the temperature
to the necessary range of values? between 6 and 8 h
27. The Science Club advisor expects that between 42 and 49 students will attend
the next Science Club field trip. The school allows $5.50 per student for
sandwiches and drinks. What is the advisor’s budget for food for the trip?
between $231 and $269.50
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Name
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Date
Practice
1-5
Form K
Solving Inequalities
Write the inequality that represents the sentence.
1. Five less than a number is at least 228. x 2 5 L 228
2. The product of a number and four is at most 210. 4x K 210
3. Six more than a quotient of a number and three is greater than 14. x3 1 6 S 14
Solve each inequality. Graph the solution.
4. 5a 2 10 . 5 a S 3
5. 25 2 2y $ 33 y K 24
To start, add 10 to each side.
7 6 5 4 3 2 1
1
2
3
4
5
6
7
6. 22(n 1 2) 1 6 # 16 n L 27
7. 2(7a 1 1) . 2a 2 10 a S 21
10 9 8 7 6 5 4
4 3 2 1
0
1
2
Solve the following problem by writing an inequality.
8. The width of a rectangle is 4 cm less than the length. The perimeter is
at most 48 cm. What are the restrictions on the dimensions of the rectangle?
To start, record what you know.
width: length 2 4
perimeter: at most 48 cm
Describe what you need to find.
restrictions on the width and
w S 0, l 2 4 S 0
length of the rectangle
2(l 2 4) 1 2l K 48; 4 R l K 14; 0 R w K 10
The length is more than 4 cm and at most 14 cm.
The width is more than 0 cm and at most 10 cm.
Is the inequality always, sometimes, or never true?
10. 2x 1 8 # 2(x 1 1) never
9. 5(x 2 2) $ 2x 1 1 sometimes
11. 6x 1 1 , 3(2x 2 4) never
12. 2(3x 1 3) . 2(3x 1 1) always
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Practice (continued)
1-5
Form K
Solving Inequalities
Solve each compound inequality. Graph the solution.
13. 2x . 24 and 4x , 12 x S 22 and x R 3
To start, simplify each inequality.
x . 22 and x , 3
2 1
0
1
2
3
4
5 4 3 2 1
0
1
4
5
6
5 4 3 2 1
0
1
5 4 3 2 1
0
1
2 1
3
4
Remember, “and” means that a solution makes BOTH inequalities true.
14. 3x $ 212 and 5x # 5 x L 24 and x K 1
15. 6x . 6 and 9x # 45 x S 1 and x K 5
0
1
2
3
Solve each compound inequality. Graph the solution.
16. 3x , 29 or 8x . 28 x R 23 or x S 21
To start, simplify each inequality.
x , 23 or x . 21
Remember, “or” means that a solution makes EITHER inequality true.
17. 7x # 228 or 2x . 22 x K 24 or x S 21
18. 3x . 3
or
5x , 2x 2 3 x S 1 or x R 21
0
1
Write an inequality to represent each sentence.
19. The average of Shondra’s test scores in Physics is between 88 and 93.
88 R x R 93 OR x S 88 and x R 93
20. The Morgans are buying a new house. They want to buy either a house more
the 75 years old or a house less than 10 years old. x S 75 or x R 10
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2
Name
Class
1-5
Date
Standardized Test Prep
Solving Inequalities
Multiple Choice
For Exercises 1–5, choose the correct letter.
1. What is the solution of 4t 2 (3 1 t) # t 1 7? B
t # 52
t#5
t#2
t#1
r , 24
r . 24
2. What is the solution of 217 2 2r , 3(r 1 1)? I
r.4
r . 220
3
3. Which graph best represents the solution of 4(m 1 4) . m 1 3? A
1
3 2 1
0
1
2
3
1
3 2 1
0
1
2
3
8
9
10 11 12
1
3 2 1
0
1
6
7
4. What is the solution of the compound inequality 4x , 28 or 9x . 18?
x , 2 or x . 22
x.2
x , 22
x , 22 or x . 2
2
3
I
5. What is the solution of the compound inequality 22x # 6 and 23x . 227? C
x # 23 and x . 9
x $ 23 and x , 9
x $ 3 and x , 29
x # 3 and x . 29
Short Response
6. Geometry The lengths of the sides of a triangle are in the ratio 3 : 4 : 5.
Describe the length of the longest side if the perimeter is not more than 72 in.
Solve the inequality 3x 1 4x 1 5x " 72 or 12x " 72 or x " 6. The longest side,
5x, is not more than 5 ∙ 6 5 30 in. [2] All student calculations are correct.
[1] minor incorrect calculation [0] no answer given
7. Between 8.5% and 9.4% of the city’s population uses the municipal transit
system daily. According to the latest census, the city’s population is 785,000.
How many people use the transit system daily?
[2] between 66,725 and 73,790 people [1] minor incorrect miscalculation of
one or both values in the range [0] no answer given
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Name
Class
Date
Enrichment
1-5
Solving Inequalities
F
R
I
E
D
R
I
C
H
1
2
3
4
5
6
7
8
9
10
B
E
S
S
E
L
11
12
13
14
15
16
He was the first person to measure the distance to a star successfully. He also
discovered that certain mathematical functions play a key role in models of
physical phenomena. These functions were named after him.
To find his name, solve each of the following inequalities.Then use the solutions and
the table below to determine the position in which to write the associated letter.
Solutions
Position
Solutions
Position
x1
1
1
x
2
6
x4
11
x2
2
x5
7
x0
12
x 1
3
5
x
2
8
x 2
4
x0
9
4
x
3
x 1
2
all real numbers
5
10
x2
15
x3
16
B
2(x 1 1) 2 (2 2 x) , 12 x R 4
C
3x 1 6 , 16 2 x x R
Solutions
5
2
D 3x 1 5 1 3(x 1 5) . 6x 1 15 all real numbers
E
8x 2 2 2 6(x 2 3) , 16 x R 0
E
3(x 1 2) 1 2x . 24 x S 22
E
(3 1 x) 1 (3 2 3x) , x x S 2
F
3x 2 5 1 (x 1 9) . 8 x S 1
H 5x 2 3(x 2 1) . 3 x S 0
I
5x 2 3 2 3 Q x 1 73 R . 0 x S 5
I
2(x 1 4) 2 (5 2 x) . 0 x S 21
L
(x 1 1) 2 (1 2 x) , (x 2 1) 2 (2 2 2x) x S 3
R
7x 2 5 2 (21 2 x) . 0 x S
R
2x 1 5 1 2(x 1 5) , 23 x R 2
S
2(x 2 1) , 2 1 2(1 1 7x) x S 2 12
S
3 2 x , 5 1 2(x 1 1) x S 2 43
1
2
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13
14
Name
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Date
Reteaching
Solving Inequalities
As with an equation, the solutions of an inequality are numbers that make it true.
The procedure for solving a linear inequality is much like the one for solving linear
equations. To isolate the variable on one side of the inequality, perform the same
algebraic operation on each side of the inequality symbol.
The Addition and Subtraction Properties of Inequality state that adding or
subtracting the same number from both sides of the inequality does not change
the inequality.
If a , b, then a 1 c , b 1 c.
If a , b, then a 2 c , b 2 c.
The Multiplication and Division Properties of Inequality state that multiplying or
dividing both sides of the inequality by the same positive number does not change
the inequality.
If a , b and c . 0, then ac , bc.
a
If a , b and c . 0, then c , bc .
Problem
What is the solution of 3(x 1 2) 2 5 # 21 2 x? Graph the solution.
Justify each line in the solution by naming one of the properties of inequalities.
3x 1 6 2 5 # 21 2 x
3x 1 1 # 21 2 x
4x 1 1 # 21
4x # 20
x#5
Distributive Property
Simplify.
Addition Property of Inequality
Subtraction Property of Inequality
Division Property of Inequality
To graph the solution, locate the boundary point. Plot a point at x 5 5. Because the
inequality is “less than or equal to,” the boundary point is part of the solution set.
Therefore, use a closed dot to graph the boundary point. Shade the number line to
the left of the boundary point because the inequality is “less than.”
Graph the solution on a number line.
2
3
4
5
6
7
8
Exercises
Solve each inequality. Graph the solution.
1. 2x 1 4(x 2 2) . 4 x + 2
3 2 1
0
1
2
2. 4 2 (2x 2 4) $ 5 2 (4x 1 3) x # 23
5 4 3 2 1
3
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1
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Date
Reteaching (continued)
Solving Inequalities
The procedure for solving an inequality is similar to the procedure for solving an
equation but with one important exception.
The Multiplication and Division Properties of Equality also state that, when you
multiply or divide each side of an inequality by a negative number, you must
reverse the inequality symbol.
If a , b and c , 0, then ac . bc.
a
If a , b and c , 0, then c . bc .
Problem
What is the solution of 2x 2 3(x 2 1) , x 1 5? Graph the solution.
Justify each line in the solution by naming one of the properties of inequalities.
2x 2 3(x 2 1) , x 1 5
2x 2 3x 1 3 , x 1 5
2x 1 3 , x 1 5
22x , 2
x . 21
Distributive Property
Simplify.
Subtraction Property of Inequality
Division Property of Inequality
The direction of the inequality changed in the last step because we divided both
sides of the inequality by a negative number.
Graph the solution on a number line.
1
3 2 1
0
1
2
3
Exercises
Solve each inequality.
3. x 2 1 # 24(22 2 x) x # 23
4. 7 2 7(x 2 7) . 24 1 5x x * 5
5. 7(x 1 4) 2 13 $ 12 1 13(3 1 x) x " 26
6. 4x 2 1 , 6x 2 5 x + 2
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Additional Vocabulary Support
Absolute Value Equations and Inequalities
Concept List
»x… R 3
»x… S 3
»x… K 3
»x… L 3
»x… 5 3
Choose the concept from the list below that best represents the item in
each box.
1. numbers more than
2. numbers three units
3 units away from zero
3.
away from zero or more
than three units away
from 0
»x… S 3
⫺6 ⫺4 ⫺2
0
2
4
6
»x… R 3
»x… L 3
5.
4. numbers less than
6. numbers 3 units away
3 units away from zero
from zero
⫺6 ⫺4 ⫺2
»x… R 3
0
2
4
6
7.
8. numbers three units
⫺6 ⫺4 ⫺2
»x… L 3
»x… 5 3
»x… S 3
0
2
4
6
away from zero or less
than three units away
from 0
9.
⫺6 ⫺4 ⫺2
»x… K 3
»x… K 3
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2
4
6
Name
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Date
Think About a Plan
Absolute Value Equations and Inequalities
Write an absolute value inequality to represent the situation.
Cooking Suppose you used an oven thermometer while baking and discovered
that the oven temperature varied between 15 and 25 degrees from the setting. If
your oven is set to 350°, let t be the actual temperature.
1. How do you have to think to solve this problem?
If I subtract the set temperature from the real temperature, the result
should be between 25° and 5°.
2. Write a compound inequality that represents the actual oven temperature t.
345 K t K 355
3. It often helps to draw a picture. Graph this compound inequality on a
number line.
345
350
355
t
4. What is the definition of tolerance?
Tolerance is the difference between a desired measurement and its
maximum and minimum allowable values. It equals half of the difference
between the maximum and minimum values.
5°
5. What is the tolerance of the oven? _________
6. Use the tolerance to write an inequality without absolute values.
25 K t 2 350 K 5
7. Rewrite the inequality as an absolute value inequality.
»t 2 350… K 5
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Practice
Form G
Absolute Value Equations and Inequalities
Solve each equation. Check your answers.
1. u23x u 5 18 x 5 6 or x 5 26
2. u 5y u 5 35 y 5 7 or y 5 27
3. u t 1 5 u 5 8 t 5 3 or t 5 213
4. 3 u z 1 7 u 5 12 z 5 23 or z 5 211
5. u 2x 2 1 u 5 5 x 5 3 or x 5 22
6. u 4 2 2y u 1 5 5 9 y 5 4 or y 5 0
Solve each equation. Check for extraneous solutions.
7. u x 1 5 u 5 3x 2 7 x 5 6
8. u 2t 2 3 u 5 3t 2 2 t 5 1
5
9. u 4w 1 3 u 2 2 5 5 w 5 1 or w 5 22
10. 2 u z 1 1 u 2 3 5 z 2 2
z 5 21
Solve each inequality. Graph the solution.
11. 5 u y 1 3 u , 15 26 R y R 0
⫺8 ⫺6 ⫺4 ⫺2
0
2
12. u 2t 2 3 u # 5 21 K t K 4
⫺2 ⫺1
4
1
14. 2 u 2w
13. u 4b u 2 3 . 9 b < 23 or b > 3
⫺9 ⫺6 ⫺3
0
3
6
0
1
2
1
2
3
4
2 1 u 2 3 $ 1 w K 272 or w L
⫺6 ⫺4 ⫺2
9
1
15. 2 u 4x 1 1 u 2 5 # 1 21 K x K 2
⫺3 ⫺2 ⫺1
0
0
2
4
9
2
6
2
16. u 3z 2 2 u 1 5 . 9 z R 23 or z S 2
⫺3 ⫺2 ⫺1
3
0
1
2
3
Write each compound inequality as an absolute value inequality.
17. 27.3 # a # 7.3 »a» K 7.3
18. 11 # m # 19 »m 2 15» K 4
19. 28.6 # F # 29.2 »F 2 28.9» K 0.3
20. 0.0015 # t # 0.0018 »t 2 0.00165» K 0.00015
Write an absolute value equation or inequality to describe each graph.
21.
⫺6 ⫺4 ⫺2
0
2
4
22.
6
»x» 5 6
⫺3 ⫺2 ⫺1
0
1
»x» S 2.5
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3
Name
1-6
Class
Practice
Date
Form G
(continued)
Absolute Value Equations and Inequalities
Solve each equation.
23. 3 u 2x 1 5 u 5 9x 2 6 x 5 7
3
24. u 4 2 3m u 5 m 1 10 m 5 7 or m 5 2 2
25. 2 u 4w 2 5 u 5 12w 2 18 w 5 2
3
26. 4 u8t 2 12 u 5 6(t 2 1) t 5 5
4
27. u 5p 1 3 u 2 4 5 2p p 5 13 or p 5 21
28. u 7y 2 3 u 1 1 5 0 no solution
Solve each inequality. Graph the solution.
29. 23 u 2t 1 1 u , 9 all real numbers
30. u 22x 1 4 u $ 4 x K 0 or x L 4
0
⫺3 ⫺2 ⫺1
y 1 2
31. ` 3 ` 2 1 , 2
1
1
32. 7 u 4z 1 5 u 1 2 . 5
⫺16 ⫺12 ⫺8 ⫺4
0
2
⫺1
3
211 R y R 7
4
0
1
⫺9 ⫺6 ⫺3
8
2
0
3
3
4
5
z R 213
2 or z S 4
6
9
Write an absolute value inequality to represent each situation.
33. To become a potential volunteer donor listed on the National Marrow Donor
Program registry, a person must be between the ages of 18 and 60. Let a
represent the age of a person on the registry. »a 2 39» K 21
34. Two friends are hiking in Death Valley National Park. Their elevation ranges
from 228 ft below sea level at Badwater to 690 ft above sea level at Zabriskie
»x 2 231» K 459
Point. Let x represent their elevation.
35. The outdoor temperature ranged between 37°F and 62°F in a 24-hour period.
Let t represent the temperature during this time period. »t 2 49.5» K 12.5
The diameter of a ball bearing in a wheel assembly must be between 1.758 cm and
1.764 cm.
36. What is the tolerance? 0.003 cm
37. What absolute value inequality represents the diameter of the ball bearing?
Let d represent the diameter in cm. »d 2 1.761» R 0.003
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Practice
Form K
Absolute Value Equations and Inequalities
Solve each equation. Check your answers. Graph the solution.
1. u 22x u 5 12 x 5 6 or x 5 26
⫺6 ⫺4 ⫺2
0
2
4
2. u 7y u 5 28 y 5 4 or y 5 24
⫺6 ⫺4 ⫺2
6
0
2
4
6
Solve each equation. Check your answers.
3. u t 1 7 u 5 1 t 5 26 or t 5 28
t 1 7 5 1 or t 1 7 5 21
To start, rewrite the absolute value
equation as two equations.
4. 4 u z 1 1 u 5 24 z 5 5 or z 5 27
5. u 2w 1 1 u 5 5 w 5 2 or w 5 23
6. u 2x 2 2 u 5 4 x 5 3 or x 5 21
7. u 5 2 2y u 1 3 5 8 y 5 0 or y 5 5
Solve each equation. Check for extraneous solutions.
8. u 2z 2 9 u 5 z 2 3 z 5 6 or z 5 4
To start, rewrite as two equations.
2z 2 9 5 z 2 3 or 2z 2 9 5 2(z 2 3)
9. u x 1 6 u 5 2x 2 3 x 5 9
10. u 2t 2 5 u 5 3t 2 10 t 5 5
11. 2 u 4y 1 1 u 5 4y 1 10 y 5 2 or y 5 21
12. u w 1 1 u 2 5 5 2w w 5 22
Write an absolute value equation to describe each graph.
13.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
1
2
3
14.
4
Variable may vary. Sample: »x… 5 3
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
Variable may vary. Sample: »x 1 3… 5 1
Is the absolute value equation always, sometimes, or never true? Explain.
15. u w u 5 22
16. u z u 1 1 5 z 1 1
never; the distance between a number
w and 0 is never negative
sometimes; the equation is true only
for z L 0
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Practice (continued)
Form K
Absolute Value Equations and Inequalities
Solve each inequality. Graph the solution.
17. 2 u x 1 5 u # 8
29 K x K 21
⫺9 ⫺8 ⫺7 ⫺6 ⫺5⫺4 ⫺3 ⫺2 ⫺1
ux 1 5u # 4
To start, divide each side by 2.
x 1 5 is greater than or equal to 24 and less than or equal to 4.
18. u x 1 1 u 2 3 # 1 25 K x K 3
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
1
2
0
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
3
0
1
2
3
21. u y 2 3 u 1 2 $ 4 y K 1 or y L 5
0
1
22. u 2t 1 2 u 1 5 # 9 23 K t K 1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
z R 23 or z S 1
19. u 2z 1 2 u 2 1 . 3
20. 2 u w 1 3 u 2 1 , 1 24 R w R 22
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
24 # x 1 5 # 4
1
2
3
4
5
6
23. u 2s 1 1 u . 3 s R 22 or s S 1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
1
1
2
3
Write each compound inequality as an absolute value inequality.
24. 1.2 # a # 2.4 »a 2 1.8… K 0.6
To start, find the tolerance.
2.4 2 1.2
1.2
5 2 5 0.6
2
25. 22 , x , 4 »x 2 1… R 3
26. 1 # m # 2 »m 2 1.5… K 0.5
27. 20 # y # 30 »y 2 25… K 5
28. 23 , t , 17 »t 2 7… R 10
Write an absolute value inequality to represent each situation.
29. In order to enter the kiddie rides at the amusement park, a child must be
between the ages of 4 and 10. Let a represent the age of a child who may go
on the kiddie rides.
»a 2 7… K 3
30. The outdoor temperature ranged between 42°F and 60°F in a 24-hour period.
Let t represent the temperature during this time period.
»t 2 51… K 9
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Standardized Test Prep
Absolute Value Equations and Inequalities
Multiple Choice
For Exercises 1–5, choose the correct letter.
1. What is the solution of u 5t 2 3 u 5 8? C
t 5 8 or t 5 28
t 5 11
5 or t 5 21
t 5 1 or t 5 211
5
t 5 5 or t 5 23
8
2. What is the solution of u 3z 2 2 u # 8? F
10
22 # z # 3
10
23 #z#2
8
10
z # 22 or z $ 3
10
z # 2 3 or z $ 2
1
3. What is the solution of 2 u 2x 1 3 u 2 1 . 1? B
7
22 , x , 12
7
x , 22 or x . 12
7
x . 2 or x , 212
7
x , 12 or x . 22
4. Which absolute value inequality is equivalent to the compound inequality
23 # T # 45? I
u T 2 11 u # 34
u T 2 45 u # 22
u T 2 24 u # 1
u T 2 34 u # 11
5. Which is the correct graph for the solution of u 2b 1 1 u 2 3 # 2? C
⫺3 ⫺2 ⫺1
0
1
2
3
⫺3 ⫺2 ⫺1
0
1
2
3
⫺3 ⫺2 ⫺1
0
1
2
3
⫺3 ⫺2 ⫺1
0
1
2
3
Short Response
6. An employee’s monthly earnings at an electronics store are based on a salary
plus commissions on her sales. Her earnings can range from $2500 to $3200,
depending on her commission. Write a compound inequality to describe
E, the amount of her monthly earnings. Then rewrite your inequality as an
absolute value inequality.
2500 K E K 3200,»E 2 2850… K 350
[2] correct compound inequality and correct absolute value inequality
[1] incorrect compound inequality OR incorrect absolute value inequality
[0] no answer given
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Enrichment
Absolute Value Equations and Inequalities
When is the distance from six to ten less than the distance from one to two? When
the distance is traveled on a word ladder! A word ladder is a sequence of words in
which only one letter in each word changes. To find a word ladder beginning with
the word ONE and ending with the word TWO, solve each of the following
absolute value inequalities. Write the solutions in the form a , x , b, where a
and b are integers.
Associated with each inequality is a pair of letters. Fill in the word ladder by
placing the first letter of the pair on the line numbered by a and the second letter
on the line numbered by b.
O ____
W ____
E
____
3
2
1
O ____
W ____
L
____
6
5
4
L
O ____
I ____
____
9
8
7
I ____
L
A ____
____
10 11 12
I ____
R
A ____
____
13 14 15
I ____
R
F ____
____
16 17 18
F ____
I ____
N
____
19 20 21
T ____
I ____
N
____
22 23 24
N
T ____
O ____
____
27
26
25
O
T ____
O ____
____
30
28 29
O
N
E
OO
u 3x 2 51 u , 39 4 R x R 30
WL
3 , 10 2 u 11 2 2x u 2 R x R 9
RI
u 95 2 5x u 1 3 , 23 15 R x R 23
EF
u 4x 2 38 u 2 14 , 12 3 R x R 16
IO
u x 2 20 u , 9 11 R x R 29
II
2 , 5 2 u x 2 17 u 14 R x R 20
WO
u 93 2 6x u , 63 5 R x R 26
LL
u 2x 2 18 u 1 4 , 10 6 R x R 12
II
u 8x 2 100 u , 36 8 R x R 17
AA
26 , 2u 46 2 4x u 10 R x R 13
RF
u 4x 2 74 u 1 7 , 9 18 R x R 19
NN
5 , 20 2 u 225 2 10x u 21 R x R 24
TT
3 1 u 106 2 4x u , 9 25 R x R 28
ON
20 , 70 2 u 5x 2 85 u 7 R x R 27
OT
u 92 2 8x u 2 40 , 44 1 R x R 22
T
W
O
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Reteaching
1-6
Absolute Value Equations and Inequalities
Solving absolute value equations require solving two equations separately. Recall
that for a real number x, u x u is the distance from zero to x on the number line. The
equation u x u 5 p means that either x 5 p or x 5 2p because both are p units
from 0.
Problem
What is the solution set for the equation u 5x 1 1 u 2 3 5 4?
The first step in solving an absolute value equation is to isolate the absolute value
on one side of the equal sign.
u 5x 1 1 u 2 3 5 4
u 5x 1 1 u 2 3 1 3 5 4 1 3
Add 3 to each side.
u 5x 1 1 u 5 7
Simplify.
Next, rewrite the absolute value as two equations and solve each of them
separately.
5x 1 1 5 7
5x 5 6
x5
6
5
or
or
5x 1 1 5 27
5x 5 28
Definition of absolute value
or
285
Division Property of Equality
x5
Addition Property of Equality
Notice that the same operations are performed in the same order on each of the
two equations. However, do not try to “simplify” the process by solving a single
equation. This leads to errors.
The solutions are x 5 65 or x 5 285 . Check each solution in the original equation:
Check
6
P5 ? 5 1 1P 2 3 5 4
u6 1 1u 2 3 5 4
4 5 43
8
P 5 ? Q25 R 1 1 P 2 3 5 4
u 28 1 1 u 2 3 5 4
4 5 43
Exercises
Solve each absolute value equation. Check your work.
1. u 2x 2 3 u 2 4 5 3 x 5 22 or x 5 5
2. u 3x 2 6 u 1 1 5 13 x 5 6 or x 5 22
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Reteaching (continued)
Absolute Value Equations and Inequalities
To solve an absolute value inequality, keep in mind that u x u is the distance from
zero to x on the number line. So, if u x u , p, then x is less than p units from 0, so
u x u , p 1 2p , x , p.
And, if u x u . p, then x is greater than p units from 0, so
u x u . p 1 x , 2p or x . p.
In this case, we need to rewrite the absolute value inequality as two separate
inequalities. Do not try to combine them into one inequality.
Problem
What is the solution set for the inequality u 2x 1 3 u . 11?
Because the inequality is ., use u x u . p 1 x , 2p or x . p.
Begin by rewriting the absolute value as two equations and solve each of them
separately.
2x 1 3 , 211
or
2x 1 3 . 11
2x , 214
or
2x . 8
x . 27
or
x.4
Rewrite as a compound inequality.
Subtract 3 from each side.
Divide each side by 2.
The solution set is x , 27 or x . 4.
Exercises
Complete the steps to solve the inequality P 2x 2 4 P K 3 .
3.
4.
5.
23
u
1
u
2
u
#
x
224
#
#
x
2
#
#
x
#
3
u
7
u
14
u
Rewrite as a compound inequality.
u
Add 4 to each part.
u
Multiply each part by 2 .
6. What is the solution? 2 K x K 14
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Chapter 1 Quiz 1
Form G
Lessons 1-1 through 1-3
Do you know HOW?
1. Describe a rule for the pattern.
Answers may vary. Sample: Stack one
triangle on top of the leftmost triangle;
add two triangles to the right of the
rightmost triangle.
3
18
2. Name all the integers in the list: 0, 22, 5 , p, !7 , 121, !9 ,2 6 .
0, 22, 121, !9,218
6
Name the property of real numbers illustrated by each equation.
4. 3 1 p 5 p 1 3
Commutative Property of Addition
3. 2(3x 2 y) 5 6x 2 2y
Distributive Property
Write an algebraic expression to model the word phrase.
5. the sum of g and the quotient of 3 and h
g1
6. six times the difference of x and 22
6(x 2 22)
3
h
Evaluate the expression for the given value of the variable.
7. 12x 2 9(x 2 1); x 5 2 15
8.
t(2t 1 3)
1
t 1 6 ; t 5 22 2
Do you UNDERSTAND?
9. Writing How can you use a graph to find a pattern?
Answers may vary. Sample: Choose some points on the graph; use these points to
make a table of input and output values; look for a pattern in the process column.
10. Vocabulary What is another name for an additive inverse? opposite
11. Reasoning Is there a Multiplication Property of Closure that applies to
irrational numbers? Justify your answer.
No. Answers may vary. Sample: !2 ? !2 5 2 , which is not irrational.
12. Compare and Contrast What is the difference between simplifying an
expression and evaluating an expression?
Answers may vary. Sample: Simplifying an expression is rewriting it using the properties
of real numbers and combining like terms, resulting in a simpler expression. Evaluating
an expression is substituting values for the variables, resulting in a numeric value.
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Date
Chapter 1 Quiz 2
Form G
Lessons 1-4 through 1-6
Do you know HOW?
Solve each equation. Check your answer.
1
2. 2 (3 2 w) 5 w 1 9 w 5 25
1. 3(t 2 2) 5 2t 1 11 t 5 17
3. Write an equation to solve the problem. Then solve the equation.
Maria is 3 years older than her brother, Luis. Luis is 2 years older than their
younger sister, Karla. The sum of their ages is 55. How old are the three siblings?
Let L be Luis’ age: (L 1 3) 1 L 1 (L 2 2) 5 55; Maria is 21, Luis is 18, Karla is 16
Solve each inequality. Graph the solution.
5. 26a # 18 and 5 2 3a . 2 23 " a * 1
4. 2(3 2 2n) , 2 2 (n 1 5) n + 3
0
1
2
3
4
5
⫺4 ⫺3 ⫺2 ⫺1
6
0
1
2
6. The city’s registrar of voters estimates that not less than 48% and no more
than 63% of voters will vote in the next mayoral election. Currently, there are
238,000 registered voters in the city. How many people will vote in the next
election? between 114,240 and 149,940, inclusive
Solve each equation. Check for extraneous solutions.
7. |2p 2 7| 1 6 5 11 p 5 1 or p 5 6
9. Solve the inequality
8. |m 2 2| 5 2m 1 5 m 5 21
u t 22 3 u 1 1 . 4 and graph the solution.
t * 23 or t + 9
⫺6 ⫺3
0
3
6
9
12
10. In a chemical process, between 77 g and 85 g of carbon is added to a mixture.
Let C be the amount of carbon. Write an absolute value inequality describing
the amount of carbon added. |C 2 81| * 4
Do you UNDERSTAND?
11. Writing Suppose the sum of three consecutive even integers is given. How
can you find the three numbers?
Answers may vary. Sample: Let n be the first integer. Even integers differ by 2, so the next
one is n 1 2 and the third is n 1 4. Therefore, n 1 (n 1 2) 1 (n 1 4) 5 the given sum. Solve
the equation for n to get the first number. The other two numbers are n 1 2 and n 1 4.
12. Compare and Contrast How do the solutions to |x| # 1, |x| # 0, and |x| # 21
differ? Answers may vary. Sample: |x| " 1 has an infinite number of solutions,
21 " x " 1; |x| " 0 has a single unique solution, x 5 0; and |x| " 21 has no solution.
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Chapter 1 Test
Form G
Do you know HOW?
Identify a pattern and find the next three numbers in the pattern.
1. 24, 21, 2, 5, . . .
2. 3, 21, 147, 1029, . . .
Each term is 3 more than the previous
Each term is 7 times more than the
term; 8, 11, 14
previous term; 7203; 50,421; 352,947
3. Justify each step by naming the property used.
a.
4 Q 23 ? 14 R 5 4 Q 14 ? (23) R
Commutative Property of Multiplication
b.
1
5 Q 4 ? 4 R ? (23) Associative Property of Multiplication
c.
5 1 ? (23)
Inverse Property of Multiplication
d.
5 23
Identity Property of Multiplication
Evaluate the expression for the given value of the variable.
4. 2a2 1 4a 2 17; a 5 5 222
5.
6(s 2 2) 2 4(s 1 1)
; s 5 3 21
3s 1 1
6. The expression 19.95 1 0.02x models the daily cost in dollars of renting a car.
In the expression, x represents the number of miles the car is driven. What is
the cost of renting a car for a day when the car is driven 50 miles? $20.95
Solve each equation.
7. 3r 1 3.7 5 5r 2 2.5 3.1
8. 3(5t 1 2) 5 36 2
Solve each equation for x. State any restrictions on the variables.
3t
9. tx 2 ux 5 3t x 5 t 2 u , t u u
10.
x23
6 1 3 5 a x 5 6a 2 15
Solve each formula for the indicated variable.
p
12. P 5 2/ 1 2w, for / < 5 2 2 w or < 5 P 22 2w
1
11. R 5 2 (r1 1 r2), for r2 r2 5 2R 2 r1
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Chapter 1 Test (continued)
Form G
Write an equation and solve the problem.
13. Two buses leave Dallas at the same time and travel in opposite directions. One
bus averages 58 mi/h, and the other bus averages 52 mi/h. When will they be
363 mi apart? 3 h 18 min later
Solve each inequality. Graph the solution.
3
15. 4a . 3(a 1 1) 2 Q 7 2 2a R a R 8
14. 3m 1 7 $ 4 m L 21
⫺3 ⫺2 ⫺1
0
1
2
3
5
6
7
8
9
10 11
Solve each compound inequality. Graph the solutions.
16. 3x 2 1 # 5 or 2x 2 4 $ x
17. 23t # 12 and 22t . 26
x K 2 or x L 4
0
1
2
3
24 K t R 3
4
5
⫺6 ⫺4 ⫺2
6
0
2
4
6
Solve each equation. Check for extraneous solutions.
18. u 2x 1 3 u 5 5 x 5 1 or x 5 24
19. u x 1 6 u 5 2x x 5 6
20. The temperature T of a refrigerator is at least 35°F and at most 41°F. Write a
compound inequality and an absolute value inequality for the temperature of
the refrigerator. 35 " T " 41; |T 2 38| " 3
Do you UNDERSTAND?
21. Open-Ended There is no Closure Property of Division that applies to integers.
For example, 2 4 3 is not an integer. What is another example of a set of
real numbers that does not have a Closure Property for one of the basic
operations? Give a specific example to illustrate your claim.
Answers may vary. Sample: Closure Property of Multiplication for negative numbers;
(22) ? (24) 5 8, which is not a negative number.
22. Reasoning Explain in words why |x| , 0 has no solution.
Answers may vary. Sample: |x| * 0 represents the set of numbers x that are fewer than
0 units from 0 on the number line. Since there are no numbers less than 0 units from 0
on the number line, the inequality |x| * 0 has no solution.
23. Writing Explain the difference between the solution(s) to an equation and
the solutions to an inequality.
Answers may vary. Sample: Equations generally have a finite (or countable) number of
solutions, whereas inequalities typically have an infinite number of solutions.
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Date
Chapter 1 Quiz 1
Form K
Lessons 1–1 through 1–3
Do you know HOW?
1. Describe a rule for the pattern.
Answers may vary. Sample: Start with a
column of two circles. Add one circle to
the right of the top row; add two circles to
the right of the bottom row.
7
6
2. Identify the integers in the list: 21, 9, 0, p, !25, 81, !5, 22. 21, 0, !25, 81, 2 62
Name the property of real numbers illustrated by each equation.
3. 3(2 ? 5) 5 (3 ? 2) ? 5
4. 5(x 1 2) 5 5 ? x 1 5 ? 2
Associative Property of Multiplication
Distributive Property
Write an algebraic expression to model the word phrase.
5. the sum of y and the product of 7 and x y 1 7x
6. eight more than the quotient of t and 2 2t 1 8
Evaluate the expression for the given value of the variable.
7. 2b 1 6(b 2 4); b 5 8 40
8. x 1 2(x 2 1); x 5 24 214
Do you UNDERSTAND?
9. Writing Give an example of a number that is an irrational number. Explain
why it is irrational.
Answers may vary. Sample: π ; π is an irrational number because 3.141592 . . . neither
terminates nor repeats; π cannot be written as a quotient of integers.
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Chapter 1 Quiz 2
Form K
Lessons 1–4 through 1–6
Do you know HOW?
Solve each equation. Check your answer.
1. 2(x 2 1) 5 4x 2 12 5
2. 3(2 2 2y) 5 y 2 8 2
3. Write an equation to solve the problem. Then solve the equation.
Brandon has 3 more dollars than Mark. Ashley has 10 dollars less than Mark.
They have 35 dollars together. How many dollars does each person have?
Variable may vary. Sample: Let d be the number of dollars Mark has:
(d 1 3) 1 d 1 (d 2 10) 5 35; Brandon has $17, Mark has $14, Ashley has $4.
Solve each inequality. Graph the solution.
4. 18 2 2a $ 26 a K 24
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
5. 5x , 215 or 3x . 23 x R 23 or x S 21
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
0
1
Solve each equation. Check for extraneous solutions.
7
7. u 2x 1 2 u 5 x 1 5 x 5 3 or x 5 23
6. 3 u z 1 1 u 5 21 z 5 6 or z 5 28
8. Solve the inequality 3u w 1 1 u 2 2 , 1 and graph the solution. 22 R w R 0
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
9. Carla buys a package of flower seeds to plant in her garden. The flower seed
package advertizes there are between 75 and 83 seeds in each package. Let
s be the number of seeds in the package. Write an absolute value inequality
describing the number of seeds in the package. »s 2 79… K 4
Do you UNDERSTAND?
10. Writing Write an inequality that has no solution. Explain why it does not have
a solution.
Answers may vary. Sample: 6x 1 1 R 3(2x 2 4); Using the Distributive Property,
6x 1 1 R 6x 2 12; subtract 6x from each side; 1 R 212. Since 1 R 212 is never true,
the inequality has no solution.
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Chapter 1 Test
Form K
Do you know HOW?
Identify a pattern and find the next three numbers in the pattern.
1. 25, 21, 3, 7, c
Each term is 4 more than the previous
term; 11, 15, 19
2. 64, 32, 16, 8, c
Each term is half of the previous
term; 4, 2, 1
3. What properties of real numbers are illustrated by each equation below?
a. 28 1 3 5 3 1 (28) Commutative Property of Addition
b. 4 1 (24) 5 0 Inverse Property of Addition
c. 2(8 1 t) 5 2 ? 8 1 2 ? t Distributive Property
7
8
d. 8 ? 7 5 1 Inverse Property of Multiplication
Evaluate the expression for the given value of the variable.
4. a2 2 2(a 1 1); a 5 3 1
5. 5(2s 2 1) 2 3(s 1 2); s 5 4 17
6. The expression 15 1 5x models the daily cost in dollars of renting scuba gear
from the water sports store. In the expression, x represents the number of
hours the scuba gear is used. What is the cost of renting scuba gear for a day
when the gear is used for 3 hours? $30
Solve each equation.
7. 2r 1 2 5 3r 2 5 7
8. 8(t 1 1) 5 64 7
Solve each equation for x. State any restrictions on the variables.
9.
xt 2 1
135a
8
10.
x 5 8a 2t 23 ; t u 0
6x 2 1
5y
5
x5
5y 1 1
6
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Chapter 1 Test (continued)
Form K
Write an equation and solve the problem.
11. Two buses leave Columbus, Ohio at the same time and travel in opposite
directions. One bus averages 55 mi/h and the other bus averages 48 mi/h.
When will they be 618 mi apart? 6 h
Solve each inequality. Graph the solution.
13. 2a 1 5 , 6a 1 1 a S 1
12. 2n 1 1 $ 7 n L 3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Solve each compound inequality. Graph the solutions.
14. 3x # 26 or 2x 1 1 $ 3 x K 22 or x L 115. 22t 1 2 , 4 and 2t , 6 21 R t R 3
⫺3 ⫺2 ⫺1
0
1
2
⫺3 ⫺2 ⫺1
3
0
1
2
3
Solve each equation. Check for extraneous solutions.
16. u 3x 1 3 u 5 18 x 5 5 or x 5 27
17. u b 1 2 u 5 2b b 5 2
18. The weatherman announced that the temperature T over the next few weeks
will be at least 648F and at most 788F. Write an absolute value inequality for the
temperature over the next few weeks. »T 2 71… K 7
Do you UNDERSTAND?
19. What is another name for the multiplicative inverse? reciprocal
20. Reasoning Explain in words why 2 u x u , 24 has no solution.
Answers may vary. Sample: Dividing both sides by 2 gives |x| R 22. The absolute value
of any number must be nonnegative, so the inequality has no solution.
21. Open-Ended What is the difference between simplifying an expression and
evaluating an expression?
Answers may vary. Sample: Simplifying an expression is rewriting it using the
properties of real numbers and combining like terms, resulting in a simpler expression.
Evaluating an expression is substituting values for the variables, resulting in a
numerical value.
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Chapter 1 Performance Tasks
Give complete answers.
Task 1
Write an algebraic expression that requires each of the following properties of real
numbers to simplify, in the order given: the Distributive Property, the Associative
Property of Addition, the Commutative Property of Addition, and the Distributive
Property. Simplify the expression showing the use of each property.
Answers may vary. Check students’ work.
[4] Student writes an algebraic expression satisfying the conditions and correctly simplifies
the expression, labeling each step with the appropriate property.
[3] Student writes an algebraic expression satisfying the conditions. Simplification contains
only minor errors.
[2] Student writes an algebraic expression that satisfies only two or three of the conditions.
Simplification contains errors.
[1] Student writes an algebraic expression that satisfies one or none of the conditions.
Simplification contains significant errors or is missing.
[0] Response is missing or inappropriate.
Task 2
a. Explain how you determine the total number of points scored by a basketball
player who makes free-throws (worth 1 point), two-point goals (2 points), and
three-point goals (3 points) in a game.
b. Use your method from part a to find the total points scored by each of the
following three players.
Free-throws
2-Point Goals
3-Point Goals
Player A
3
5
1
Player B
0
7
3
Player C
2
6
0
a. The total number of points is the sum of 1 times the number of free-throws, 2 times the
number of two-point goals, and 3 times the number of three-point goals.
b. A: 16 points, B: 23 points, C: 14 points
[4] Student gives a detailed explanation of how to find the total. Steps in finding the total
for three players are correct.
[3] Student gives an adequate explanation of how to find the total. Steps in finding the
total for three players contain minor errors.
[2] Student does not fully explain how to find the total. Steps in finding the total for three
players contain errors.
[1] Student does not correctly or completely explain how to find the total. Steps in finding
the total for three players contain major errors.
[0] Response is missing or inappropriate.
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Chapter 1 Performance Tasks (continued)
Task 3
Explain how the properties of inequalities differ from the properties of equality
and how the solutions of an inequality differ from the solutions of an equation.
Use the following equation and inequality as part of your explanation.
25x 5 10
25x . 10
Answers may vary. Sample: When multiplying or dividing both sides of an inequality by a
negative value, the inequality symbol must reverse direction. The solution of the inequality
is x * 22, whereas the solution of the equation is x 5 22. The solutions of the inequality
do not include the solution of the equation, but consist of all numbers less than it.
[4] Student gives specific details of the differences and effectively uses the examples to
illustrate these differences. Steps in solving the equation and inequality are correct.
[3] Student gives adequate details of the differences and makes reference to the examples.
Steps in solving the equation and inequality contain minor errors.
[2] Student solves the equation and the inequality but does not explain the differences.
Steps in solving the equation and inequality contain errors.
[1] Student does not correctly or completely solve the equation and inequality. Steps in
solving the equation and inequality contain major errors.
[0] Response is missing or inappropriate.
Task 4
a. Write an inequality using an absolute value that can be rewritten as a
compound inequality with or.
b. Solve and graph the inequality in part a.
c. Write an inequality using an absolute value that can be rewritten as a
compound inequality with and.
d. Solve and graph the inequality in part c.
a. Answers may vary. Sample: |x| + 2
b. Answers may vary. Sample: x + 2 or x * 22;
⫺6 ⫺4 ⫺2 0 2 4 6
c. Answers may vary. Sample: |x| * 2
d. Answers may vary. Sample: 22 * x * 2;
⫺6 ⫺4 ⫺2 0 2 4 6
[4] Student writes and solves inequalities using an absolute value with no mistakes. Graphs
are accurate and detailed.
[3] Student correctly creates inequalities using an absolute value. Graphs or solutions
contain minor errors.
[2] Student makes significant errors in writing or solving the inequalities. Graphs contain
major errors.
[1] Student does not write or solve all inequalities. Graph is incomplete or missing.
[0] Response is missing or inappropriate.
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Chapter 1 Cumulative Review
Multiple Choice
For Exercises 1–10, choose the correct letter.
1. What is the next number in the pattern? 1, 2, 4, 7, 11, … D
12
13
15
16
2. Which of the following numbers is an integer but not a natural number? G
223
0
!9
7
3. Which of the following lists of numbers is ordered from least to greatest? C
3
3
3
0, 21, 22, 23, 24
21, 22, 0, 1, 2
0, 1, !2, 2, !3
0, 21, 1, 22, 2
4. Which of the following expressions is not equivalent to 4(1 2 2x) 1 2(5 2 x)? H
14 2 10x
4 2 8x 1 10 2 2x
4 2 8x 1 10 2 x
4 1 10(1 2 x)
5. What is the value of |x 1 3| 1 5x 2 7 for x 5 29? A
246
258
64
240
6. The expression 20,000 2 1250t models the value, in dollars, of a piece of
equipment t years after purchase. What is the value of the equipment after
8 years? I
$15,000
$0
$150,000
$10,000
7. What is the solution of 4[x 2 (3 2 2x)] 1 5 5 3(x 1 11)? B
31
40
2
3
9
5
8. Which number is not a solution of the inequality 227 # 3(1 2 2x) # 3? G
1
0
21
4
2
9. Which number is not a solution of the equation |x 1 5| 5 x 1 5? D
5
0
25
210
10. Which of the following equations have more than one solution? F
II. 2 Q x 1 12 R 5 2x 1 1 III. |x 12| 5 21
I. |x 2 3| 5 4
I and II only
I and III only
IV.
II and III only
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3x 1 5 5 14
I, II, III, and IV
Name
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Date
Chapter 1 Cumulative Review (continued)
Short Response
Number of Teachers
11. Derek has noticed there are fewer students per class at
his friend’s private school than at his public school. He
talks to the principal of the private school and learns that
the number of students is the same in every class. Derek
surveys the classes and makes the graph at the right,
comparing the number of students and the number of
teachers.
a. How many teachers are required for 85 students at the
private school? 5
b. How many teachers are required for 153 students? 9
4
3
2
1
17 34 51 68 85
Number of Students
√ 11 5 13
1
12. Graph the numbers 23, 53, and !11 on a number line.
1
13. What are the opposite and the reciprocal of 224 ?
5
⫺9 ⫺6 ⫺3
0
3
6
9
opposite: 214; reciprocal: 249
14. Write an equation to solve the problem. Find three consecutive even numbers
whose sum is 144. n 1 (n 1 2) 1 (n 1 4) 5 144
15. Solve ax 2 3x 1 5 5 a 1 b for x. State any restrictions on the variables. x 5
a 1 b25
a23 ,
au3
Solve each inequality. Graph the solution.
1
16. 3(t 2 2) 1 5 # 4t 1 2
t L 232
⫺3 ⫺2 ⫺1
0
1
2
18. u 3x 2 1 u , 7
17. 2x 1 5 , 3 or 6 2 x # 5
x R 21 or x L 1
3
⫺3 ⫺2 ⫺1
0
1
2
22 R x R 83
3
⫺3 ⫺2 ⫺1
0
1
2
19. Write the compound inequality 3.1 # m # 4.7 as an absolute value inequality. |m 2 3.9| K 0.8
Extended Response
1
20. Writing What is the value of the expression 0 ? 0? Explain.
It is undefined; because division by 0 is undefined, the entire expression is undefined.
[4] appropriate methods and explanation with no computational errors
[3] appropriate methods and explanation, but with one computational error
[2] incorrect value OR correct value without explanation
[1] correct value, without work shown or explanation
[0] answer missing or no attempt made
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3
TEACHER INSTRUCTIONS
Chapter 1 Project Teacher Notes: Buy the Hour
About the Project
The Chapter Project gives students an opportunity to use expressions, equations,
inequalities, and graphs to model real-life situations. Students write and simplify
expressions by using the Distributive Property and combining like terms, evaluate
expressions, and write and solve equations and inequalities in one variable.
Introducing the Project
• Encourage students to keep all project-related materials in a separate folder.
• Ask students what the term minimum wage means.
• Have students explain the similarities and differences among the terms
expression, equation, and inequality. Remind students that it is important to
define the variable when writing expressions, equations, and inequalities.
Activity 1: Researching
Students research federal and state minimum wages.
Activity 2: Modeling
Students write and evaluate expressions modeling amounts of money earned
based on minimum wages found in Activity 1.
Activity 3: Solving
Students write and solve equations and inequalities that model relationships
between amounts of money earned. Students also graph the solutions of their
inequalities.
Finishing the Project
You may wish to plan a project day on which students share their completed
projects. Encourage students to explain their processes as well as their results.
• Have students review their methods for writing and evaluating expressions;
for writing and solving equations; and for writing, solving, and graphing
inequalities.
• Ask groups to share their insights that resulted from completing the project,
such as shortcuts they found for solving equations and inequalities or for
researching data.
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Chapter 1 Project: Buy the Hour
Beginning the Chapter Project
Have you had your first job yet? If so, you were probably paid an hourly wage. The
amount of money you earned for each hour you worked may have been the minimum
wage. This amount, set by the U.S. Department of Labor, is the minimum amount for
one hour of work an employer is allowed to pay to employees who meet certain specific
criteria. Each state may set its own minimum wage, but where federal and state laws
set different rates, the employer is required to pay the greater of the two amounts to all
employees to whom the conditions of the federal law apply.*
In this project, you will write expressions that model amounts of money earned. You will
write equations and inequalities to determine the number of hours that must be worked
to satisfy certain conditions. You will also research the current federal and state minimum
wage laws.
Activities
Activity 1: Researching
Research the current federal minimum wage. Then find out whether the state in
which you live has set its own minimum wage. If so, what is that wage? Select a
state other than the state in which you live. Research the minimum wage for that
state. You might find it helpful to contact the U.S. Department of Labor and state
labor commissioners, or to use the Internet to find this data. Check students’ work.
Activity 2: Modeling
Suppose you earn the minimum wage determined in Activity 1 for the state other than your own.
• Suppose that next week you plan to work h hours. Write an expression that models the
amount of money you will earn. Answers may vary. Sample (based on minimum
wage of $7.25 effective July 24, 2009): 7.25h
• Suppose that your friend earns the same hourly wage that you earn, but works
in a job for which he receives tips. Write an expression that models your
friend’s total earnings for a week during which he works n hours and receives
$15 in tips. Then, evaluate the expression for n 5 10 and explain what this
number means. 7.25n 1 15, $87.50, the total earned when your friend works 10 h
• Write an expression that models the sum of your earnings for 3 weeks and your friend’s
earnings for 2 weeks if you each work r hours per week and your friend receives $15 in
tips per week. Simplify the expression. 3(7.25r) 1 2(7.25r 1 15) 5 36.25r 1 30
• Write an expression that models the difference between your earnings
and your friend’s earnings for a week during which you work h hours, your
friend works n hours, and your friend earns t dollars in tips. (Hint: Be sure to
consider the fact that you do not know who earns more money!) |7.25h 2 (7.25n 1 t)|
*Source: http://www.dol.gov/dol/topic/wages/minimumwage.htm
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Chapter 1 Project: Buy the Hour (continued)
Activity 3: Solving
Round numbers of hours to the nearest tenth if necessary.
• Suppose that last week your employer gave you a $.50/h raise and a $20
bonus as a reward for good work. You earned a total of $80 for the week. Let
x represent the number of hours you worked that week. Write an equation to
model this situation. Then solve your equation and explain the meaning of
your solution. Answers may vary. Sample (based on minimum wage of $7.25 effective
July 24, 2009): 7.75x 1 20 5 80; 7.7; you worked about 7.7 h last week to earn $80.
• Suppose your friend (still earning minimum wage) receives $20 in tips, and that you
(earning $.50/h more than your friend) have earned the same amount of money at the
end of a week during which you worked the same number of hours as your friend. Write
an equation to model this situation. Then solve your equation and explain the meaning
of your solution. 7.25h 1 20 5 7.75h, where h is the number of hours each worked; 40;
each worked 40 h.
• Suppose that your friend wants to earn at least $95 next week and he expects
to earn $15 in tips. Write an inequality that models this situation. Then solve
and graph your inequality. Explain the meaning of your solution.
7.25x 1 15 # 95; x # 11.0; your friend must work at least 11.0 h to earn at least
$95 next week.
Finishing the Project
8
9
10 11 12 13 14
The answers to the activities should help you to complete your project. You
should prepare a presentation for the class describing your results. Your
presentation should include the data you researched; the expressions, equations,
and inequality you used to model the given situations; and the graph of your
inequality.
Reflect and Revise
Ask a classmate to review your project. After you have reviewed each other’s
presentations, decide if your work is clear, complete, and convincing. If needed,
make changes to improve your presentation.
Extending the Project
Research the minimum wages set by other states. If they differ from the minimum
wage of your state, determine possible factors that might contribute to the
differences. Find out what conditions might exist that would allow an employer to
pay an employee less than the federal minimum wage.
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Name
Class
Date
Chapter 1 Project Manager: Buy the Hour
Getting Started
Read the project. As you work on the project, you will need a calculator and
materials on which you can record your results and make calculations. Keep all of
your work for the project in a folder.
Checklist
Suggestions
☐ Activity 1: researching
minimum wages
☐ Select a state in which you are interested.
☐ Activity 2: writing algebraic
expressions
☐ Substitute reasonable values for the variables to
determine if the expressions make sense.
☐ Activity 3: writing and solving
equations and inequalities
☐ Check that your answers are reasonable.
☐ algebraic models
☐ Have you defined the variables in your expressions,
equations, and inequality? How does the graph of an
equation differ from the graph of an inequality? What
does this mean in terms of your solution?
Scoring Rubric
4
The expressions, the equations, and the inequality are correct. The graph and
all calculations are accurate. Explanations are thorough and well thought
out. The presentation is clear and complete.
3
The expressions, the equations, and the inequality have minor errors. The
graph and calculations are mostly correct. The explanations and presentation
lack detail or contain small errors.
2
The expressions, the equations, and the inequality have major errors.
The graph and calculations contain minor errors. The explanations and
presentation contain minor inaccuracies.
1
The expressions, the equations, and the inequality are not correct. The graph
is not accurate. Calculations contain major errors or are incomplete. The
explanations and presentation are inaccurate or incomplete.
0
Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Teacher’s Evaluation of the Project
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Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
76