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School District of Palm Beach County Summer Packet Grade 8 Readiness
(Outgoing 7th Grade )
Summer 2013 Students and Parents, This Summer Packet for Grade 8 Readiness (with a focus on Algebra skills) is designed to provide an opportunity to review and remediate foundational skills from Grade 7 in preparation for success in Grade 8 Pre‐Algebra. These materials include instruction and problem solving on each worksheet. The focus of each selected worksheet is a foundational skill from Grade 7 designed to prepare students for Mathematics in Grade 8. The source of the worksheets is the Glencoe Middle School Mathematics series. All of the contents of this packet have been copied with permission. We hope you are able to utilize the resources included in this packet to make your summer both educational as well as relaxing. Thank you! 1-1
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Integers and Absolute Value
Integers less than zero are negative integers. Integers greater than zero are positive integers.
negative integers
positive integers
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8
zero is neither
positive nor negative
The absolute value of an integer is the distance the number is from zero on a number line. Two
vertical bars are used to represent absolute value. The symbol for absolute value of 3 is 3 .
Example 1
Write an integer that represents 160 feet below sea level.
Because it represents below sea level, the integer is -160.
Example 2
Evaluate | -2 |.
On the number line, the point -2 is
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 units away from 0. So, -2 = 2.
-4 -3 -2 -1 0
1
2
3
4
Exercises
Write an integer for each situation.
1. 12°C above zero
2. a loss of $24
3. a gain of 20 pounds
4. falling 6 feet
Evaluate each expression.
5. 12 6. -150 7. -8 + 2
8. 6 + 5 9. -19 - 17
Chapter 1
10. 84 - -62 13
Course 2
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
The Coordinate Plane
The coordinate plane is used to locate points. The horizontal number line is the x-axis. The vertical
number line is the y-axis. Their intersection is the origin.
Points are located using ordered pairs. The first number in an ordered pair is the x-coordinate; the
second number is the y-coordinate.
The coordinate plane is separated into four regions called quadrants.
Example 1
Write the ordered pair that corresponds to point P. Then state the
quadrant in which P is located.
• Start at the origin.
• Move 4 units left along the x-axis.
• Move 3 units up on the y-axis.
The ordered pair for point P is (-4, 3).
P is in the upper left quadrant or Quadrant II.
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
•
•
•
•
4
3
2
1
1
y
1 2 3 4x
-4 -3-2 O
-2
-3
.
-4
Graph and label point M at (0, -4).
Start at the origin.
Move 0 units along the x-axis.
Move 4 units down on the y-axis.
Draw a dot and label it M.
Exercises
Write the ordered pair corresponding to each point
graphed at the right. Then state the quadrant or axis
on which each point is located.
1. P
3. R
2. Q
4. S
4
3
2
1
6. B(0, -3)
7. C(3, 2)
8. D(-3, -1)
9. E(1, -2)
Chapter 1
10. F(1, 3)
19
3
4
1 2 3 4x
-4 -3-2 O
2
Graph and label each point on the coordinate plane.
5. A(-1, 1)
y
-2
-3
-4
4
3
2
1
-4 -3-2
O
1
y
1 2 3 4x
-2
-3
-4
Course 2
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Add Integers
To add integers with the same sign, add their absolute values. The sum is:
•
positive if both integers are positive.
•
negative if both integers are negative.
To add integers with different signs, subtract their absolute values. The sum is:
•
positive if the positive integer’s absolute value is greater.
•
negative if the negative integer’s absolute value is greater.
To add integers, it is helpful to use a number line.
Example 1
Find 4 + (–6).
Example 2
Use a number line.
Use a number line.
•
•
•
Find –2 + (–3).
Start at 0.
Move 4 units right.
Then move 6 units left.
•
•
•
Start at 0.
Move 2 units left.
Move another 3 units left.
-3
-6
-2
+4
-3 -2 -1 0 1 2 3
4 + (-6) = -2
4
-6 -5 -4 -3 -2 -1 0 1 2 3
-2 + (-3) = -5
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Add.
1. -5 + (-2)
2. 8 + 1
3. -7 + 10
4. 16 + (-11)
5. -22 + (-7)
6. -50 + 50
7. -10 + (-10)
8. 100 + (-25)
9. -35 + (-20)
10. -7 + (-3) + 10
11. -42 + 36 + (-36)
12. -17 + 17 + 9
Write an addition expression to describe each situation. Then find
each sum.
13. HAWK A hawk is in a tree 100 feet above the ground. It flies down to the
ground.
14. RUNNING Leah ran 6 blocks north then back 4 blocks south.
Chapter 1
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Subtract Integers
To subtract an integer, add its opposite.
Example 1
Find 6 - 9.
6 - 9 = 6 + (-9)
= -3
Example 2
Simplify.
Find -10 - (-12).
-10 - (-12) = -10 + 12
=2
Example 3
To subtract 9, add -9.
To subtract -12, add 12.
Simplify.
Evaluate a - b if a = -3 and b = 7.
a - b = -3 - 7
= -3 + (-7)
= -10
Replace a with -3 and b with 7.
To subtract 7, add -7.
Simplify.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Subtract.
1. 7 - 9
2. 20 - (-6)
3. -10 - 4
4. 0 - 12
5. -7 - 8
6. 13 - 18
7. -20 - (-5)
8. -8 - (-6)
9. 25 - (-14)
11. 15 - 65
10. -75 - 50
12. 19 - (-10)
Evaluate each expression if m = -2, n = 10, and p = 5.
13. m - 6
14. 9 - n
15. p - (-8)
16. p - m
17. m - n
18. -25 - p
Chapter 1
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NAME ________________________________________ DATE _____________ PERIOD _____
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Multiply Integers
The product of two integers with different signs is negative.
The product of two integers with the same sign is positive.
Example 1
Find 5(-2).
5(-2) = -10
Example 2
The integers have different signs. The product is negative.
Find -3(7).
-3(7) = -21
Example 3
The integers have different signs. The product is negative.
Find -6(-9).
-6(-9) = 54
Example 4
The integers have the same sign. The product is positive.
Find (-7)2.
(-7)2 = (-7)(-7)
= 49
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 5
abc = –2(–3)(4)
= 6(4)
= 24
There are 2 factors of -7.
The product is positive.
Evaluate abc if a = –2, b = –3, and c = 4.
Replace a with –2, b with –3, and c with 4.
Multiply –2 and –3.
Multiply 6 and 4.
Exercises
Multiply.
1. -5(8)
2. -3(-7)
3. 10(-8)
4. -8(3)
5. -12(-12)
6. (-8)2
7. -5(7)
8. 3(-2)
9. -6(-3)
ALGEBRA Evaluate each expression if a = -3, b = -4, and c = 5.
10. 5bc
11. -4b
12. 2ac
13. -2a
14. 9b
15. ab
16. -3ac
17. -2c2
18. abc
Chapter 1
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NAME ________________________________________ DATE _____________ PERIOD _____
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Divide Integers
The quotient of two integers with different signs is negative.
The quotient of two integers with the same sign is positive.
Example 1
Find 30 ÷ (-5).
30 ÷ (-5)
The integers have different signs.
30 ÷ (-5) = -6
The quotient is negative.
Example 2
Find -100 ÷ (-5).
-100 ÷ (-5)
The integers have the same sign.
-100 ÷ (-5) = 20
The quotient is positive.
Exercises
Divide.
2. -14 ÷ (-7)
18
3. −
4. -6 ÷ (-3)
5. -10 ÷ 10
-80
6. −
7. 350 ÷ (-25)
8. -420 ÷ (-3)
-2
540
9. −
45
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. -12 ÷ 4
-20
-256
10. −
16
ALGEBRA Evaluate each expression if d = -24, e = -4, and f = 8.
11. 12 ÷ e
12. 40 ÷ f
13. d ÷ 6
14. d ÷ e
15. f ÷ e
16. e2 ÷ f
-d
17. −
e
18. ef ÷ 2
f+8
-4
19. −
d-e
20. −
Chapter 1
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2-2
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A
Add and Subtract Like Fractions
Like fractions are fractions that have the same denominator. To add or subtract like fractions, add or
subtract the numerators and write the result over the denominator.
Simplify if necessary.
Example 1
3+1
3
1
−
+−
=−
4
4
4
4
=−
2
1
2-1
−
-−
=−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Add the numerators.
0
1
4
1
2
3
4
1
0
1
3
2
3
1
Simplify.
Example 2
1
=−
3
4
the denominator.
=1
3
4
Write the sum over
4
3
3
1
Find −
+−
. Write in simplest form.
3
2
1
Find −
-−
. Write in simplest form.
3
3
Subtract the numerators.
Write the difference over
the denominator.
Exercises
Add or subtract. Write in simplest form.
5
1
1. −
+−
8
7
2
2. −
-−
8
9
9
3
1
3. - −
+−
7
5
4. −
-−
5
5
5. −
+−
3
1
6. – −
-−
3
1
7. −
+−
10
2
1
8. −
-−
7
4
9. −
+−
7
8
10. −
-−
4
4
9
10
15
Chapter 2
9
15
8
8
8
5
9
25
8
5
9
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
2-2
Reteach
C
Add and Subtract Unlike Fractions
To add or subtract fractions with different denominators,
• Rename the fractions using the least common denominator (LCD).
• Add or subtract as with like fractions.
• If necessary, simplify the sum or difference.
2
1
Find −
+−
.
Example
3
4
Method 1 Use a Model.
2
−
3
1
+ −
4
11
−
12
1
3
1
3
1
4
1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12
Use the LCD.
Method 2
1 3
+−
·−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
1
2 4
−
+−
=−
·−
3
4
3 4
8
=−
+
12
4
3
3
11
− or −
12
12
Rename using the LCD, 12.
Add the fractions.
Exercises
Add or subtract. Write in simplest form.
3
1
+−
1. −
2
3
1
2. −
-−
4
8
( 6)
7
5
+ -−
3. −
15
(
)
5
5
+ -−
5. −
9
12
( 3)
2
2
1
4. −
-−
5
3
3
11
6. −
-−
12
4
7
1
– -−
7. −
7
1
8. −
-−
3
7
+−
9. −
3
2
10. −
+−
8
10
Chapter 2
12
9
5
31
2
3
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
2-2
Reteach
D
Add and Subtract Mixed Numbers
To add or subtract mixed numbers:
• Add or subtract the fractions. Rename using the LCD if necessary.
• Then, add or subtract the whole numbers.
• Simplify if necessary.
3
1
Find 6 −
+ 2−
. Write in simplest form.
Example 1
+
10
1
6−
10
3
2−
10
10
Add the whole numbers and the fractions separately.
4
2
or 8 −
8−
10
Simplify.
5
2
1
Find 8 −
– 6−
.
Example 2
2
8−
3
1
-6 −
2
4
8−
6
3
6−
6
3
2
Rename the fractions using the LCD.
1
2−
Subtract.
6
3
1
Find 4 −
- 2−
.
5
4−
1
4−
4
3
-2 −
5
20
12
2−
20
4
25
3−
20
12
2−
20
13
1−
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 3
5
5
25
Rename 4 −
as 3 −
.
20
20
Subtract the whole numbers and then the fractions.
Exercises
Add or subtract. Write in simplest form.
3
1
1. 1 −
+ 4−
5
5
1
2. 2 −
– 1−
2
1
+ 2−
3. 3 −
3
1
4. 5 −
– 3−
7
5. 8 – 6 −
3
4
6. 1 −
+−
5
3
2
8
Chapter 2
6
4
5
36
6
6
10
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
2-3
Reteach
B
Multiply Fractions
To multiply fractions, multiply the numerators and multiply the denominators.
5
3
5×3
15
1
−
×−
=−
=−
=−
6×5
5
6
30
2
To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply the
fractions.
8
5
40
2
1
1
2−
× 1−
=−
×−
=−
= 3−
3
Example 1
3×5
5
4
12
3
3
5
← Multiply the numerators.
← Multiply the denominators.
8
=−
15
Example 2
3
2
3
2
4
Find −
×−
. Write in simplest form.
2×4
2
4
−
×−
=−
3
4
4
5
Simplify.
1
1
Find −
× 2−
. Write in simplest form.
5
1
1
1
−
× 2−
=−
×−
3
2
3
2
1×5
=−
3×2
5
=−
6
3
2
5
1
Rename 2−
as an improper fraction, −
.
2
2
Multiply.
Simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Multiply. Write in simplest form.
2
2
1. −
×−
7
1
2. −
×−
5
4. −
×4
3
2
5. 1−
× -−
3
3
9
7
10. −
×8
5
Chapter 2
3
3
1
3. - −
×−
8
3
( 5)
3
3
2
7. −
× 1−
4
2
(
3
1
6. 3−
× 1−
4
1
1
8. -3−
× -2−
3
2
)
6
1
1
9. 4−
×−
5
7
3
1
12. −
× 2−
1
4
11. -2−
×−
3
5
6
8
42
4
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
2-3
Reteach
D
Divide Fractions
To divide by a fraction, multiply by its multiplicative inverse or reciprocal. To divide by a mixed number,
rename the mixed number as an improper fraction.
1
2
Find 3−
÷−
. Write in simplest form.
Example
3
9
10
1
2
2
3− ÷ − = − ÷ − Rename 3 −13 as an improper fraction.
3
9
3
9
10 9
=−
·−
3
5
2
9
2
Multiply by the reciprocal of −
, which is −
.
9
2
3
10
/ · /−9
=−
Divide out common factors.
= 15
Multiply.
/3 12/
1
Exercises
Divide. Write in simplest form.
5
2
2. −
÷−
1
1
3. - −
÷−
1
4. 5 ÷ - −
5
5. −
÷ 10
1
6. 7−
÷2
5
1
7. −
÷ 3−
1
8. 36 ÷ 1−
1
9. -2−
÷ (-10)
2
4
10. 5−
÷ 1−
2
1
11. 6 −
÷ 3−
6
3
13. 4 −
÷ 2−
1
14. 12 ÷ -2−
2
1
1. −
÷−
3
4
5
6
2
5
7
Chapter 2
2
8
3
2
5
7
3
2
3
1
12. 4−
÷−
9
(
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
( 2)
6
4
2
)
52
8
1
1
15. 4−
÷ 3−
6
6
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
2-3
Reteach
E
Powers and Exponents
exponent
34 =
base
3 3 3 3 = 81
common factors
The exponent tells you how many times the base is used as a factor.
Example 1
Write 63 as a product of the same factor.
The base is 6. The exponent 3 means that 6 is used as a factor 3 times.
63 = 6 · 6 · 6
Example 2
Evaluate 54.
54 = 5 · 5 · 5 · 5
= 625
Example 3
1
1
1
1
1
Write −
· −
· −
in exponential form.
· −
· −
4
4
4
4
4
1
The base is −. It is used as a factor 5 times, so the exponent is 5.
4
1
1
1
1 5
1
1
− · − · − · −
· −
= −
4
4
4
4
4
4
()
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Write each power as a product of the same factor.
1. 73
(3)
2
3. −
2. 27
4
4. 154
Evaluate each expression.
5. 35
6. 73
(5)
1
8. −
7. 84
3
Write each product in exponential form.
9. 2 · 2 · 2 · 2
10. 7 · 7 · 7 · 7 · 7 · 7
3
3
3
· −
· −
11. −
12. 9 · 9 · 9 · 9 · 9
13. 12 · 12 · 12
2
2
2
2
14. −
· −
· −
· −
15. 6 · 6 · 6 · 6 · 6
16. 1 · 1 · 1 · 1 · 1 · 1 · 1 · 1
Chapter 2
57
10
10
10
5
5
5
5
Course 2
3-1
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Solve One-Step Addition and Subtraction Equations
Remember, equations must always remain balanced. If you subtract the same number from each side
of an equation, the two sides remain equal. Also, if you add the same number to each side of an
equation, the two sides remain equal.
Example 1
x + 5 = 11
- 5 = -5
x
= 6
Solve x + 5 = 11. Check your solution.
Write the equation.
Subtract 5 from each side.
Simplify.
Check x + 5 = 11
6 + 5 11
11 = 11 Write the original equation.
Replace x with 6.
This sentence is true.
The solution is 6.
Example 2
15 = t - 12
+12 = + 12
27 = t
Solve 15 = t - 12. Check your solution.
Write the equation.
Add 12 to each side.
Simplify.
Write the original equation.
Replace t with 27.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Check 15 = t - 12
15 27 - 12
15 = 15 This sentence is true.
The solution is 27.
Exercises
Solve each equation. Check your solution.
1. h + 3 = 14
2. m + 8 = 22
3. p + 5 = 15
4. 17 = y + 8
5. w + 4 = -1
6. k + 5 = -3
7. 25 = 14 + r
8. 57 + z = 97
9. b - 3 = 6
13. -9 = w - 12
Chapter 3
10. 7 = c - 5
11. j - 12 = 18
12. v - 4 = 18
14. y - 8 = -12
15. 14 = f - 2
16. 23 = n - 12
18
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Solve One-Step Multiplication and Division Equations
Use the Division Property of Equality to solve multiplication equations and the Multiplication Property
of Equality to solve division equations.
The Division Property of Equality states that if you divide each side of an equation by the same
nonzero number, the two sides remain equal.
The Multiplication Property of Equality states that if you multiply each side of an equation by the
same number, the two sides remain equal.
Example 1
Solve 30 = 6x.
30 = 6x
Write the equation.
30
6x
−
=−
Divide each side of the equation by 6.
6
6
5=x
30
÷ 6 = 5.
The solution is 5.
Example 2
x
Solve −
= –2.
x
−
= –2
–5
x
−
(–5) = –2(–5)
–5
x = 10
–5
Write the equation.
Multiply each side of the equation by –5.
– 2(– 5) = 10.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The solution is 10.
Exercises
Solve each equation. Check your solution.
1. 3x = 12
2. 9k = –360
3. –15a = –45
4. 14 = 2b
x
5. −
= 12
a
6. 16 = −
c
7. −
=7
8. –7y = 42
5
–2
m
9. −
= –4
6
Chapter 3
3
b
10. –2 = −
–9
25
Course 2
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3-2
Reteach
D
Solve Equations with Rational Coefficents
Multiplicative inverses, or reciprocals, are two numbers whose product is 1. To solve an equation in
which the coefficient is a fraction, multiply each side of the equation by the reciprocal of the coefficient.
1
Find the multiplicative inverse of 3 −
.
Example 1
4
13
1
3−
=−
4
Rename the mixed number as an improper fraction.
4
13 4
−
·−=1
4
13
4
Multiply −
by −
to get the product 1.
13
4
13
1
4
The multiplicative inverse of 3 −
is −
.
4
13
Example 2
4
Solve −
x = 8. Check your solution.
4
−
x=8
5
5 4
5
−
−x= −
8
4 5
4
()
()
x = 10
5
Write the equation.
4 5
Multiply each side by the reciprocal of −
, −.
5 4
Simplify.
The solution is 10.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Find the multiplicative inverse of each number.
4
1. −
12
2. −
9
13
15
3. - −
1
4. 6 −
7
4
Solve each equation. Check your solution.
3
5. −
x = 12
10
6. 16 = −
a
15
8. −
y=3
9. 21 = 0.75a
5
7
Chapter 3
7. 9 = 0.3n
3
31
7
14
b
10. −
= -−
3
9
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
3-3
Reteach
B
Solve Two-Step Equations
To solve a two-step equation, undo the addition or subtraction first. Then undo the multiplication or
division.
Solve 7v - 3 = 25. Check your solution.
Example 1
7v - 3 = 25
+3 = +3
7v
= 28
Write the equation.
Undo the subtraction by adding 3 to each side.
Simplify.
7v
28
−
=−
7
Undo the multiplication by dividing each side by 7.
7
v=4
Simplify.
7v - 3¬= 25
7(4) - 3¬ 25
28 - 3¬ 25
25¬= 25 Check
Write the original equation.
Replace v with 4.
Multiply.
The solution checks.
The solution is 4.
Solve -10 = 8 + 3x. Check your solution.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
-10 = 8 + 3x
-8 = -8
Write the equation.
Undo the addition by subtracting 8 from each side.
-18 =
Simplify.
3x
-18
3x
−
=−
3
3
Undo the multiplication by dividing each side by 3.
-6 = x
Simplify.
-10 = 8 + 3x
-10 8 + 3(-6)
-10 8 + (-18)
-10 = -10 Check
Write the original equation.
Replace x with -6.
Multiply.
The solution checks.
The solution is -6.
Exercises
Solve each equation. Check your solution.
1. 4y + 1 = 13
2. 6x + 2 = 26
3. -3 = 5k + 7
2
4. −
n + 4 = -26
5. 7 = -3c - 2
6. -8p + 3 = -29
7. -5 = -5t - 5
8. -9r + 12 = -24
7
9. 11 + −
n=4
3
10. 35 = 7 + 4b
4
11. -15 + −
p=9
12. 49 = 16 + 3y
13. 2 = 4t - 14
14. -9x - 10 = 62
15. 30 = 12z - 18
16. 7 + 4g = 7
4
17. 2 + −
x = -12
18. 50 = 16q + 2
2
1
19. 7c - −
=−
20. 9y + 4 = 22
9
9
Chapter 3
5
3
37
2
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NAME ________________________________________ DATE _____________ PERIOD _____
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D
Solve Equations with Variables on Each Side
To solve an equation with variables on each side, use the Properties of Equality to write an equivalent
equation with the variables on one side.
Then solve the equation.
Express 6x + 10 = x + 8 as another equivalent equation.
Example 1
6x + 10 – 10 = x + 8 – 10
6x = x – 2
Subtract 10 from each side.
Simplify.
Solve 4x = x + 27.
Example 2
4x = x + 27
Write the equation.
4x – x = x – x + 27
Subtraction Property of Equality
3x = 27
Simplify.
3x
27
−
=−
3
3
Division Property of Equality
x=9
Simplify. Check your solution.
Solve 3x – 16 = 5x – 4.
Example 3
3x – 16 = 5x – 4
–16 = 2x – 4
–16 + 4 = 2x – 4 + 4
Subtraction Property of Equality
Simplify.
Addition Property of Equality
–12 = 2x
Simplify.
2x
12
–−
=−
Division Property of Equality
2
2
–6 = x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3x – 3x – 16 = 5x – 3x – 4
Write the equation.
Simplify. Check your solution.
Exercises
Express each equation as another equivalent equation. Justify your
answer.
1. –11 + x = 13 – x
2. 10 + 2x = 7x + 3
Solve each equation. Check your solution.
3. 2x + 6 = x + 15
4. 3x – 2 = 8x – 32
5. –12 – x = 8 – 3x
Chapter 3
44
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B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Equations and Functions
The solution of an equation with two variables consists of two numbers, one for each variable that
makes the equation true. When a relationship assigns exactly one output value for each input value, it
is called a function. Function tables help to organize input numbers, output numbers, and function
rules.
Example
Complete a function table for y = 5x. Then identify the domain
and range.
Choose four values for x. Substitute the values for x into the expression. Then evaluate to
find the y value.
x
5x
y
0
5(0)
0
1
5(1)
5
2
5(2)
10
3
5(3)
15
The domain is {0, 1, 2, 3}. The range is {0, 5, 10, 15}.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete the following function tables. Then identify the domain and range.
1. y = 4x
x
2. y = 10x
4x
0
1
1
2
2
3
3
4
3. y = -0.5x
x
Chapter 5
x
y
10x
y
3x
y
4. y = 3x
-0.5x
x
y
2
10
3
11
4
12
5
13
13
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Functions and Graphs
The solution of an equation with two variables consists of two numbers, one for each variable, that
make the equation true. The solution is usually written as an ordered pair (x, y), which can be
graphed. If the graph for an equation is a straight line, then the equation is a linear equation.
Graph y = 3x - 2.
Example
Select any four values for the input x. We chose
2, 1, 0, and -1. Substitute these values for x to
find the output y.
y
x
3x - 2
y
(x, y)
2
3(2) - 2
4
(2, 4)
1
3(1) - 2
1
(1, 1)
0
3(0) - 2
-2
(0, -2)
-1
3(-1) - 2
-5
(-1, -5)
O
x
Four solutions are (2, 4), (1, 1), (0, -2), and (-1, -5).
The graph is shown at the right.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each equation.
1. y = x - 1
2. y = x + 2
y
y
O
x
4. y = 4x
y
O
x
6. y = 2x
y
x
O
x
O
5. y = 2x + 4
y
Chapter 5
3. y = -x
y
x
O
19
O
x
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
5-2
Reteach
B
Constant Rate of Change
A rate of change is a rate that describes how one quantity changes in relation to another.
A constant rate of change is the rate of change of a linear relationship.
Example 1
Find the constant rate of change for the table.
Students
Number of Textbooks
5
15
10
30
15
45
20
60
The change in the number of textbooks is 15. The change in the number of students is 5.
change in number of textbooks
15 textbooks
−− = −
change in number of students
5 students
The number of textbooks increased by
15 for every 5 students.
3 textbooks
= −
Write as a unit rate.
1 student
Example 2
To find the rate of change, pick any two points on the line,
such as (8, 25) and (10, 35).
change in number
(35 – 25)
10
−− = − = −
or 5 T-shirts per hour
change in time
(10 – 8)
35
30
25
20
15
10
5
0
2
8 P.M. 9 P.M. 10 P.M.
Time
Exercises
Find the each constant rate of change.
Side Length
Perimeter
1
4
2
8
3
12
4
16
2.
96
88
Miles
1.
80
72
0
10 A.M. 12 P.M. 2 P.M.
Time
Chapter 5
26
Course 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The graph represents the number of
T-shirts sold at a band concert. Use
the graph to find the constant rate
of change in number per hour.
T-shirts
So, the number of textbooks increases by 3 textbooks per student.
NAME ________________________________________ DATE _____________ PERIOD _____
5-2
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C
Slope
Slope is the rate of change between any two points on a line.
change in y
vertical change
rise
slope = − = −− or −
run
change in x
horizontal change
Example
The table shows the length of a patio as blocks are added.
Number of Patio Blocks
0
1
2
3
4
Length (in.)
0
8
16
24
32
Graph the data. Then find the slope of the line.
change in y
change in x
24 – 8
=−
3–1
16
=−
2
8
=−
1
slope = −
Length (in.)
Explain what the slope represents.
Definition of slope
Use (1, 8) and (3, 24).
length
number
−
Simplify.
0
1 2 3 4 5 6
Number
So, for every 8 inches, there is 1 patio block.
Exercises
Draw a graph and find the slope of the line. Explain
what the slope represents.
Cases
1
2
3
4
Juice Bottles
12
24
36
48
Bottles
1. The table shows the number of juice bottles per case.
48
42
36
30
24
18
12
6
0
1 2 3 4 5 6 7 8
Cases
2. At 6 A.M., the retention pond had 28 inches of water in it.
The water receded so that at 10 A.M. there were 16 inches
of water left.
Inches
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
36
32
28
24
20
16
12
8
4
32
28
24
20
16
12
8
4
0
6 A.M 8 A.M 10 A.M 12 P.M
Time
Chapter 5
31
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Reteach
Problem-Solving Investigation: Use a Graph
The graph shows the results of a survey of teachers’ years of
experience and number of honors level classes taught at a high
school. How many honors classes would you predict a teacher with
24 years’ experience would teach?
Number of
Honors Classes
Example
NAME ________________________________________ DATE _____________ PERIOD _____
6
5
4
3
2
1
0
4 8 12 16 20 24
Years Experience
Understand
You know the number of honors level classes taught and the
number of years of experience from the graph.
Plan
Look at the trends in the data on the graph.
Solve
Using the line, you can predict that a teacher with 24 years’
experience would teach 5 honors level classes.
Check
Draw a line that is as close to as many of the points as possible.
The estimate is close to the line, so the prediction is reasonable.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Use the graph below. Each point on the graph represents one person
in a group training for a long-distance bicycle ride. The point shows
the number of miles that person cycles each day and the number of
weeks that person has been in training.
Distance (mi)
1. Draw a line that is close to as many of the points as possible.
35
30
25
20
15
10
5
0
1 2 3 4 5 6 7
Weeks Training
2. Does the number of miles bicycled each day increase as the
number of weeks in training increases?
3. Predict the number of miles bicycled each day for someone
who has been training for 9 weeks.
Chapter 5
36
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C
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Direct Variation
When two variable quantities have a constant ratio, their relationship is called a direct variation.
The constant ratio is called the constant of variation.
The time it takes Lucia to pick pints of
blackberries is shown in the graph.
Determine the rate in minutes per pint.
Since the graph forms a line, the rate of change is constant.
Use the graph to find the constant ratio.
minutes
15
−
=−
number of pints
1
30
15
−
or −
2
60
45
Minutes
Example 1
15
45
15
−
or −
1
3
1
0
It takes 15 minutes for Lucia to pick 1 pint of blackberries.
Example 2
1 2 3 4 5 6 7
Pints
There are 12 trading cards in a package. Make a table and graph to
show the number of cards in 1, 2, 3, and 4 packages. Is there a
constant rate? a direct variation?
Numbers of Packages
1
2
3
4
Number of Cards
12
24
36
48
Cards
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
30
96
84
72
60
48
36
24
12
0
1 2 3 4 5 6 7
Packages
Because there is a constant increase of 12 cards, there is a constant rate
of change. The equation relating the variables is y = 12x, where y is the
number of cards and x is the number of packages. This is a direct variation.
The constant of variation is 12.
Exercises
1. SOAP Wilhema bought 6 bars of soap for $12. The next day, Sophia bought
10 bars of the same kind of soap for $20. What is the cost of 1 bar of soap?
2. COOKING Franklin is cooking a 3-pound turkey breast for 6 people. If the
number of pounds of turkey varies directly with the number of people,
make a table to show the number of pounds of turkey for 2, 4, and 8 people.
Chapter 5
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NAME ________________________________________ DATE _____________ PERIOD _____
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Inverse Variation
An inverse variation is a relationship where when x increases in value, y decreases in value, or as x
decreases in value, y increases in value. The product of x and y is a constant k.
k
An inverse variation function is of the form xy = k or y = −
x.
Example
TICKETS When tickets to the picnic cost $6 each, 50
people attend. When tickets are reduced to $4 each, 75 people attend.
Complete a table and graph for prices of $10, $5, $3, and $2.
k
Find the value of k by using the equation y = −
x.
k
Write the equation.
y=−
x
k
50 = −
Replace y with 50 and x with 6.
6
k
50(6) = −
(6)
6
()
300 = k
Simplify.
Use the value for k to complete the table.
Cost
$10
$5
$3
$2
Number of Tickets
30
60
100 150
150
100
50
0
2
4 6 8 10 12 14
Cost ($)
Exercise
Make a table and graph to complete Exercise 1.
1. INVESTMENT The time it takes to double the balance
in an account varies inversely with the interest rate.
If you invest $1,000 at 6% it will take 12 years to
double your money. Find the time it will take to double
your money at 4%.
Time (yrs)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The graph shows that this is an inverse variation.
Number of Tickets
Multiply both sides by 6.
18
16
14
12
10
8
6
4
2
0
1 2 3 4 5 6 7 8
Interest Rate (%)
Chapter 5
47
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NAME ________________________________________ DATE _____________ PERIOD _____
6-1
Reteach
B
Percent of a Number
To find the percent of a number, you can write the percent as a fraction and then multiply or write the
percent as a decimal and then multiply.
Example 1
Find 25% of 80.
25
1
25% = −
or −
100
4
1
1
− of 80 = − × 80 or 20
4
4
Write 25% as a fraction, and reduce to lowest terms.
Multiply.
So, 25% of 80 is 20.
Example 2
What number is 15% of 200?
15% of 200 = 15% × 200
Write a multiplication expression.
= 0.15 × 200
Write 15% as a decimal.
= 30
Multiply.
So, 15% of 200 is 30.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each number.
1. Find 20% of 50.
2. What is 55% of $400?
3. 5% of 1,500 is what number?
4. Find 190% of 20.
5. What is 24% of $500?
6. 8% of $300 is how much?
7. What is 12.5% of 60?
8. Find 0.2% of 40.
9. Find 3% of $800.
10. What is 0.5% of 180?
11. 0.25% of 42 is what number?
12. What is 0.02% of 280?
Chapter 6
13
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
6-2
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B
The Percent Proportion
A percent proportion compares part of a quantity to a whole quantity for one ratio and lists the
percent as a number over 100 for the other ratio.
percent
100
part
whole
− = −
Example 1
part
whole
What percent of 24 is 18?
percent
100
− = −
Percent proportion
Let n represent the percent.
18
n
−
=−
24
100
18 × 100 = 24 × n
Write the proportion.
Find the cross products.
1,800 = 24n
Simplify.
1,800
24n
−=−
24
24
Divide each side by 24.
75 = n
So, 18 is 75% of 24.
Example 2
part
whole
What number is 60% of 150?
percent
100
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
− = −
Percent proportion
Let a represent the percent.
a
60
−
=−
150
100
a × 100 = 150 × 60
Write the proportion.
Find the cross products.
100a = 9,000
Simplify.
9,000
100a
−
=−
100
100
Divide each side by 100.
a = 90
So, 90 is 60% of 150.
Exercises
Find each number. Round to the nearest tenth if necessary.
1. What number is 25% of 20?
2. What percent of 50 is 30?
3. 30 is 75% of what number?
4. 40% of what number is 36?
5. What number is 20% of 625?
6. 12 is what percent of 30?
Chapter 6
25
Course 2
NAME ________________________________________ DATE _____________ PERIOD _____
6-2
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C
The Percent Equation
To solve any type of percent problem, you can use the percent equation, part = percent · whole,
where the percent is written as a decimal.
Example 1
600 is what percent of 750?
600 is the part and 750 is the whole. Let n represent the percent.
⎧
⎫
⎧
⎫
⎧
⎫
part = percent · whole
600 =
n
· 750
600
750n
−
=−
750
Write the percent equation.
Divide each side by 750.
750
0.8 = n
Simplify.
80% = n
Write 0.8 as a percent.
So, 600 is 80% of 750.
Example 2
45 is 90% of what number?
45 is the part and 90% or 0.9 is the percent. Let n represent the whole.
⎧
⎫
⎧
⎫
⎧
⎫
part = percent · whole
45 =
0.9
·
n
45
0.9n
−
=−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0.9
0.9
50 = n
Write the percent equation.
Divide each side by 0.9.
Simplify.
So, 45 is 90% of 50.
Exercises
Write an equation for each problem. Then solve. Round to the nearest
tenth if necessary.
1. What percent of 56 is 14?
2. 36 is what percent of 40?
3. 80 is 40% of what number?
4. 65% of what number is 78?
5. What percent of 2,000 is 8?
6. What is 110% of 80?
7. 85 is what percent of 170?
8. Find 30% of 70.
Chapter 6
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B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Percent of Change
A percent of change is a ratio that compares the change in quantity to the original amount. If the
original quantity is increased, it is a percent of increase. If the original quantity is decreased, it is a
percent of decrease.
Example 1
Last year, 2,376 people attended the rodeo. This year, attendance
was 2,950. What was the percent of change in attendance to the
nearest whole percent?
Since this year’s attendance is greater than last year’s attendance, this is a percent of
increase.
The amount of change is 2,950 – 2,376 or 574.
amount of change
original amount
574
=−
2,376
percent of change = −−
≈ 0.24 or 24%
Substitution
Simplify.
The percent of change is about 24%.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
Che’s grade on the first math exam was 94. His grade on the second
math exam was 86. What was the percent of change in Che’s grade
to the nearest whole percent?
Since the second grade is less than the first grade, this is a percent of decrease. The amount
of change is 86 - 94 or -8.
amount of change
original amount
8
= -−
94
percent of change = −−
≈ -0.09 or -9%
Substitution
Simplify.
The percent of change is -9%.
Exercises
Find each percent of change. Round to the nearest whole percent if
necessary. State whether the percent of change is an increase or
decrease.
1. original: 4
new: 5
2. original: 1.0
new: 1.3
3. original: 15
new: 12
4. original: $30
new: $18
5. original: 60
new: 63
6. original: 160
new: 136
7. original: 7.7
new: 10.5
8. original: 9.6
new: 5.9
Chapter 6
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Sales Tax and Tips
Sales Tax is a percent of the purchase price and is an amount paid in addition to the purchase price.
Tip, or gratuity, is a small amount of money in return for service.
Example 1
SOCCER Find the total cost of a $17.75 soccer ball if the
sales tax is 6%.
Method 1
Method 2
First, find the sales tax.
6% of $17.75 = 0.06 · 17.75
100% + 6% = 106%
≈ 1.07
The sales tax is $1.07.
Add the percent of tax
to 100%.
The total cost is 106% of the regular price.
106% of $17.75 = 1.06 · 17.75
≈ 18.82
Next, add the sales tax to the regular price.
1.07 + 17.75 = 18.82
The total cost of the soccer ball is $18.82.
Example 2
MEAL A customer wants to leave a 15% tip on a bill for
$18.50 at a restaurant.
Method 1 Add tip to regular price.
100% + 15% = 115% Add the percent of tip
to 100%.
= 2.78
Next, add the tip to the bill total.
$18.50 + $2.78 = $21.28
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
First, find the tip.
15% of $18.50 = 0.15 · 18.50
Method 2 Add the percent of tip to
100%.
The total cost is 115% of the bill.
115% of $18.50 = 1.15 · 18.50
= 21.28
The total cost of the bill is $21.28.
Exercises
Find the total cost to the nearest cent.
1. $22.95 shirt, 6% tax
2. $24 lunch, 15% tip
3. $10.85 book, 4% tax
4. $97.55 business breakfast, 18% tip
5. $59.99 DVD box set, 6.5% tax
6. $37.65 dinner, 15% tip
Chapter 6
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NAME ________________________________________ DATE _____________ PERIOD _____
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Discount
Discount is the amount by which the regular price of an item is reduced. The sale price is
the regular price minus the discount.
Example
TENNIS Find the price of a $69.50 tennis racket that is on
sale for 20% off.
Method 1: Subtract the discount from the regular price.
First, find the amount of the discount.
20% of $69.50 = 0.2 · $69.50
= $13.90
Write 20% as a decimal.
The discount is $13.90.
Next, subtract the discount from the regular price.
$69.50 - $13.90 = $55.60.
Method 2: Subtract the percent of discount from 100%.
100% - 20% = 80%
Subtract the discount from 100%.
The sale price is 80% of the regular price.
80% of $69.50 = 0.80 · 69.50
= 55.60
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The sale price of the tennis racket is $55.60.
Exercises
Find the sale price to the nearest cent.
1. $32.45 shirt; 15% discount
2. $128.79 watch; 30% discount
3. $40.00 jeans; 20% discount
4. $74.00 sweatshirt; 25% discount
5. $28.00 basketball; 50% discount
6. $98.00 tent; 40% discount
Chapter 6
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NAME ________________________________________ DATE _____________ PERIOD _____
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Simple Interest
Simple interest is the amount of money paid or earned for the use of money. To find simple interest I,
use the formula I = prt. Principal p is the amount of money deposited or invested. Rate r is the annual
interest rate written as a decimal. Time t is the amount of time the money is invested in years.
Example 1
Find the simple interest earned in a savings account where
$136 is deposited for 2 years if the interest rate is 7.5% per year.
I = prt
Formula for simple interest
I = 136 · 0.075 · 2
Replace p with $136, r with 0.075, and t with 2.
I = 20.40
Simplify.
The simple interest earned is $20.40.
Example 2
Find the simple interest for $600 invested at 8.5% for 6 months.
6
or 0.5 year
6 months = −
Write the time in years.
I = prt
Formula for simple interest
12
I = 600 · 0.085 · 0.5
p = $600, r = 0.085, t = 0.5
I = 25.50
Simplify.
The simple interest is $25.50.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Find the simple interest earned to the nearest cent for each principal,
interest rate, and time.
1. $300, 5%, 2 years
2. $650, 8%, 3 years
3. $575, 4.5%, 4 years
1
4. $735, 7%, 2 −
years
5. $1,665, 6.75%, 3 years
3
6. $2,105, 11%, 1 −
years
7. $903, 8.75%, 18 months
8. $4,275, 19%, 3 months
Chapter 6
2
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