Chapter 3
Chapter 3 Opener
Section 3.1
Try It Yourself (p. 79)
3.1 Activity (pp. 80 –81)
1. a.
 1
 4
1
= −1
2
1. 2 xy = 2 − (3)
Value when x = 0
Expression
3(0) + 2 − 0 + 4
A.
3x + 2 − x + 4
= 0+ 2−0+ 4
= 6
 1
 4
= −3 − 9
2. 12 x − 3 y = 12 −  − 3(3)
B.
= −12
5(0 − 3) + 2 = 5( − 3) + 2
5( x − 3) + 2
= −15 + 2
= −13
 1
3. −4 x − y + 4 = −4 −  − 3 + 4
 4
0 + 3 − 2(0) + 1
=1−3+ 4
C.
x + 3 − ( 2 x + 1)
= 2
D.
8. 8 ÷ h
9. 3t + 4
− 4 x + 2 − x + 3x
− 4(0) + 2 − 0 + 3(0)
= 0+ 2−0+ 0
= 2
= −14
6. n − 9
= 0 + 3−1
= 2
2
 1
4. 8 x − y 2 − 3 = 8 −  − (3) − 3
4
 
= −2 − 9 − 3
5. 3q + 5
= 0 + 3 − (0 + 1)
7. 6 p
10. 7c − 2
E.
− (1 − x) + 3
− (1 − 0) + 3 = −1 + 3 = 2
F.
2 x + x − 3x + 4
2(0) + 0 − 3(0) + 4
= 0+ 0−0+ 4
= 4
4 − 3 + 2(0 − 1)
G.
4 − 3 + 2( x − 1)
= 4 − 3 + 2( −1)
= 4 − 3 + ( − 2)
= 1 + ( − 2)
= −1
H.
2(1 − x + 4)
2(1 − 0 + 4) = 2(5) = 10
5 − 4 − 0 + 2(0)
I.
5 − ( 4 − x + 2 x)
= 5 − ( 4 − 0 + 0)
= 5− 4
=1
5(0) − 2(0) + 4 − 0
J.
5 x − ( 2 x + 4 − x)
= 0 − ( 0 + 4 − 0)
= 0− 4
= −4
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51
Chapter 3
Value when x = 1
Expression
Expression
Value when x = 0
a.
4
4
b.
−x + 1
− (0) + 1 = 0 + 1 = 1
c.
4x − 4
4(0) − 4 = 0 − 4 = − 4
d.
2x + 6
2(0) + 6 = 0 + 6 = 6
e.
5 x − 13
5(0) − 13 = 0 − 13 = −13
= 1 + 3 − ( 2 + 1)
f.
− 2 x + 10
− 2(0) + 10 = 0 + 10 = 10
=1+3−3
g.
x + 2
0+ 2 = 2
h.
2x − 1
2(0) − 1 = 0 − 1 = −1
i.
− 2x + 2
− 2(0) + 2 = 0 + 2 = 2
j.
−x + 2
− (0) + 2 = 0 + 2 = 2
Expression
Value when x = 1
a.
4
4
b.
−x + 1
− (1) + 1 = −1 + 1 = 0
= 4 − 3 + 2(0)
c.
4x − 4
4(1) − 4 = 4 − 4 = 0
= 4−3+ 0
d.
2x + 6
2(1) + 6 = 2 + 6 = 8
e.
5 x − 13
5(1) − 13 = 5 − 13 = − 8
f.
− 2 x + 10
− 2(1) + 10 = − 2 + 10 = 8
= 5 − ( 4 − 1 + 2)
g.
x + 2
1+ 2 = 3
= 5−5
h.
2x − 1
2(1) − 1 = 2 − 1 = 1
i.
− 2x + 2
− 2(1) + 2 = − 2 + 2 = 0
j.
−x + 2
− (1) + 2 = −1 + 2 = 1
3(1) + 2 − 1 + 4
A.
3x + 2 − x + 4
= 3+ 2 −1+ 4
= 8
B.
5( x − 3) + 2
5(1 − 3) + 2 = 5( − 2) + 2
= −10 + 2
= −8
1 + 3 − 2(1) + 1
C.
x + 3 − ( 2 x + 1)
=1
D.
− 4 x + 2 − x + 3x
− 4(1) + 2 − 1 + 3(1)
= −4 + 2 − 1 + 3
= 0
E.
− (1 − x) + 3
F.
2 x + x − 3x + 4
− (1 − 1) + 3 = 0 + 3 = 3
2(1) + 1 − 3(1) + 4
= 2+1−3+ 4
= 4
4 − 3 + 2(1 − 1)
G.
4 − 3 + 2( x − 1)
=1
H.
2(1 − x + 4)
2(1 − 1 + 4) = 2( 4) = 8
5 − 4 − 1 + 2(1)
I.
5 − ( 4 − x + 2 x)
= 0
5(1) − 2(1) + 4 − 1
J.
5 x − ( 2 x + 4 − x)
= 5 − ( 2 + 4 − 1)
= 5−5
= 0
A matches with d, B matches with e, C matches with j,
D matches with i, E matches with g, F matches with a,
G matches with h, H matches with f, I matches with b,
J matches with c.
b. Sample answer: The expressions in the right column
have no parentheses and do not have more than one
x-term or more than one constant term. To obtain the
equivalent expressions, the like terms are combined.
52
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Chapter 3
2. Answer should include, but is not limited to: Students will
write a lesson on simplifying an expression. They will
describe the steps you can use to simplify an expression,
write 3 examples using expressions from Activity 1, and
write 3 exercises using expressions different from the
ones in Activity 1.
3. Sample answer: You can simplify an algebraic
expression by: (1) using the Distributive Property to get
rid of parentheses, (2) use the Commutative and
Associative Properties of Addition to get the variable
terms together and the terms without variables together,
(3) combine the variable terms and combine the terms
without variables.
4. Sample answer: It may be easier to evaluate an algebraic
expression for a given value when the expression is
simplified. For real-life problems, it may be easier to
recognize a value, like a total cost, based on the parts of a
simplified algebraic expression.
3.1 On Your Own (pp. 82 – 83)
1. Terms: y , 10, −
3
y
2
2
2. Terms: 2r , 7 r , − r , − 9
Like terms: 2r 2 and − r 2
3. Terms: 7, 4 p, − 5, p, 2q
Like terms: 7 and − 5, 4 p and p
4. 14 − 3 z + 8 + z = 14 + 8 − 3 z + z
= 22 − 3 z + z
= 22 + ( − 3 + 1) z
= 22 − 2 z
= − 2 z + 22
5. 2.5 x + 4.3 x − 5 = ( 2.5 + 4.3) x − 5
= 6.8 x − 5
6.
= −2 g − 8 + 7 g
= − 2g + 7 g − 8
= ( − 2 + 7) g − 8
= 5g − 8
3
4
9. 7 − 4 x −
1
3 
 = 7 − 4 x  −
4
4 
= 7 − 3x + 1
3
3
3 3
b − b =  − b
8
4
8 4
3
= − b
8
7. 3(q + 1) − 4 = 3(q) + 3(1) − 4
= 3q + 3 − 4
= 3q − 1
 1
4 − 
 4
= − 3x + 7 + 1
= −3 x + 8
10. Words:
Variable:
Each is
each is
and each
is
ticket $7.50, large $3.50,
small
$3.
drink
popcorn
The same number of each item is purchased.
So, x can represent the number of tickets, the
number of large drinks, and the number of
small popcorns.
Expression: 7.50x
3
Like terms: y and − y
2
2
8. −2( g + 4) + 7 g = −2( g ) − 2(4) + 7 g
+
$3.50x +
3x
7.50 x + 3.50 x + 3x = (7.50 + 3.50 + 3) x
= 14 x
The expression 14x represents the amount of money the
group spends at the movies. The total cost per person is
$14, which is $0.25 less per person.
3.1 Exercises (pp. 84 –85)
Vocabulary and Concept Check
1. The terms of an expression are separated by addition.
Rewrite the expression as 3 y + ( − 4) + ( − 5 y )
The terms are 3 y , − 4, and − 5 y.
2. Use the Distributive Property to add the coefficients of
the like terms 3n and 4n.
3n + 4n − 2 = (3 + 4)n − 2 = 7n − 2
3. no; The expression contains like terms, 3x and 2x, that
should be combined.
3x + 2 x − 4 = (3 + 2) x − 4
= 5x − 4
4. The algebraic expression 5 x − 4 + 6 y is in simplest
form because there are no like terms and no parentheses.
Practice and Problem Solving
5. Terms: t , 8, 3t
Like terms: t and 3t
6. Terms: 3 z , 4, 2, 4 z
Like terms: 3z and 4 z , 4 and 2
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53
Chapter 3
7. Terms: 2n, − n, − 4, 7 n
19. 4 p − 5( p + 6) = 4 p − 5( p ) − 5(6)
= 4 p − 5 p − 30
Like terms: 2n, − n, and 7 n
= ( 4 − 5) p − 30
8. Terms: − x, − 9 x 2 , 12 x 2 , 7
2
Like terms: − 9 x and 12 x
= − p − 30
2
2
3
2
2
(−9) + 14c
3
3
= −8c + 6 + 14c
20. − (12c − 9) + 14c = − (12c) −
9. Terms: 1.4 y, 5, − 4.2, − 5 y 2 , z
Like terms: 5 and − 4.2
10. Terms:
= − 8c + 14c + 6
= (−8 + 14)c + 6
1
3 1
s, − 4, s, , − s 3
2
4 8
Like terms:
1
3
1
s and s, − 4 and
2
4
8
11. Like terms have the same variables raised to the same
exponents. The term 2x 2 is not a like term because x is
squared. The like terms are 3x and 9x.
12. 12 g + 9 g = (12 + 9) g
= 21g
13. 11x + 9 − 7 = 11x + 2
14. 8s − 11s + 6 = (8 − 11) s + 6
= −3s + 6
15. 4.2v − 5 − 6.5v = 4.2v − 6.5v − 5
= ( 4.2 − 6.5)v − 5
= 6c + 6
21.
Each weighs each
weighs and each weighs
sleeping 3.4 lb, backpack 4.6 lb,
water 2.2 lb.
bag
bottle
x represents the number of hikers.
3.4 x + 4.6 x + 2.2 x = (3.4 + 4.6 + 2.2) x
= 10.2 x
The expression 10.2x indicates that the weight carried by
each hiker is 10.2 pounds.
22. Before simplifying:
−8 x + 5 − 2 x − 4 + 5 x = −8(2) + 5 − 2(2) − 4 + 5(2)
= −16 + 5 − 4 − 4 + 10
= −9
After simplifying:
−8 x + 5 − 2 x − 4 + 5 x = − 8 x − 2 x + 5 x + 5 − 4
= (−8 − 2 + 5) x + 5 − 4
= −5 x + 5 − 4
= −2.3v − 5
= −5 x + 1
16. 8 + 4a + 6.2 − 9a = 8 + 6.2 + 4a − 9a
= −5(2) + 1
= 8 + 6.2 + ( 4 − 9)a
= −10 + 1
= 14.2 − 5a
17.
2
9
2
9
y − 4+ 7 −
y = −4 + 7 + y −
y
5
10
5
10
9
2
= −4 + 7 +  −
y
 5 10 
9
4
= −4 + 7 + 
−
y
 10 10 
5
= 3−
y
10
1
= 3− y
2
18. 4(b − 6) + 19 = 4(b) − 4(6) + 19
= 4b − 24 + 19
= 4b − 5
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= −9
Sample answer: Simplifying the expression first is easier
because you only have to substitute once instead of
substituting three times.
(
)
( )
23. 8 x 2 + 3 x 2 + y = 8 x 2 + 3 x 2 + 3( y )
2
= 8 x + 3x 2 + 3 y
= (8 + 3) x 2 + 3 y
= 11x 2 + 3 y
7 x2 + 7 y + 4x2 − 4 y = 7 x2 + 4x2 + 7 y − 4 y
= ( 7 + 4) x 2 + ( 7 − 4) y
= 11x 2 + 3 y
Because both expressions simplify to 11x 2 + 3 y, they
are equivalent.
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Chapter 3
24. B; 6 − 4(2 − 5 x) = 6 − 4(2) − 4(−5 x)
28. Sample answer:
= 6 − 8 + 20 x
= −2 + 20 x
Solution A: The expression inside the parentheses was
simplified first, but 2 and − 5x are not like terms.
Solution C: The solution evaluated the expression from
left to right by subtracting 4 from 6, instead of using
order of operations and distributing the − 4 first.
Solution D: The solution did not distribute the negative
sign to both of the terms inside the parentheses.
x
2x
1
4
(4 + 1)( x
+ 2 x) = (5)(3x)
= 15 x
29. a. Area of
25. A =  w
=
Total
Area of
−
area
red strips
= (3 + x)(3)
gold foil
= 3(3 + x)
The total area is: A =  w
= 3(3) + 3( x)
= ( 20)(12)
= 9 + 3x
= 240 square inches.
The expression 9 + 3x represents the area of the banner
in square feet.
26. The earnings
the earnings
the earnings
is
is
is
for washing
for waxing
for washing
$8,
$12,
$10,
each car
each car
each truck
the earnings
is
and for waxing
$15.
each truck
= (12 + 20) x − x 2
8 x + 12 x + 10 y + 15 y = (8 + 12) x + (10 + 15) y
= 20 x + 25 y
The expression 20 x + 25 y represents the earnings for
washing and waxing x cars and y trucks.
= 32 x − x 2 square inches.
Therefore, the area of the gold foil is:
Area = 240 − (32 x − x 2 )
= 240 − 32 x − ( − x 2 )
= 240 − 32 x + x 2 square inches.
The expression 240 − 32 x + x 2 represents the area
of the gold foil in square inches.
27. Sample answer:
x
A =  w = 12 x square inches, respectively. Together,
the two strips intersect. The intersection has an area of
x 2 square inches. So, the area of the red strips is:
Area = 12 x + 20 x − x 2
x represents the number of cars. y represents the number
of trucks.
2
Separately, the areas of the horizontal and vertical
red strips are A =  w = 20 x square inches and
3
b. When x = 3, the area of the gold foil is 153 square
inches.
240 − 32 x + x 2 = 240 − 32(3) + 32
5
= 240 − 32(3) + 9
= 240 − 96 + 9
5( 2 + x + 3) = 5( x + 2 + 3)
= 5( x + 5)
= 153
c. Sample answer: England
= 5( x) + 5(5)
Fair Game Review
= 5 x + 25
30. 14.5 in., 14.8 in., 15 in., 15.3 in., 15.8 in.
31. 0.52 m, 0.545 m, 0.55 m, 0.6 m, 0.65 m
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55
Chapter 3
3. a. Sample answer: ( x + 3) − ( x + 1)
32. C; 12 inches = 1 foot
Elevation of bird’s nest: 12 ft
(
Elevation of mole’s den: −1 ft
) (
)=(
12 − ( −1) = 13
=
The difference in height is 13 feet.
=
Section 3.2
( x + 3) − ( x
1. a. Sample answer: x + 3
+ 1) = 2
b. Sample answer: ( x − 4) − ( x − 3)
b. Sample answer: 2 x − 2
(
c. Sample answer: 2 x + 3
) (
)=(
) (
=
)=
(x
+ 2) + ( x + 4) = 2 x + 6
) (
− 4) − ( x − 3) = −1
c. Sample answer: ( 2 x + 5) − ( x − 1)
b. Sample answer: ( x − 5) + ( x − 2)
(x
)=
(
)
(
)=
c. Sample answer: ( x + 5) + ( 2 x − 3)
)
(
(
)
(
)
=
− 5) + ( x − 2) = 2 x − 7
(
)
=
2. a. Sample answer: ( x + 2) + ( x + 4)
(
) (
=
d. Sample answer: 3x − 4
(x
)
=
3.2 Activity (pp. 86 –87)
(
) (
)
=
=
=
(2 x
=
+ 5) − ( x − 1) = x + 6
d. Sample answer: (3 x − 7) − ( 2 x + 3)
=
(x
+ 5) + ( 2 x − 3) = 3x + 2
(
)
)
(
)
)
=
(
)
(
)
=
d. Sample answer: ( 2 x − 8) + (3 x + 5)
(
(
=
=
=
=
(3 x
− 7) − ( 2 x + 3) = x − 10
=
(2 x
56
− 8) + (3x + 5) = 5 x − 3
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Chapter 3
4. a. Sample answer: ( 2 x + 1) + ( x − 1)
(
)
(
5. Sample answer: Model the algebraic expressions using
algebra tiles. To add: Combine “like” tiles and remove
zero pairs. Write the resulting algebraic expression
modeled by the remaining tiles. To subtract: Remove
any “like” tiles from the first expression and the second
expression. If tiles remain in the second expression, add
the opposites of the tiles. Write the resulting algebraic
expression modeled by the remaining tiles.
)=
=
6. Sample answer: ( − x + 3) − ( − 2 x − 2)
=
(2 x
(
)
+ 1) + ( x − 1) = 3x
)
( )
)
=(
)
(
)
=
b. Sample answer: ( 2 x − 6) + (3 x + 2)
(
(
=
=
=
=
(− x
+ 3) − ( − 2 x − 2) = x + 5
3.2 On Your Own (pp. 88 –89)
=
(2 x
1. ( x + 3) + (2 x − 1) = x + 3 + 2 x − 1
= x + 2x + 3 − 1
− 6) + (3x + 2) = 5 x − 4
= 3x + 2
c. Sample answer: ( 2 x + 4) − ( x + 2)
(
)
(
)=
(
2. ( −8 z + 4) + (8 z − 7) = −8 z + 4 + 8 z − 7
)
(
= −8 z + 8 z + 4 − 7
)
= −3
3. (4 − n) + 2(−5n + 3) = 4 − n − 10n + 6
=
= −n − 10n + 4 + 6
= −11n + 10
=
4.
=
(2 x + 4) − ( x
+ 2) = x − 2
d. Sample answer: ( 4 x + 3) − ( 2 x − 1)
(
) (
) (
=
=
) (
)
1
1
1
1
( w − 6) + ( w + 12) = w − 3 + w + 3
2
4
2
4
1
1
= w+ w−3+3
2
4
2
1
= w+ w−3+3
4
4
3
= w
4
5. (m − 3) − (− m + 12) = m − 3 + m − 12
= m + m − 3 − 12
=
=
(4 x + 3) − (2 x − 1) = 2 x + 4
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= 2m − 15
6. −2(c + 2.5) − 5(1.2c + 4) = −2c − 5 − 6c − 20
= −2c − 6c − 5 − 20
= −8c − 25
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57
Chapter 3
7. Sample answer: ( 2 x + 7) − ( 2 x − 4)
7. earnings = selling price − purchase price
= ( d + 2) − ( d − 2)
(
= ( d + 2) + ( − d + 2)
)(
= d + 2−d + 2
)(
)(
=
)
=
= d −d + 2+ 2
= 4
=
You earn $4.
=
3.2 Exercises (pp. 90 –91)
(2 x
Vocabulary and Concept Check
1. The algebraic expression is not linear because one
variable has an exponent with a value not equal to 1.
+ 7) − ( 2 x − 4) = 11
8. (n + 8) + (n − 12) = n + 8 + n − 12
= n + n + 8 − 12
2. The algebraic expression is linear because the variable
= 2n − 4
has an exponent with a value of 1.
3. The algebraic expression is not linear because one
9. (7 − b) + (3b + 2) = 7 − b + 3b + 2
= −b + 3b + 7 + 2
variable has an exponent with a value not equal to 1.
4. Vertical method: Align like terms vertically and add or
subtract; Horizontal method: Use properties of operations
to group like terms and simplify.
= 2b + 9
10. (2 w − 9) + (−4 w − 5) = 2 w − 9 − 4 w − 5
= 2w − 4w − 9 − 5
= −2 w − 14
5. The question "What is x more than 3x − 1?" is different.
11. (2 x − 6) + 4( x − 3) = 2 x − 6 + 4 x − 12
(3 x − 1) − ( x) = 3 x − 1 − x
= 2 x + 4 x − 6 − 12
= 2x − 1
= 6 x − 18
(3 x − 1) + ( x) = 3 x − 1 + x
= 4x − 1
12. 5(−3.4k − 7) + (3k + 21) = −17 k − 35 + 3k + 21
= −17 k + 3k − 35 + 21
Practice and Problem Solving
6. Sample answer: ( 2 x − 6) + ( x + 5)
(
)
(
)=
= −14k − 14
13. (1 − 5q ) + 2(2.5q + 8) = 1 − 5q + 5q + 16
= − 5q + 5q + 1 + 16
= 17
=
14. 3(2 − 0.9h) + (−1.3h − 4) = 6 − 2.7 h − 1.3h − 4
= −2.7h − 1.3h + 6 − 4
= −4h + 2
=
15.
(2 x
− 6) + ( x + 5) = 3x − 1
1
1
(9 − 6m) + (12m − 8) = 3 − 2m + 3m − 2
3
4
= −2m + 3m + 3 − 2
= m +1
1
2
16. − (7 z + 4) +
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1
7
(5 z − 15) = − z − 2 + z − 3
5
2
7
= − z + z − 2−3
2
1
= −2 z − 5
2
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Chapter 3
17.
total =
amount in
savings account
= (10w + 120)
+
amount in
checking account
+
(45w +
25)
= 10 w + 120 + 45w + 25
= 10 w + 45w + 120 + 25
= 55w + 145
So, you have a total of 55w + 145 in both accounts.
18. a.
total =
=
amount you
catch
+
amount your
friend catches
(3m
+
( 4m
+ 13)
+ 6)
= 3m + 13 + 4m + 6
24.
3
1
9
(3 x + 6) − (5 x − 24) = x
4
4
4
9
= x
4
9
= x
4
4
= x
4
9
5
− x + 6
2
4
5
9
− x + + 6
4
2
5
9 12
− x + +
4
2
2
21
+
2
1
= x + 10
2
25. −3 • −5 = 15, not −15
(4m + 9) − 3(2m − 5) = 4m + 9 − 6m + 15
= 4m − 6m + 9 + 15
= 3m + 4m + 13 + 6
= 7 m + 19
So, you and your friend catch a total of 7 m + 19
fireflies.
b. Amount you catch: 3(5) + 13 = 15 + 13 = 28
= −2m + 24
26. a. From Exercise 18, you and your friend catch
7 m + 19 fireflies after m minutes. Make a table that
shows the number of fireflies caught after each
minute.
Amount your friend catches:
4(5) + 6 = 20 + 6 = 26
Because 28 > 26, you catch more fireflies.
19. (−2 g + 7) − ( g + 11) = − 2 g + 7 − g − 11
= − 2 g − g + 7 − 11
= − 3g − 4
20. (6d + 5) − (2 − 3d ) = 6d + 5 − 2 + 3d
= 6d + 3d + 5 − 2
= 9d + 3
21. (4 − 5 y ) − 2(3.5 y − 8) = 4 − 5 y − 7 y + 16
= −5 y − 7 y + 4 + 16
= −12 y + 20
22. (2n − 9) − 5(−2.4n + 4) = 2n − 9 + 12n − 20
= 2n + 12n − 9 − 20
= 14n − 29
1
1
23.
(− 8c + 16) − (6 + 3c) = − c + 2 − 2 − c
8
3
= −c − c + 2 − 2
= − 2c
+
+1
+1
+1
+1
Time
(minutes)
Number of
fireflies caught
0
7(0) + 19 = 19
1
7(1) + 19 = 26
2
7( 2) + 19 = 33
3
7(3) + 19 = 40
4
7( 4) + 19 = 47
5
7(5) + 19 = 54
+1
+7
+7
+7
+7
+7
The number of fireflies caught increases by 7 each
minute. So, 7 fireflies are caught each minute during
the competition.
b. At 0 minutes into the competition, you and your friend
already have caught 7(0) + 19 = 19 fireflies.
27. no; If the variable terms are opposites, the sum is a
numerical expression.
28. Add the two given sides:
(5n + 6) + (4n + 5) = 5n + 6 + 4n + 5
= 5n + 4 n + 6 + 5
= 9n + 11
Subtract the result from above from the perimeter:
(17 n + 11) − (9n + 11) = 17 n + 11 − 9n − 11
= 17 n − 9n + 11 − 11
= 8n
So, the expression that represents the measure of the third
side is 8n.
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59
Chapter 3
29. difference = Cab Cruiser Charge − taxi Express charge
Let x be the cost per mile.
Cab Cruiser: 2.75 + 3.9x
=
Taxi Express: 2.6 + 3.65x
(2.75 + 3.9 x) − (2.6 + 3.65 x)
=
= 2.75 + 3.9 x − 2.6 + 3.65 x
= 3.9 x − 3.65 x + 2.75 − 2.6
=
= 0.25 x + 0.15
=
30. Let x be the width of the floor.
Let x + 10 be the length of the floor.
2( x) + 2( x + 10) = 2 x + 2 x + 20
= 4 x + 20
Because this expression counts the 4 corner tiles twice,
you must subtract 4 from the expression to find the
number of tiles.
So, 4 x + 20 − 4 = 4 x + 16 is the expression that
represents the number of tiles along the outside of the
room.
31.
 4
1
 5
 3
4
4
+
5
3
12
20
+
15 15
12 + 20
15
32
2
, or 2
15
15
34. − x + 4 y = −  −  + 4 
35. D; Use a net to find the area of each face.
Top: 5 • 5 = 25
Bottom: 5 • 5 = 25
Front: 5 • 5 = 25
Back: 5 • 5 = 25
Side: 5 • 5 = 25
Side: 5 • 5 = 25
5 ft
(2 x − 4) − ( x − 1) = 2 x − 4 − x + 1
= 2x − x − 4 + 1
top
= x −3
5 ft
5 ft
5 ft
5 ft
side
front
When x = 3: 3 − 3 = 0 = 0
side
When x = −3: −3 − 3 = −6 = 6
back
5 ft
Fair Game Review
4 1
+
5
3
12
5
= −
+
15 15
−12 + 5
=
15
7
= −
15
bottom
32. x + y = −
 4
1
 5
 3
8
6
− +
5
3
24
30
−
+
15
15
− 24 + 30
15
6
2
, or
15
5
33. 2 x + 6 y = 2 −  + 6 
=
=
=
=
Surface = Area + Area + Area + Area + Area + Area
of
of
of a
of a
Area
of
of
bottom front back
side
side
top
S
= 25 +
+ 25 + 25 + 25 + 25
25
= 150
So, the surface area is 150 square feet.
3.2 Extension (pp. 92–93)
Practice
1. 9 = 3 ∙ 3
21 = 3 ∙ 7
The GCF is 3.
9 + 21 = 3(3) + 3(7)
= 3(3 + 7)
2. 32 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
48 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3
The GCF is 2 • 2 • 2 • 2 = 16.
32 − 48 = 16(2) − 16(3)
= 16(2 − 3)
60
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Chapter 3
3. 8x = 2 ∙ 2 ∙ 2 ∙ x
2=2
1
2
13. Factor out − .
The GCF is 2.
1
1
− x + 6 = − ( x − 12)
2
2
8 x + 2 = 2(4 x) + 2(1)
= 2(4 x + 1)
4.
3y = 3 ∙ y
24 = 2 ∙ 2 ∙ 2 ∙ 3
1
4
14. Factor out − .
1
5
1
− x − y = − (2 x + 5 y)
2
4
4
The GCF is 3.
3 y − 24 = 3( y ) − 3(8)
= 3( y − 8)
P = 4s
15.
(12 x − 32) = 4 s
5. 20z = 2 ∙ 2 ∙ 5 ∙ z
8=2∙2∙2
The GCF is 2 • 2 = 4.
20 z − 8 = 4(5 z ) − 4(2)
= 4(5 z − 2)
4(3 x − 8) = 4 s
4(3 x − 8)
4s
=
4
4
(3 x − 8) = s
So, the side length of the mat is (3x − 8) feet.
6. 15w = 5 ∙ 3 ∙ w
65 = 5 ∙ 13
16. a.
The GCF is 5.
Table
15w + 65 = 5(3w) + 5(13)
Extended Table
3 ft
= 5(3w + 13)
6 ft
3 ft
Leaves
3 ft
7. 36a = 2 ∙ 2 ∙ 3 ∙ 3 ∙ a
16b = 2 ∙ 2 ∙ 2 ∙ 2 ∙ b
36a + 16b = 4(9a ) + 4(4b)
= 4(9a + 4b)
8. 21m = 3 ∙ 7 ∙ m
49n = 7 ∙ 7 ∙ n
The GCF is 7.
21m − 49n = 7(3m) − 7(7 n)
= 7(3m − 7 n)
9. Factor out
1
.
3
1
1
1
b − = (b − 1)
3
3
3
10. Factor out
b.
3 ft
3 ft
3 ft
The GCF is 2 • 2 = 4.
Leaves
A = w
(18 + 6 x) =  • 3
6(3 + x) =  • 3
•3
6(3 + x)
=
3
3
2(3 + x) = 
6 + 2x = 
So, the expression that represents the length of the
extended table is (6 + 2 x). The x represents the
shorter dimension of each leaf.
3
.
8
3
3
3
d +
= ( d + 2)
8
4
8
11. Factor out 2.2.
2.2 x + 4.4 = 2.2( x + 2)
12. Factor out 4.
3

4h − 3 = 4 h − 
4

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61
Chapter 3
17.
1
h(b1 + b2 )
2
1
1 1 
3
 x −  =  (b1 + b2 )
4
4
2 2 


A =
1 1
3
 x −  =  (b1 + b2 )
4
4  4

3
1
1
x =  (b1 + b2 ) +
4
4
 4
9. ( s + 12) + (3s − 8) = s + 12 + 3s − 8
= s + 3s + 12 − 8
= 4s + 4
10. (9t + 5) + (3t − 6) = 9t + 3t + 5 − 6
= 12t − 1
11. (2 − k ) + 3(−4k + 2) = 2 − k − 12k + 6
= −k − 12k + 2 + 6
 1 
3
1
4 • x = 4 •   (b1 + b2 ) + 
4
4
4
 
3x = (b1 + b2 ) + 1
3x − 1 = (b1 + b2 )
Sample answer: One possible pair of expressions is 2x
and x − 1. A second possible pair is 2 x − 1 and x.
Study Help
Available at BigIdeasMath.com.
Quiz 3.1–3.2
= −13k + 8
12.
1
1
1
1
(q − 12) + (q + 9) = q − 3 + q + 3
4
3
4
3
1
1
= q + q −3+3
4
3
3
4
=
q + q
12
12
7
=
q
12
13. (n − 8) − (−2n + 2) = n − 8 + 2n − 2
1. Terms: 11x, 2 x
= n + 2n − 8 − 2
Like terms: 11x and 2 x
= 3n − 10
2. Terms: 9 x, − 5 x
14. −3(h − 4) − 2(−6h + 5) = −3h + 12 + 12h − 10
= −3h + 12h + 12 − 10
Like terms: 9 x and − 5 x
= 9h + 2
3. Terms: 21x, 6, − x, − 5
Like terms: 21x and − x, 6 and − 5
15. Factor out 5.
5c − 15 = 5(c − 3)
4. Terms: 8 x, 14, − 3 x, 1
Like terms: 8 x and − 3 x, 14 and 1
5. 2(3x + x) = 2(3x) + 2( x)
= 6x + 2x
= (6 + 2) x
2
.
9
2
2
2
j +
= ( j + 3)
9
3
9
17. Factor out 2.4.
2.4n + 9.6 = 2.4( n + 4)
= 8x
6. − 7 + 3x + 4 x = − 7 + (3 + 4) x
= −7 + 7x
7. 2 x + 4 − 3x + 2 + 3 x = 2 x − 3x + 3 x + 4 + 2
= ( 2 − 3 + 3) x + 4 + 2
= 2x + 6
8. 7 x + 6 + 3x − 2 − 5 x = 7 x + 3x − 5 x + 6 − 2
= (7 + 3 − 5) x + 6 − 2
= 5x + 4
62
16. Factor out
Big Ideas Math Red
Worked-Out Solutions
18. Factor out − 6.
−6 z + 12 = −6( z − 2)
19. Each
costs each costs
each
costs
and
brush $3.99, roller $6.89,
paint can $21.79.
Let x represent the number of brushes, the number of
rollers, and the number of paint cans.
3.99 x + 6.89 x + 21.79 x = (3.99 + 6.89 + 21.79) x
= 32.67 x
The expression 32.67x represents the total amount of
money you spend for painting supplies.
Copyright © Big Ideas Learning, LLC
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Chapter 3
20.
total apples
apples you
apples your friend
=
+
picked
pick
picks
c. p − 7 = −3
=
= ( 2n − 3) + ( n + 4)
= 2n − 3 + n + 4
= 2n + n − 3 + 4
=
= 3n + 1
So, you and your friend pick (3n + 1) apples.
The total number of apples you and your friend picked is
one more than 3 full baskets.
=
21. P = 2 + 2 w
= 2(3w) + 2 w
=
= 6w + 2w
So, p = 4.
= ( 6 + 2) w
d. −15 = t − 5
= 8w
The expression 8w represents the perimeter of the
exercise mat.
=
Section 3.3
=
3.3 Activity (pp. 96 –97)
1. a. To get the variable tile by itself, remove the red tiles on
the left side by adding three yellow tiles to each side.
How many zero pairs can you remove from each side?
three
=
=
=
=
=
So, −10 = t or t = −10.
2. a. Remove the yellow tiles on the right side by adding
=
So, x = −1.
two red tiles to each side.
How many zero pairs can you remove from the right
side? two
b. z − 6 = 2
=
=
=
=
=
=
=
So, n = − 7.
=
So, z = 8.
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63
Chapter 3
3. a. − 4 = x + 1
b. y + 10 = −5
=
=
=
=
=
=
=
=
So, −5 = x or x = −5.
So, y = −15.
b. x − 3 = 3
c. 7 + b = −1
=
=
=
=
=
=
=
=
So, x = 6.
So, b = − 8.
c. x − 5 = − 4
d. 8 = 12 + z
=
=
=
=
=
=
=
So, x = 1.
=
So, − 4 = z or z = − 4.
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Chapter 3
d. 5 = x − 2
2. w + 13.2 =
− 13.2
=
10.4
?
−2.8 + 13.2 = 10.4
w = −2.8
10.4 = 10.4 
The solution is w = −2.8.
=
5
1
= −
6
6
?
4 5
1
− =−
6 6
6
1
1
− = −
6
6
5
1
= −
6
6
5
5
+
+
6
6
Check: x −
3. x −
=
4
2
=
6
3
x =
=
The solution is x =
So, 7 = x or x = 7.
4. Words:
x + 57 = 19
4. a.
x + 57 − 57 = 19 − 57
2
.
3
120.50
=
P − 145.25
The equation is 120.50 = P − 145.25.
So, x = − 38°F.
b. Sample answer: The melting point of bromine is 19
degrees Fahrenheit. This is 57 degrees Fahrenheit
more than the melting point of mercury. So, subtract
57 from 19 to get 19 − 57 = −38. So, the melting
point of mercury is − 38°F .
The method in part (a) is the algebraic solution
because it assigns a variable to the unknown quantity.
The method in part (b) is the arithmetic solution
because it is a direct numerical solution that does not
use any variables.
5. Sample answer: Use algebra tiles to model the equation.
Then get the variable tile by itself by adding tiles and
removing zero pairs.
6. Sample answer: To solve an addition equation, subtract to
get the variable by itself. To solve a subtraction equation,
add to get the variable by itself.
7. Sample answer: x = 2 is a solution of one equation and
x = 3 is a solution to a different equation.
8. Sample answer: The value of a stock varies from day to
day. The weights of golden retrievers vary from dog to
dog.
3.3 On Your Own (pp. 98 –99)
1. p − 5 = −2

The profit is $145.25 less than
today
the profit yesterday.
Equation:
x = −38
+5
w + 13.2 = 10.4
Check:
− 13.2
Check: p − 5 = −2
+5
p = 3
5. Words:
Your score is 27 points less than
your friend’s score.
Variable: Let s be your friend’s score.
Equation:
−12
=
s − 27
−12 = s − 27
+ 27
+ 27
15 = s
Your friend’s score is 15 points.
3.3 Exercises (pp. 100 –101)
Vocabulary and Concept Check
1. To solve m + 6 = − 4, use the Subtraction Property of
Equality.
2. Addition and subtraction are inverse operations.
3. Check m = −2 in the first equation.
m + 3 = −5
?
−2 + 3 = −5
1 ≠ −5
Because you get a false statement, the two equations are
not equivalent.
?
3 − 5 = −2
− 2 = −2 
The solution is p = 3.
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65
Chapter 3
4. Solve the equations.
x + 3 = −1
−3 −3
x = −4
11.
x + 1 = −5
−1 −1
x = −6
1
2
= q +
2
3
2
2
−
−
3
3
−
x − 2 = −6 x − 9 = −13
+2 +2
+9
+9
x = −4
x = −4
1
= q
6
Check:
The equation x + 1 = −5 does not belong because it has
a solution of x = −6, whereas the other three equations
have the solution x = − 4.
Practice and Problem Solving
5. a − 6 =
+6
13
Check: a − 6 = 13
+6
?
19 − 6 = 13
a = 19
13 = 13 
The solution is a = 19.
6.
−3 = z − 8
+8
Check: −3 = z − 8
+8
?
−3 = 5 − 8
5 = z
−3 = −3 
The solution is z = 5.
7. −14 = k + 6
−6
Check: −14 = k + 6
−6
?
−14 = −20 + 6
−20 = k
−14 = −14 
The solution is k = −20.
8. x + 4 = −14
−4
Check:
−4
+ 7.6
10. −10.1 = w + 5.3
− 5.3
2
3
2
3
4
6
c − 7.6 = − 4
?
−4 = −4 
1
5 19
15 19 
 1
= − + 
 −2 + 3 = − +
6
2
6
6
6
 2
1
2
1
−2
2
1
−2
2
1
−2
2
1
−2
2
1
−2 
2
= −2
?
=
?
=
?
=
?
=
=
The solution is p =
Check: −10.1 = w + 5.3
− 5.3
−15.4 = w
The solution is w = −15.4.
66
1
6
1
−3
6
19
−
6
19
−
6
15
−
6
1
−2
2
Check: p − 3
x + 4 = −14
The solution is c = 3.6.
2
3
4
6
4
6
2
p =
3
3.6 − 7.6 = − 4
c = 3.6
1
+
6
?
1
=− +
6
? 3
=
6
1
= 
2
?
p =
−14 = −14 
+ 7.6
2
3
=−
1
1
= −2
6
2
1
1
+3
+3
6
6
?
Check:
= q +
12. p − 3
The solution is x = −18.
9. c − 7.6 = − 4
1
2
1
2
1
2
1
2
1
2
1
The solution is q = − .
6
−18 + 4 = −14
x = −18
3 4
1 2
=
− 
 −
2
3
6
6

Big Ideas Math Red
Worked-Out Solutions
?
−10.1 = −15.4 + 5.3
−10.1 = −10.1 
13. g − 9 = −19
+9
2
.
3
Check:
g − 9 = −19
?
+9
−10 − 9 = −19
g = −10
−19 = −19 
The solution is g = −10.
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Chapter 3
14.
−9.3 = d − 3.4
+ 3.4
Check: −9.3 = d − 3.4
+ 3.4
?
−9.3 = −5.9 − 3.4
−5.9 = d
−9.3 = −9.3 
The solution is d = −5.9.
15.
4.58 + y =
− 4.58
2.5
4.58 + y = 2.5
?
4.58 − 2.08 = 2.5
y = −2.08
1
4
= r −
4
5
4
4
+
+
5
5
−2
29
= r
20
9
−1
= r
20
2.5 = 2.5 
1
4
1
−2
4
1
−2
4
1
−2
4
1
−2
4
Check: −2
The solution is y = −2.08.
16. x − 5.2 = −18.73
+ 5.2
+ 5.2
x = −13.53
x − 5.2 = −18.73
Check:
?
−13.53 − 5.2 = −18.73
−18.73 = −18.73 
The solution is x = −13.53.
5
1
=
9
6
5
5
−
−
9
9
17. q +
q = −
Check:
3
10 
1 5
=
− 
 −
18 18 
6 9
7
18
5
9
7
5
−
+
18 9
7
10
−
+
18 18
3
18
1
6
q +
=
?
=
?
=
?
=
=
1
6
1
6
1
6
1
6
1
6
The solution is q = −
7
.
18
4
5
9
4
−1 −
20 5
29 16
−
−
20 20
45
−
20
1
−2 
4
= r −
?
=
?
=
?
=
=
The solution is r = −1
9
.
20
3
5
= 1
8
6
3
3  5
3
11 27
44
81 
−3
− 3 1 − 3 =
−
=
−

8
8  6
8
6
8
24
24 
37
13
= −1
w = −
24
24
19. w + 3
Check:

4
9 4
45 16 
 1
= − +
= −
+
 −2 +

4
5
4
5
20
20 

−
Check:
− 4.58
18.
3
8
3
+3
8
27
+
8
81
+
24
44
24
5
1
6
w+3
13
−1
24
37
−
24
37
−
24
5
6
? 5
=1
6
? 5
=1
6
? 5
=1
6
? 5
=1
6
5
=1
6
=1

13
The solution is w = −1 .
24
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Big Ideas Math Red
Worked-Out Solutions
67
Chapter 3
20.
2
2
4 + k = −3
5
11
26. Words:
2
35 22 
 2
−3 − 4 = −
−
2  11
5
11
5 
−4 
5 
175 242 
=−
−


55
55 

417
32
k = −
= −7
55
55
2
−4
5
Check:
4
2
5
22
5
242
55
4
2
+ k
5
32
−7
55
32
−7
55
417
−
55
175
−
55
2
−3
11
2
= −3
11
?
2
= −3
11
?
2
= −3
11
?
2
= −3
11
?
2
= −3
11
2
= −3 
11
The solution is k = −7
The temperature is 184.9°F less
of dry ice
than the outside
temperature.
Variable: Let t be the outside temperature.
Equation:
+ 184.9
+ 184.9
75.6 = t
The outside temperature is 75.6°F.
27. Words:
A company’s is
profit
Equation:
22. n − 4 = −15
+4
n = −11
The solution is n = −11.
− 10
3
− 10
c = −7
The solution is c = −7.
24. y + ( −3) = −8
y − 3 = −8
+3
+3
y = −5
The solution is y = −5.
25. p − 6 = −14
+6
P + 2.54
− 2.54
−1.16 = P
The company’s profit last year was −$1.16 million.
The difference in elevation
of a helicopter and a
is
submarine
Words:
 3
h − −7 
 4
Equation:
x = 2
23. c + 10 =
=
1
18 meters.
2
Variable: Let h be the elevation of the helicopter.
−8
+4
1.38
1.38 = P + 2.54
− 2.54
each side.
−8
$2.54 million more
than last year.
Variable: Let P be the company’s profit last year.
28.
x + 8 = 10
t − 184.9
−109.3 = t − 184.9
32
.
55
21. You need to use an inverse operation. Subtract 8 from
=
−109.3
+6
p = −8
=
18
1
2
1
 3
h −  − 7  = 18
2
 4
3
1
h + 7 = 18
4
2
3
3 1
3
37 31
74 31 
−7
− 7 18 − 7 =
−
=
− 
4
4 2
4
2
4
4
4
43
h
=
4
3
h
= 10
4
3
The elevation of the helicopter is 10 meters.
4
29. Let x be the unknown side length.
x + 3 + 5 = 12
x + 8 = 12
−8
−8
x = 4
The unknown side length is 4 centimeters.
The solution is p = −8.
68
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Chapter 3
30. Let x be the unknown side length.
x + 8.3 + 3.8 + 8.3 =
x + 20.4 =
− 20.4
34. Words:
Total
is
distance
24.2
24.2
x = 3.8
Equation:
65
The unknown side length is 3.8 inches.
65
31. Let x be the unknown side length.
x + 6.4 + 11.1 + 5.2 =
34.6
x + 22.7 =
34.6
The height plus the height is
of the
of the
statue
pedestal
305 feet.
h
h + 153 =
− 153
+
153
=
305
− 71.1
Equation:
−200
is
The missing angle measures 108.9°.
1
50 feet higher
6
than your second
jump.
=
2
1
1002 301 

−200 − 50 = −
−
1 
5
6
5
6 
− 50 
6 
6012 1505 
= −
−


30
30 

7517
= j
30
17
−250
= j
30
−
17
Your second jump is −250 feet.
30
36. Let p be your score in the fourth round. Because the
leader has 311.62 points, you need a total score of
311.62 points to tie.
63.43 + 87.15 + 81.96 + p = 311.62
232.54 + p =
1
j + 50
6
2
1
= j + 50
5
6
1
− 50
6
− 71.1
m = 108.9
Variable: Let j be the height of your second jump.
2
−200
5
1
3
m + 71.1 = 180
− 153
Your first
jump
24
35. m + 30.3 + 40.8 = 180
305
The height of the statue is 152 feet.
Words:
+
d
619
= d
15
4
41 = d
15
h = 152
33.
=
4
The bus has traveled 41 kilometers.
15
Variable: Let h be the height of the statue.
Equation:
3
5
1
328 73 
 3
65 − 24 =
−
1  5
3
5
3 
− 24 
3 
984 365 
=
−


15
15 

11.9
The unknown side length is 11.9 feet.
32. Words:
distance
remaining.
3
1
= d + 24
5
3
1
− 24
3
− 22.7
x =
plus
Variable: Let d be the distance the bus already traveled.
− 20.4
− 22.7
distance
already
traveled
− 232.54
311.62
− 232.54
p = 79.08
You need more than 79.08 points in the fourth round
to win.
37. First solve x + 6 = 2. Then substitute the result in the
expression 2 x − 1.
x+6 =
−6
2
−6
x = −4
2 x − 1 = 2( − 4) − 1 = −8 − 1 = −9
So, the value of 2 x − 1 is −9.
38. The distance between 0 and 2 on a number line is 2. The
distance between 0 and −2 on a number line is also 2.
So, x = 2 or x = −2.
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Big Ideas Math Red
Worked-Out Solutions
69
Chapter 3
39.
x −2 =
b. 2k = −8
4
+2
+2
=
x = 6
The distance between 0 and 6 on a number line is 6. The
distance between 0 and −6 on a number line is also 6.
So, x = 6 or x = −6.
40.
=
=
x + 5 = 18
−5
=
So, k = − 4.
−5
c. −15 = 3t
x = 13
The distance between 0 and 13 on a number line is 13.
The distance between 0 and −13 on a number line is
also 13.
=
=
So, x = 13 or x = −13.
=
Fair Game Review
41. −7 × 8 = −56
42. 6 × ( −12) = −72
43. 18 ÷ ( −2) = −9
44. −26 ÷ 4 = −6.5
45. B; Students who voted for you =
3
× 144
4
=
3 × 144
4
=
=
So, −5 = t or t = −5.
d. −20 = 5m
36
=
1
= 108
Female students who voted for you =
=
5
× 108
9
=
12
=
=
5 × 108
9
=
1
=
= 60
So, 60 female students voted for you.
Section 3.4
=
So, − 4 = m or m = − 4.
3.4 Activity (pp. 102–103)
e. 4h = −16
1. a. Your goal is to get one variable tile by itself. Because
there are 3 variable tiles, divide the 12 tiles into
3 equal groups.
=
=
=
=
=
So, x = − 4.
70
Big Ideas Math Red
Worked-Out Solutions
=
=
=
=
=
=
So, h = − 4.
Copyright © Big Ideas Learning, LLC
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Chapter 3
2. a. 8 = 4x
d. 3 x = −18
=
=
=
=
=
=
=
=
=
=
=
So, x = −6.
So, x = 2.
3. a. 32 x = 400
b. 6 x = −12
32 x
400
=
32
32
x = 12.5
=
So, it takes 12.5 seconds.
Divide the distance by the rate to get
=
=
b.
=
400
= 12.5.
32
x
= −96
3.6
x
• 3.6 = −96 • 3.6
3.6
x = −345.6
=
=
=
So, the melting point of nitrogen is − 345.6°F .
=
Multiply the melting point of radon by 3.6 to get
− 96 • 3.6 = − 345.6.
So, x = −2.
c. −10 = 2x
c.
=
=
3
x = −6
4
4 3
4
• x = • ( − 6)
3 4
3
24
x = − , or − 8
3
So, last year's profit is − $8 million.
=
Multiply this year’s profit by the reciprocal of
=
get −6 •
So, −5 = x or x = −5.
Copyright © Big Ideas Learning, LLC
All rights reserved.
3
to
4
4
= −8.
3
Sample answer: The method that uses the equation is
the algebraic solution because it assigns a variable to
the unknown quantity. The other method is the
arithmetic solution because it is a direct numerical
solution that does not use any variables.
Big Ideas Math Red
Worked-Out Solutions
71
Chapter 3
4. If you have an equation involving multiplication, you
divide each side by the same (nonzero) number. If you
have an equation involving division, you multiply each
side by the same number.
Sample answer:
3x = −9
3x
−9
=
3
3
x = −3
x
= −2
5
x
5•
= 5 • ( −2)
5
x = −10
Check:
7. Words:
−a
−24
=
−1
−1
a = 24
x
= −2
5
−10 ?
= −2
5
−a = −24
?
−( 24) = − 24
−24 = −24 
The solution is a = 24.
3.
Check: 3 = −1.5n
3 = −1.5n
3
−1.5n
=
−1.5
−1.5
−2 = n
?
3 = −1.5( −2)
3 = 3
The solution is n = −2.
2
−14 = x
4.
3
3
3
• ( −14) =
•
2
2
−21 = x
2
Check: −14 = x
3
? 2
2 
−14 = ( −21)
 x
3
3 
−14 = −14 
The solution is x = −21.
5.
8
8
− b = 5
− b = 5
Check:
5
5
8  25  ?
5  8 
5
− • −  = 5
− •  − b = − • 5
5  8
8  5 
8
5 = 5
25
b = −
8
1
= −3
8
1
The solution is b = −3 .
8
72
Big Ideas Math Red
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?
3
(−24) = −9
8
−9 = −9 
The record is
low in
Hawaii
−0.15
times
the record
low in
Alaska.
Variable: Let t be the record low in Alaska.
The solution is x = −10.
Check:
3
h = −9
8
Check:
The solution is h = −24.
−2 = −2 
2. −a = −24
3
h = −9
8
8 3 
8
•  h  = • ( −9)
3 8 
3
h = −24
x
= −1
2
x
2•
= 2 • ( −1)
2
x = −2
3.4 On Your Own (pp. 104–105)
1.
6.
Equation:
=
12
− 0.15
×
t
12 = −0.15t
12
−0.15t
=
−0.15
−0.15
−80 = t
The record low temperature in Alaska is −80°F.
3.4 Exercises (pp. 106 –107)
Vocabulary and Concept Check
1. You can use multiplication to solve equations involving
division because multiplication and division are inverse
operations.
2. Sample answer: Opening a door and then closing the
door are inverse operations. Walking from home to the
store and walking from the store back home are inverse
operations.
3. Dividing by 5 will undo multiplying by 5.
4. Adding 12 will undo subtracting 12.
5. Multiplying by −8 will undo dividing by −8.
6. Subtracting −6 will undo adding −6.
Practice and Problem Solving
7. 3h = 15
Check:
3h
15
=
3
3
h = 5
3h = 15
?
3(5) = 15
15 = 15 
The solution is h = 5.
8. −5t = − 45
Check:
−5t
− 45
=
−5
−5
t = 9
−5t = − 45
?
−5(9) = − 45
− 45 = − 45 
The solution is t = 9.
Copyright © Big Ideas Learning, LLC
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Chapter 3
9.
n
= −7
2
n
2•
= 2 • ( −7)
2
n = −14
Check:
n
= −7
2
−14 ?
= −7
2
−7 = −7 
17.
The solution is n = −14.
10.
k
= 9
−3
k
−3 •
= −3 • 9
−3
k = −27
Check:
Check:
5m
−10
=
5
5
m = −2
−14.4
−0.6 p
=
−0.6
−0.6
24 = p
19.
5m = −10
?
−10 = −10

?
−14.4 = −0.6( 24)
−14.4 = −14.4 
3
3
g = −12
g = −12
Check:
4
4
?
3
4 3 
4
•  g  = • ( −12)
( −16) = −12
4
3 4 
3
−12 = −12 
g = −16
The solution is g = −16.
Check:
8t = −32
20.
?
8( − 4) = −32
−32 = −32

The solution is t = − 4.
2
8 = − c
5
5
5
− •8 = − •
2
2
 2 
− c
 5 
−20 = c
Check:
1.6
−0.2 x
=
−0.2
−0.2
x = −8
?
−0.2( −8) = 1.6
1.6 = 1.6
b
14.
Check: −10 = −
4
?
40
 b
−10 = −
− 4 • ( −10) = − 4 •  − 
4
 4
−10 = −10
40 = b
21.


The solution is b = 40.
−6 p
48
=
−6
−6
p = −8
4
− f = −3
9
9  4 
9
− •  − f  = − • ( −3)
4  9 
4
27
4
3
= 6
4
4
− f = −3
Check:
9
4  27  ?
− •   = −3
9  4
f =
b
−10 = −
4
Check:
2
Check: 8 = − c
5
?
2
8 = − ( −20)
5
8 = 8
The solution is c = −20.
−0.2 x = 1.6
The solution is x = −8.
15. −6 p = 48

The solution is p = 24.
5( −2) = − 10
8t
−32
=
8
8
t = −4
13. −0.2 x = 1.6
n
= 5
1.6
8 ?
= 5
1.6
5 = 5
Check: −14.4 = −0.6 p
18. −14.4 = −0.6 p
k
= 9
−3
−27 ?
= 9
−3
9 = 9
The solution is m = −2.
12. 8t = −32
Check:
The solution is n = 8.
The solution is k = −27.
11. 5m = −10
n
= 5
1.6
n
1.6 •
= 1.6 • 5
1.6
n = 8
−6 p = 48
−3 = −3
?
−6( −8) = 48
48 = 48


3
The solution is f = 6 .
4
The solution is p = −8.
16. −72 = 8d
Check: −72 = 8d
8d
−72
=
8
8
−9 = d
?
−72 = 8( −9)
−72 = −72 
The solution is d = −9.
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73
Chapter 3
22.
8
26 = − y
5
5
5  8 
− • 26 = − •  − y 
8
8  5 
27. Let x be the number. An equation is
x
= 21
−1.5
x
−1.5 •
= −1.5 • 21
−1.5
x = −31.5
65
= y
4
1
−16 = y
4
−
8
Check: 26 = − y
5
?
8
26 = − •
5
x
= 21.
−1.5
The solution is x = −31.5.
28. Words:
 65 
− 
 4
Amount times number of is
earned
newspapers
the cost
of the
cleats.
Variable: Let n be the number of newspapers.
26 = 26 
Equation:
1
The solution is y = −16 .
4
•
0.75
=
n
36
0.75n = 36
−4.2 x = 21
0.75n
36
=
0.75
0.75
n = 48
21
− 4.2 x
=
− 4.2
− 4.2
You have to sell 48 newspapers to buy the exercise
cleats.
23. You need to divide each side by − 4.2.
x = −5
29. Words:
24. Let x be the number. An equation is
x
= −16.
−9
x
= −16
−9
x
−9 •
= −9 • ( −16)
−9
x = 144
The solution is x = 144.
2
3
=
.
25. Let x be the number. An equation is x •
5
20
2
3
x•
=
5
20
2 5
3
5
x• •
=
•
5 2
20 2
3
x =
8
The solution is x =
3
.
8
26. Let x be the number. An equation is 15 x = −75.
15 x = −75
15 x
−75
=
15
15
x = −5
Rate equals distance divided by time.
Variable: Let x be the distance.
Equation: 12
3
5
3
30 • 12
5
63
30 •
5
378
12
=
3
5
x
30
=
x
30
= 30 •
x
30
= x
= x
The rock climber climbs 378 feet in 30 minutes.
30. Sample answer:
a. 6 x = −18
6x
−18
=
6
6
x = −3
b.
x
= 2
−1.5
x
−1.5 •
= −1.5 • 2
−1.5
x = −3
The solution is x = − 5.
74
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Chapter 3
31. Sample answer:
35.
a. −2 x = 4.4
b.
Stock
return
times number is
of days
the total
return.
−2 x
4.4
=
−2
−2
x = −2.2
Variable: Let n be the number of days.
x
= −2
1.1
x
1.1 •
= 1.1 • ( −2)
1.1
x = −2.2
−1.26n = −10.08
−10.08
After 8 days, the total return is −$10.08.
−8 x
4
=
−8
−8
1
x = −
2
Portion of times the
equals the
students
number
number
that voted
of students
of ballots.
Variable: Let s be the number of students.
1
x
b.
= −
3
6
x
= 3•
3•
3
3
4
Equation:
1
2
a. − 4 x = 5
5
− 4x
=
−4
−4
1
4
x
5
=
8
−2
x
5
−2 •
= −2 •
8
−2
5
x = −
4
1
= −1
4
2
2
34. To solve − c = 16, you can divide each side by − or
3
3
3
multiply each side by − or multiply each side by 3,
2
then divide each side by −2.
Copyright © Big Ideas Learning, LLC
All rights reserved.
•
s
=
1464
3
s = 1464
4
4 3
4
• s =
• 1464
3 4
3
s = 1952
 1
− 
 6
There are 1952 students.
33. Sample answer:
x = −1
=
n
−1.26n
−10.08
=
−1.26
−1.26
n = 8
36. Words:
a. −8 x = 4
x = −
•
Equation: −1.26
32. Sample answer:
b.
Words:
37.
31
x = −62
25
25 31
25
•
• ( −62)
x =
31 25
31
x = −50
So, x = −50 feet.
38. Words:
The price of a
is
bike at Store A
5
the price at Store B.
6
Variable: Let b be the price at Store B.
Equation:
150.60
=
5
•b
6
5
b
6
6
6 5
• 150.60 = • b
5
5 6
180.72 = b
150.60 =
So, you save $180.72 − $150.60 = $30.12 by buying
the bike at Store A.
Big Ideas Math Red
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75
Chapter 3
39. −2 m = −10
Fair Game Review
41. 5 − 12 = 5 + ( −12) = −7
−2 m
−10
=
−2
−2
m = 5
42. −7 − 2 = − 7 + ( − 2) = −9
The distance between 0 and 5 on a number line is 5. The
distance between 0 and −5 on a number line is also 5.
So, m = 5 or m = −5.
5
40. Distance already traveled =
• 1250
7
6250
=
7
6
= 892 miles
7
Find the rate of travel after 4 days using the formula for
distance.
d = rt
6
892
7
6250
7
1 6250
•
4
7
3125
14
3
223
14
= r •4
= 4r
=
1
• 4r
4
43. 4 − ( −8) = 4 + 8 = 12
44. −14 − ( −5) = −14 + 5 = −9
45. B; 1 −
75
120
75
120 − 75
45
3
=
−
=
=
=
120
120 120
120
120
8
3
of the apartments have not been scheduled to receive
8
new carpet.
Section 3.5
3.5 Activity (pp. 108–109)
1. Remove the three red tiles on the left side by adding
three yellow tiles to each side.
How many zero pairs can you remove from each side?
three
Because there are two green tiles, divide the red tiles into
two equal groups.
= r
=
= r
=
3
Your rate of travel is 223 miles per day.
14
=
6
1
You have 1250 − 892 = 357 miles left to travel.
7
7
Find the time using the formula for distance.
=
=
d = rt
1
7
2500
7
14
2500
•
3125
7
8
5
3
1
5
357
3
= 223 t
14
3125
t
=
14
14
3125
t
=
•
3125
14
=
So, x = −1.
= t
= t
3
You have 1 more days until you reach your destination.
5
76
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Chapter 3
2.
2x − 3 = − 5
Write the equation.
b. −13 = 3 x − 4
=
2 x − 3 + 3 = − 5 + 3 Add 3 to each side.
2x = − 2
Simplify.
−2
2x
=
2
2
Divide each side by 2.
x = −1
=
Simplify.
So, x = −1.
=
a. The first step is adding 3 to each side.
b. You use the same steps to complete the problem in
Activity 1 as you do in Activity 2, only you do not use
algebra tiles.
=
=
3. a. 2 x + 2 = −6
=
=
=
So, −3 = x or x = −3.
Check:
=
−13 = 3 x − 4
−13 + 4 = 3 x − 4 + 4
−9 = 3 x
−9
3x
=
3
3
=
−3 = x 
=
4. b. Your account now
=
has a balance
of −$10.56.
=
So, x = − 4.
Check:
2 x + 2 = −6
2 x + 2 − 2 = −6 − 2
2 x = −8
−8
2x
=
2
2
x = −4 
−10.56
You withdrew $127.32
for groceries, so add
−10.56 + 127.32 = 116.76
$127.32 to −$10.56.
You tripled your
balance by making
a deposit, so find
1
of $116.76.
3
1
• 116.76 = 38.92
3
You had $38.92 in your account before the deposit.
5. Sample answer: Model the equation using algebra tiles.
Remove the smaller tiles by adding positive tiles to both
sides if the original tiles are negative or adding negative
tiles to both sides if the original tiles are positive and then
removing all pairs of positive and negative tiles. If there
are multiple larger tiles, divide the smaller tiles into equal
groups and circle them. Keep one of the groups. This
shows the value of x.
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Big Ideas Math Red
Worked-Out Solutions
77
Chapter 3
6. The first step is to subtract 1 from both sides of the
7.
2x
50
=
2
2
x = 25
Check:
2 x + 12 − 12 = 4 − 12
6.
2 x + 12 = 4
?
2( − 4) + 12 = 4
2 x = −8
?
−8 + 12 = 4
2x
−8
=
2
2
x = −4
4 = 4
The solution is x = − 4.
Check: −5c + 9 = −16
−5c = −25
?
−25 + 9 = −16
−16 = −16
The solution is c = 5.
Check: 3( x − 4) = 9
3(7 − 4) = 9
3x − 12 = 9
?
3(3) = 9
3x − 12 + 12 = 9 + 12
9 = 9
3x = 21

The solution is x = 7.
m
+6−6
2
m
2
m
2•
2
m
2
6
+ 4a = −
5
5
2
 2 ? 6
+ 4 −  = −
5
5
 5
2 8 ? 6
− = −
5 5
5
6
6
− = −
5
5

7. 4 − 2 y + 3 = −9
Check: 4 − 2 y + 3 = −9
7 − 2 y = −9
4 − 2(8) + 3 = −9
?
7 − 7 − 2 y = −9 − 7
3x
21
=
3
3
x = 7
m
+ 6 = 10
2

2
5
2
The solution is a = − .
5
?
3• x −3•4 = 9
4.
Check:
?
−5(5) + 9 = −16
−5c
−25
=
−5
−5
c = 5
3( x − 4) = 9

2
6
+ 4a = −
5
5
2
2
6
2
− + 4a = − −
5
5
5
5
8
4a = −
5
1
1  8
• 4a =
• − 
4
4  5
a = −
−5c + 9 − 9 = −16 − 9
3.
9 = 9
The solution is z = −12.
3.5 On Your Own (pp. 110 –111)
−5c + 9 = −16
?
4 + 5 = 9
z = −12
The steps are the same.
2.
Check: −
−
2 x = 50
2 x + 12 = 4
z
+ 5 = 9
3
?
−12
+ 5 = 9
−3
z
+5 = 9
3
z
+5−5 = 9−5
3
z
− = 4
3
 z
−3 •  −  = −3 • 4
 3
2 x − 75 = −25
2 x − 75 + 75 = −25 + 75
1.
−
5.
equation.
Check:
= 10 − 6
= 4
= 2•4
= 8
m
+ 6 = 10
2
?
8
+ 6 = 10
2
?
4 + 6 = 10
10 = 10 
?
−2 y = −16
4 − 16 + 3 = −9
−2 y
−16
=
−2
−2
y = 8
−12 + 3 = −9
?
−9 = −9

The solution is y = 8.
8. 7 x − 10 x = 15
Check:
−3x = 15
7 x − 10 x = 15
?
7( −5) − 10( −5) = 15
−3x
15
=
−3
−3
x = −5
?
−35 + 50 = 15
15 = 15

The solution is x = −5.
The solution is m = 8.
78
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Chapter 3
9.
−8 = 1.3m − 2.1m
Check: −8 = 1.3m − 2.1m
−8 = −0.8m
8.
−8 = 1.3(10) − 2.1(10)
−8
−0.8m
=
−0.8
−0.8
10 = m
−8 = −8 
10h − 100 = −5
10h = 95
9.
−6t − 7 = 17
−6t − 7 = 17
Check:
?
−6( − 4) − 7 = 17
−6t = 24
3.5 Exercises (pp. 112 –113)
?
24 − 7 = 17
−6t
24
=
−6
−6
t = −4
Vocabulary and Concept Check
1. To solve two-step equations, you undo the operations
in the reverse order of the order of operations.
17 = 17

The solution is t = − 4.
2. B; The first step in solving 4 + 4n = −12 is to
8n + 16.2 = 1.6
10.
subtract 4.
8n + 16.2 − 16.2 = 1.6 − 16.2
3. D; The first step in solving 4n = −12 is to divide by 4.
4. C; The first step in solving
n
= −12 is to multiply by 4.
4
5. A; The first step in solving
n
− 4 = −12 is to add 4.
4
8n = −14.6
−14.6
8n
=
8
8
n = −1.825
Check:
Check:
2v + 7 − 7 = 3 − 7
?
?
−14.6 + 16.2 = 1.6
2v + 7 = 3
1.6 = 1.6
?
2( −2) + 7 = 3
2v = − 4
8n + 16.2 = 1.6
8( −1.825) + 16.2 = 1.6
Practice and Problem Solving

The solution is n = −1.825.
?
−4 + 7 = 3
2v
−4
=
2
2
v = −2
3 = 3

11.
−5 g + 2.3 = −18.8
−5 g + 2.3 − 2.3 = −18.8 − 2.3
−5 g = −21.1
The solution is v = −2.
4b + 3 = −9
17 = 17 
−6t − 7 + 7 = 17 + 7
The height is 9.5 feet.
7.
?
17 = 19 − 2
4
The solution is k = 3 .
5
10h
95
=
10
10
h = 9.5
2v + 7 = 3
?
 19 
17 = 5  − 2
5
19
5k
=
5
5
19
= k
5
4
3 = k
5
10h − 100 + 100 = −5 + 100
6.
Check: 17 = 5k − 2
19 = 5k
?
−8 = 13 − 21
The solution is m = 10.
10.
17 = 5k − 2
17 + 2 = 5k − 2 + 2
?
Check:
4b + 3 − 3 = −9 − 3
4b = −12
4b
−12
=
4
4
b = −3
The solution is b = −3.
−5 g
−21.1
=
−5
−5
g = 4.22
4b + 3 = −9
?
4( −3) + 3 = −9
Check:
?
?
−12 + 3 = −9
−9 = −9
−5 g + 2.3 = −18.8

−5( 4.22) + 2.3 = −18.8
?
−21.1 + 2.3 = −18.8
−18.8 = −18.8

The solution is g = 4.22.
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Big Ideas Math Red
Worked-Out Solutions
79
Chapter 3
12.
2t − 5 = −10
15.
2t − 5 + 5 = −10 + 5
Check:
2t = −5
2.2h = −7.7
2t
−5
=
2
2
5
t = −
2
1
= −2
2
−7.7
2.2h
=
2.2
2.2
h = −3.5
Check:
?
?
2t − 5 = −10
4 − 7.7 = −3.7
?
 5
2 −  − 5 = −10
 2
−3.7 = −3.7
16.
−10 = −10 
− 4.8 f + 6.4 = −8.48
− 4.8 f + 6.4 − 6.4 = −8.48 − 6.4
1
The solution is t = −2 .
2
− 4.8 f = −14.88
− 4.8 f
−14.88
=
− 4.8
− 4.8
− 4 p + 9 = −5
f = 3.1
− 4 p + 9 − 9 = −5 − 9
Check:
− 4 p = −14
−4p
−14
=
−4
−4
?
?
−14.88 + 6.4 = −8.48
−8.48 = −8.48

The solution is f = 3.1.
7.3 y − 5.18 = −51.9
17.
− 4 p + 9 = −5
7.3 y − 5.18 + 5.18 = −51.9 + 5.18
?
7
−4  + 9 = −5
 2
7.3 y = − 46.72
7.3 y
− 46.72
=
7.3
7.3
y = −6.4
?
−14 + 9 = −5
−5 = −5 
1
The solution is p = 3 .
2
11 = −5 x − 2
Check:
7.3 y − 5.18 = − 51.9
?
Check: 11 = −5 x − 2
11 + 2 = −5 x − 2 + 2
13 = −5 x
−5 x
13
=
−5
−5
13
−
= x
5
3
−2 = x
5
3
The solution is x = −2 .
5
80
− 4.8 f + 6.4 = −8.48
− 4.8(3.1) + 6.4 = −8.48
7
p =
2
1
= 3
2
14.

The solution is h = −3.5.
−5 − 5 = −10
Check:
4 + 2.2h = −3.7
4 + 2.2( −3.5) = −3.7
?
13.
4 + 2.2h = −3.7
4 − 4 + 2.2h = −3.7 − 4
Big Ideas Math Red
Worked-Out Solutions
 13 
11 = −5 −  − 2
 5
?
?
11 = 13 − 2
11 = 11 
7.3( − 6.4) − 5.18 = − 51.9
?
− 46.72 − 5.18 =
51.9
− 51.9 = −51.9

The solution is y = − 6.4.
18. The first step should be to add 6 to each side.
− 6 + 2 x = −10
− 6 + 6 + 2 x = −10 + 6
2x = − 4
2x
−4
=
2
2
x = −2
Copyright © Big Ideas Learning, LLC
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Chapter 3
19. In the second step, you should divide each side by −3,
not 3.
−3 x + 2 = −7
−3 x + 2 − 2 = −7 − 2
−3 x
−9
=
−3
−3
x = 3
3
1
g − =
5
3
3
1 1
g − + =
5
3 3
3
g =
5
3
g =
5
5 3
• g =
3 5
g =
Check:
10
3
10 1
−
+
3
3
9
−
3
−
−3
5
• ( −3)
3
−5
3
1
10
= −
g −
5
3
3
?
3
1
10
(−5) − = −
5
3
3
1 ? 10
−3 −
= −
3
3
9 1 ? 10
− −
= −
3 3
3
10
10
−
= −
3
3
1
a 5
−
= −
4 6
2
1 5
a 5 5
− +
= − +
4 6 6
2 6
2
a
=
4
6
1
a
=
4
3
1
a
4•
= 4•
4
3
4
a =
3
1
=1
3
a 5
Check:
− =
4 6
4
?
3 − 5 =
4
6
4 1 5 ?
• − =
3 4 6
1 5 ?
− =
3 6
2 5 ?
− =
6 6
3 ?
− =
6
1
− =
2
−3 x = −9
20.
21.

The solution is g = −5.
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−
3 5
 1 5
= − + 
− +
2
6
6
6

1
2
1
2
1
−
2
1
−
2
1
−
2
1
−
2
1
−
2
−

1
The solution is a = 1 .
3
Big Ideas Math Red
Worked-Out Solutions
81
Chapter 3
22.
1
5
+ 2z = −
3
6
1 1
5 1
− + + 2z = − +
3 3
6 3
−
3
6
1
2z = −
2
1
1
• 2z =
•
2
2
24.
5
2
 5 1
= − + 
− +
6
3
6
6


Check:
 1
− 
 2
1
4
Check:
1
5
+ 2z = −
3
6
1
 1 ? 5
− + 2 −  = −
3
6
 4
−
1 1 ? 5
−
= −
3 2
6
2
3 ? 5
− − = −
6
6
6
5
5
− = −
6
6
−

1
The solution is z = − .
4
23.
25.
2−
5
4
 5
− − 2 = − − 
2
2
2

b
9
= −
3
2
 b
 9
−3 • −  = −3 • − 
3
 
 2
27
b =
2
1
= 13
2
−
Check:
b
3
27
2− 2
3
27 1
•
2−
2
3
9
2−
2
4 9
−
2
2
5
−
2
2−
= −
?
=
?
=
?
=
?
=
=
2
3
− x +
3
7
2 3  3
− −  +
3  28  7
1
3
+
14
7
1
6
+
14 14
7
14
1
2
The solution is x = −
b
5
= −
3
2
5
b
2− 2−
= − − 2
3
2
1
2
? 1
=
2
=
1
2
? 1
=
2
? 1
=
2
1
=
2
?
=
4
35 32 
7
=
−
 −

8
5
40
40 

9
3
− v =
4
40
4  9 
4
3
− • − v = − •
9  4 
9 40
Check:
1
30
9
4
7
− v +
=
4
5
8
?
9  1 4
7
− • −  +
=
4  30  5
8
3
4
+
40
5
3
32
+
40
40
35
40
7
8


3
.
28
9
4
7
− v +
=
4
5
8
9
4
4
7
4
− v + −
=
−
4
5
5
8
5
v = −
5
2
5
−
2
5
−
2
5
−
2
5
−
2
5
−
2
3
7
6
1
=
−
 −

2
7
14
14


2
1
− x =
3
14
3  2 
3
1
− • − x = − •
2  3 
2 14
3
x = −
28
2z = −
z = −
2
3
1
− x +
=
3
7
2
2
3
3
1
3
− x + −
=
−
3
7
7
2 7
The solution is v = −
7
8
? 7
=
8
? 7
=
8
7
=
8
?
=

1
.
30
1
The solution is b = 13 .
2
82
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Chapter 3
26. Words:
plus temp.
times number
Temp.
change
of
at
per hour
hours
1:00 P.M.
( − 4)
+
35
•
x
Equation:
=
Equation:
Words:
equals −1°.
Words:
Monthly plus cost
times
fee
per text
message
+
30
=
Equation:
59.50
30 + 0.25 x = 59.50
30 − 30 + 0.25 x = 59.50 − 30
35 − 4 x = −1
0.25 x = 29.50
0.25 x
29.50
=
0.25
0.25
x = 118
− 4 x = −36
− 4x
−36
=
−4
−4
You had 118 text messages.
x = 9
29. 3v − 9v = 30
It will take 9 hours.
−6v = 30
Cost
plus cost per times
to rent
game
bowling
shoes
number
of games
2.50
+
2.25
•
equals total amount you have.
Equation:
=
9.25
2.50 + 2.25 x = 9.25
2.50 − 2.50 + 2.25 x = 9.25 − 2.50
2.25 x = 6.75
3v − 9v = 30
?
3( −5) − 9( −5) = 30
30
−6v
=
−6
−6
v = −5
x
30. 12t − 8t = −52
?
−15 + 45 = 30
30 = 30
Check:
4t = −52

12t − 8t = −52
?
12( −13) − 8( −13) = −52
−52
4t
=
4
4
t = −13
?
−156 + 104 = −52
−52 = −52

The solution is t = −13.
31. −8d − 5d + 7 d = 72
−6d = 72
2.25 x
6.75
=
2.25
2.25
x = 3
You can bowl three games.
Check:
The solution is v = −5.
Variable: Let x be the number of games.
Words:
x
equals amount you owe.
35 − 35 − 4 x = −1 − 35
Equation:
•
0.25
−1
35 + ( − 4) • x = −1
27. Words:
number
of text
messages
Variable: Let x be the number of text messages.
Variable: Let x be the number of hours.
Equation:
28. Words:
−6d
72
=
−6
−6
d = −12
Check:
−8d − 5d + 7 d = 72
?
−8( −12) − 5( −12) + 7( −12) = 72
?
96 + 60 − 84 = 72
72 = 72

The solution is d = −12.
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Big Ideas Math Red
Worked-Out Solutions
83
Chapter 3
32.
6( x − 2) = −18
Check: 6( x − 2) = −18
6 • x − 6 • 2 = −18
2(3m) + 2m = 102
6( −1 − 2) = −18
6 x − 12 = −18
2l + 2 w = P
36.
?
6m + 2m = 102
?
6( −3) = −18
6 x − 12 + 12 = −18 + 12
8m = 102
−18 = −18 
6 x = −6
8m
102
=
8
8
3
m = 12
4
−6
6x
=
6
6
x = −1
3
So, m = 12 .
4
The solution is x = −1.
33.
− 4( m + 3) = 24
Check: − 4( m + 3) = 24
− 4 • m − 4 • 3 = 24
− 4( −9 + 3) = 24
− 4m − 12 = 24
?
37. You need to know the initial fee.
?
38. You need to know the coldest surface temperature on the
− 4( −6) = 24
− 4m − 12 + 12 = 24 + 12
24 = 24
− 4m = 36
moon.

39. a. Each group has nine insects.
− 4m
36
=
−4
−4
27 + 5 = 32
Five insects escape, so add 5.
The solution is m = −9.
So, you caught 32 insects.
−8( y + 9) = − 40
b.
−8 • y − 8 • 9 = − 40
−8 y − 72 = − 40
−8 y − 72 + 72 = − 40 + 72
Check:
9 • 3 = 27
so multiply by 3.
m = −9
34.
9
The insects are divided into 3 groups,
x−5
= 9
3
x−5
3•
= 3•9
3
x − 5 = 27
−8 y = 32
x − 5 + 5 = 27 + 5
32
−8 y
=
−8
−8
y = −4
So, you caught 32 insects, which is the same answer
obtained in part (a).
x = 32
−8( y + 9) = − 40
40.
?
Words:
−8( − 4 + 9) = − 40
The diver’s is
elevation
?
−8(5) = − 40
− 40 = − 40

Variable: Let d be the elevation of the entrance.
The solution is y = − 4.
35. Sample answer: You travel halfway up a ladder. Then
you climb down two feet and are 8 feet above the ground.
How long is the ladder?
1
x−2 = 8
2
1
x−2+2
2
1
x
2
1
2• x
2
x
= 8+2
= 10
1 foot deeper than
2
of the elevation
3
of the entrance.
Equation:
−15 =
−15 + 1 =
−14 =
−15
=
2
d −1
3
2
d −1
3
2
d −1+1
3
2
d
3
3 2
• d
2 3
d
= 2 • 10
3
• ( −14) =
2
−21 =
= 20
The elevation of the entrance is −21 feet.
The ladder is 20 feet long.
84
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Chapter 3
41. Let x be the amount by which you should change the
length.
Quiz 3.3–3.5
1.
P = 2 + 2 w
−6.5 + x = − 4.12
+ 6.5
54 = 2 • ( 25 + x) + 2 • 12
+ 6.5
x = 2.38
Check:
−6.5 + x = − 4.12
54 = 2 • 25 + 2 • x + 24
54 = 50 + 2 x + 24
?
−6.5 + 2.38 = − 4.12
54 = 74 + 2 x
−20 = 2 x
2x
−20
=
2
2
−10 = x
2.
You should decrease the length of the rectangle by
10 centimeters.
1
3
+ p = −5
2
4
1
23 9
23 18 
1
1  3
−4
− 4  −5 4 − 4 2 = − 4 − 2 = − 4 − 4 
2
2
4
p = −
Fair Game Review
42.
Check:
−6.2
× 5.6
372
3100
−34.72
1
3
+ p = −5
2
4
1  41  ?
3
4 +  −  = −5
2  4
4
9  41  ?
3
+  −  = −5
2  4
4
18  41  ?
3
+  −  = −5
4
4
 4
?
23
3
−
= −5
4
4
3
3
−5 = −5
4
4
1
The solution is p = −10 .
4
8  1 8  5
5  4
5  5
43.
×  −2  = ×  −  44.
÷ −  = • − 
3  2 3  2
2  5
2  4
4
=
3× 2
1
− 20
3
2
= −6
3
=
45.
6.2
3 18.6
− 18
06
−6
0
=
5 • ( −5)
2•4
−25
=
8
1
= −3
8
3.
b
= 4
7
 b
−7 •  −  = −7 • 4
 7

b
= 4
7
−28 ?
= 4
−7
4 = 4
−
Check: −
b = −28

The solution is b = −28.
So, −18.6 ÷ ( −3) = 6.2.
46. C; 0.75 =
1
41
, or − 10
4
4
4
So, −6.2 × 5.6 = −34.72.
8 × ( −5)

− 4.12 = − 4.12
The solution is x = 2.38.
54 − 74 = 74 − 74 + 2 x
75
25 × 3
3
=
=
100
25 × 4
4
15
5×3
3
=
=
20
5×4
4
9
3×3
3
=
=
12
3×4
4
6
3×2
2
=
=
9
3×3
3
4. − 2 w + 3.7 = − 0.5
− 3.7
− 3.7
− 2 w = − 4.2
− 2w
− 4.2
=
−2
−2
w = 2.1
Check:
− 2 w + 3.7 = − 0.5
?
− 2( 2.1) + 3.7 = − 0.5
?
− 4.2 + 3.7 = − 0.5
− 0.5 = − 0.5

The solution is w = 2.1.
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Big Ideas Math Red
Worked-Out Solutions
85
Chapter 3
5. b − 7.4 = − 6.8
+ 7.4
11. 3 x + 13 + 12 = 33
+ 7.4
b =
3 x + 25 = 33
3x + 25 − 25 = 33 − 25
0.6
3x = 8
The solution is b = 0.6.
2
1
15
75
6. a + 5 = 7 

5
2
2
10
2
2
27
54
−5
−5  −
 −
5
5
5
10
21
1
a =
, or 2
10
10
Current equals original minus amount
balance
balance
withdrawn.
Variable: Let b be the original balance.
3
15
= −
8
32
3 8
8  15 
x • • = • − 
8 3
3  32 
120
1
, or −1
x = −
96
4
x•
=
Equation: −20.51
−
x
29.79
−20.51 = x − 29.79
+ 29.79
+ 29.79
9.28 = x
You had $9.28 in your bank account before you withdrew
the money.
1
The solution is x = −1 .
4
8.
2
So, x = 2 .
3
12. Words:
1
The solution is 2 .
10
7.
3x
8
=
3
3
8
2
x = , or 2
3
3
13. Words:
2k
= 12
−2.6
 2k 
−2.6 • 
 = −2.6 • 12
 −2.6 
The water times number equals −16 feet.
level change
of days
per day
Variable: Let x be the number of days.
2k = −31.2
Equation:
2k
−31.2
=
2
2
k = −15.6
The solution is k = −15.6.
9. x + 9 + 10.5 + 2.8 = 26
x + 22.3 =
26
− 22.3
− 22.3
x = 3.7
−3
1
5
•
x
=
−16
1
−3 x = −16
5
16
− x = −16
5
5  16 
5
−
• − x = −
• ( −16)
16  5 
16
x = 5
It takes 5 days for the water level to change −16 feet.
So, x = 3.7.
10. x + 5.62 + 3.65 + 5.62 = 23.59
x + 14.89 =
23.59
− 14.89 = − 14.89
x = 8.7
So, x = 8.7.
86
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Chapter 3
14. Words:
Time times number plus
played
of
per
quarters
quarter
time you
did not play
b.
•
x
+
4
4
Mat length minus divided 2 equals
picture length
by
Variable: Let x be the amount to be cut.
(9.6 − 6)
Equation:
Variable: Let x be the average time played per quarter.
Equation:
Words:
1
2
Words:
equals length of game.
=
Equation:
=
2
the sum of the amount cut and 0.5.
Equation:
Words:
÷
x
(9.6 − 6) ÷ 2
+
0.5
= x + 0.5
3.6 ÷ 2 = x + 0.5
32
1.8 = x + 0.5
x•4+ 4
1.8 − 0.5 = x + 0.5 − 0.5
1
= 32
2
x = 1.3
You should cut 1.3 inches from the top and bottom.
1
2
64 9
−
2
2
55
2
1 55
•
4
2
55
7
, or 6
8
8
4 x = 32 − 4
4x =
4x =
1
• 4x =
4
x =
Chapter 3 Review
1. Terms: z , 8, − 4 z
Like terms: z and − 4 z
2. Terms: 3n, 7, − n, − 3
Like terms: 3n and − n, 7 and − 3
3. Terms: 10 x 2 , − y , 12, − 3 x 2
Your mean playing time was 6
15. a. Words:
7
minutes per quarter.
8
Mat width
divided 2 equals
minus picture by
width
Variable: Let x be the amount to be cut.
Equation:
Words:
(7.8 − 4)
÷
Like terms: 10 x 2 and − 3x 2
4. 4h − 8h = −4h
5. 6.4r − 7 − 2.9r = 6.4r − 2.9r − 7
= 3.5r − 7
6.
=
2
the sum of amount cut and 0.5.
Equation:
x
(7.8 − 4) ÷ 2
+
0.5
= x + 0.5
7. 3(2 + q) + 15 = 6 + 3q + 15
= 3q + 6 + 15
3.8 ÷ 2 = x + 0.5
= 3q + 21
1.9 = x + 0.5
1.9 − 0.5 = x + 0.5 − 0.5
1.4 = x
You should cut 1.4 inches from the left and right sides.
3
3
3
3
x + 19 −
x −7 = x −
x + 19 − 7
5
20
5
20
12
3
=
x −
x + 19 − 7
20
20
9
=
x + 12
20
8.
1
16
8
m − − 17
(16m − 8) − 17 =
8
8
8
= 2m − 1 − 17
= 2m − 18
9. −1.5(4 − n) + 2.8 = −6 + 1.5n + 2.8
= 1.5n − 6 + 2.8
= 1.5n − 3.2
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Big Ideas Math Red
Worked-Out Solutions
87
Chapter 3
10. (c − 4) + (3c + 9) = c − 4 + 3c + 9
3
1
=
4
4
3
3
−
−
4
4
20. n +
= c + 3c − 4 + 9
= 4c + 5
11.
2
2
2
20
2
12
− d −
(d − 10) − (d + 6) = d −
5
3
5
5
3
3
2
2
20 12
= d − d −
−
5
3
5
3
6
10
60 60
=
d − d −
−
15
15
15
15
4
= − d −8
15
n = −
5
7
= −
6
8
5
5
+
+
6
6
1
.
4
v = −
1
3
1
3
= y + 
y +
4
8
4
2
Check:
14. Factor out 1.7.
1.7 j − 3.4 = 1.7( j − 2)
15. Factor out − 5.
−5 p + 20 = −5( p − 4)
+3
Check: p − 3 = − 4
+3
−1 − 3 = − 4
−4 = −4 
6 + q =
−6
1
−6
6 + ( −5) = 1
23.  +15.2 =
+2
− 15.2
Check:
+2
−2 + ( −20) = −22
−22 = −22
+ 19
=
−11
Check: b − 19 = −11
+ 19
b = 8
?
8 − 19 = −11
−11 = −11
?
=
=

1
.
24
t − 3.7 = 1.2
Check:
?
4.9 − 3.7 = 1.2
4.9
1.2 = 1.2


− 4.5
 + 15.2 = − 4.5
Check:
− 15.2
?
−19.7 + 15.2 = − 4.5
− 4.5 = − 4.5

The solution is  = −19.7.

24.
The solution is j = −20.
19. b − 19
?
=
 = −19.7
−2 + j = −22
?
j = −20
?
=
The solution is t = 4.9.
1=1
The solution is q = −5.
−2 + j = −22
1.2
7
8
7
−
8
7
−
8
7
−
8
7
−
8
= −
+ 3.7
t =
6+ q =1
?
q = −5
18.
5
6
1
5
−
−
24 6
1
20
−
−
24
24
21
−
24
7
−
8
+ 3.7
Check:
1
24
v −
22. t − 3.7 =
The solution is p = −1.
17.

5
21 20 
 7
= −
+
− +

6
24
24 
 8
The solution is v = −
?
p = −1
1
4
? 1
=
4
? 1
=
4
1
=
4
=
21. v −
2b + 8 = 2(b + 4)
16. p − 3 = −4
2
1
= −
4
2
3
4
1
3
− +
2 4
2 3
− +
4 4
1
4
n+
1
The solution is n = − .
2
12. Factor out 2.
13. Factor out
Check:
x
= −8
3
x
3•
= 3 • ( −8)
3
x = −24
Check:
x
= −8
3
−24 ?
= −8
3
−8 = −8

The solution is x = −24.
The solution is b = 8.
88
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Chapter 3
25.
y
7
y
−7 • 7 =
•7
7
− 49 = y
y
7
? − 49
−7 =
7
−7 = −7 
32.
Check: − y =
−7 =
Words:
Variable: Let c be the total change.
z
3
= −
4
4
 z
− 4 • −  = − 4 •
 4
3
z
= −
4
4
3
3
− = −
4
4
Check: −
−
 3
− 
 4

The solution is z = 3.
33.
w
w
= −2.5
= −2.5
Check: −
20
20
50 ?
 w
−
= −2.5
−20 •  −  = −20 • ( −2.5)
20
 20 
−2.5 = −2.5
w = 50
−2c + 6 = −8
Check:
4 x = −8
34.
?
−14 + 6 = − 8
−8 = −8

3(3w − 4) = −20
3 • 3w − 3 • 4 = −20
?
4( −2) = −8
9w − 12 = −20

−8 = −8
9 w − 12 + 12 = −20 + 12
The solution is x = −2.
9 w = −8
9w
8
= −
9
9
8
w = −
9
Check: −10 = 2 y
−10
2y
=
2
2
−5 = y
?
−10 = 2( −5)
−10 = −10

Check:
The solution is y = −5.
Check:
−5.4 z
−32.4
=
−5.4
−5.4
z = 6
−5.4 z = −32.4
?
−5.4(6) = −32.4
−32.4 = −32.4
Check:
− 6.8w
3.4
=
− 6.8
− 6.8

w = − 0.5
−6.8w = 3.4
3.4 = 3.4
 8
 ?
3 − − 4  = − 20
 3

 20  ?
3 −  = − 20
 3
− 20 = − 20

8
The solution is w = − .
9
?
−6.8( −0.5) = 3.4
3(3w − 4) = − 20
  8
 ?
3 3 −  − 4  = − 20
  9

The solution is z = 6.
31. − 6.8w = 3.4
?
−2c
−14
=
−2
−2
c = 7

− 2c + 6 = − 8
The solution is c = 7.
4x
−8
=
4
4
x = −2
30. −5.4 z = −32.4
−3.2
=
− 2(7) + 6 = − 8
−2c = −14
The solution is w = 50.
29. −10 = 2 y
Check:
−2c + 6 − 6 = −8 − 6
−
28. 4 x = −8
5
The total change over the five-day period is −16°F.
z = 3
27.
÷
Equation:
c
c
= −3.2
5
c
5 • = 5 • ( −3.2)
5
c = −16
The solution is y = − 49.
26.
Total divided number equals mean
change by
of days
temperature
change.

The solution is w = − 0.5.
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Big Ideas Math Red
Worked-Out Solutions
89
Chapter 3
35.
w 5
+ =
6
8
w 5 5
+ − =
6
8 8
w
=
6
w
6•
=
6
w =
Check:
3
−1
8
11 5
− −
8
8
37.
Equation:
6 • ( −2)
Words:
−12
Equation:
3
−1
8
3
−1
8
3
−1
8
3
−1
8
3
−1
8
3
−1
8
−1.5
•
equals
−31 feet.
=
−31
n
+
(−14.5)
−1.5n − 14.5 = −31
−1.5n − 14.5 + 14.5 = −31 + 14.5
−1.5n = −16.5
−1.5n
−16.5
=
−1.5
−1.5
n = 11
It will take 11 years for the canyon floor to have an
elevation of −31 feet.

Chapter 3 Test
1. 8 x − 5 + 2 x = 8 x + 2 x − 5
−3 x − 4.6 = 5.9
= (8 + 2) x − 5
−3 x − 4.6 + 4.6 = 5.9 + 4.6
−3 x = 10.5
= 10 x − 5
−3 x
10.5
=
−3
−3
x = −3.5
Check:
current
elevation
−1.5 • n + ( −14.5) = −31
The solution is w = −12.
36.
Elevation times number plus
change
of years
per year
Variable: Let n be number of years.
−2
w 5
+ =
6
8
−12 5 ?
+ =
6
8
5 ?
−2 + =
8
16 5 ?
−
+ =
8
8
11 ?
−
=
8
3
−1 =
8
Words:
2. 2.5w − 3 y + 4 w = 2.5w + 4 w − 3 y
= ( 2.5 + 4) w − 3 y
= 6.5w − 3 y
− 3 x − 4.6 = 5.9
?
− 3( − 3.5) − 4.6 = 5.9
3. 3(5 − 2n) + 9n = 3(5) − 3( 2n) + 9n
?
= 15 − 6n + 9n
10.5 − 4.6 = 5.9
5.9 = 5.9
= 15 + ( − 6 + 9)n

= 15 + 3n
The solution is x = −3.5.
4.
5
9
5
9
x + 15 −
x −9 = x −
x + 15 − 9
7
14
7
14
9
5
=  −
 x + 15 − 9
 7 14 
9
 10
= 
−
 x + 15 − 9
14
14


1
=
x + 6
14
5.
(3 j
+ 11) + (8 j − 7) = 3 j + 11 + 8 j − 7
= 3 j + 8 j + 11 − 7
= 11 j + 4
90
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Chapter 3
6.
14. z +
3
3
(8 p + 12) + (16 p − 8) = 6 p + 9 + 6 p − 3
4
8
= 6p + 6p + 9 − 3
8.
−2.5( 2s − 5) − 3( 4.5s − 5.2) = − 5s + 12.5 − 13.5s + 15.6
= −18.5s + 28.1
9. Factor out 3.
16.
3n − 24 = 3(n − 8)
1
.
2
1
5
1
q +
= ( q + 5)
2
2
2
7 x = −3
 3 ?
7 •  −  = −3
 7
−3 = −3
2
3
k − =
7
8
2
3 3
k − + =
7
8 8
2
k =
7
7 2
• k =
2 7
k =
Check:

3
The solution is x = − .
7
2( x + 1) = −2
Check: 2( x + 1) = −2
2 • x + 2 • 1 = −2
2( −2 + 1) = −2
?
2 x + 2 = −2
?
2( −1) = −2
2 x + 2 − 2 = −2 − 2
−2 = −2
2x = − 4

17.
Check:
The solution is g = −36.
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2
g = −8
9
?
2
(−36) = −8
9
−8 = −8
Check: −14 = 6c
?
 7
−14 = 6 − 
 3
−14 = −14 
19
8
19 3
−
+
8
8
−
−2
7
• ( −2)
2
−7
2
3
k −
7
8
2
3
(−7) −
7
8
3
−2 −
8
16 3
−
−
8
8
19
−
8
19
8
19
−
8
19
−
8
19
−
8
19
−
8
= −
?
=
?
=
?
=
=

Each
costs
each
costs each
costs
women’s $45, women’s $12, men’s $15,
haircut
shampoo
haircut
and
The solution is x = −2.
2
g = −8
9
9 2
9
• g = • ( −8)
2 9
2
g = −36

The solution is k = −7.
−4
2x
=
2
2
x = −2
13.
5.4 = 5.4
1
The solution is c = −2 .
3
= −5s − 13.5s + 12.5 + 15.6
12.
−9.1
−14
6c
=
6
6
7
1
− = c, or c = −2
3
3
= 8r − 17
7x
−3
=
7
7
3
x = −
7
?
−9.1 + 14.5 = 5.4
15. −14 = 6c
= 2r + 6r − 13 − 4
Check:
z + 14.5 = 5.4
Check:
The solution is z = −9.1.
7. ( 2r − 13) − ( −6r + 4) = 2r − 13 + 6r − 4
11. 7 x = −3
5.4
− 14.5
z =
= 12 p + 6
10. Factor out
14.5 =
− 14.5
each
costs
men’s
$7.
shampoo
w represents the number of women.
m represents the number of men.

45w + 12 w + 15m + 7 m = ( 45 + 12) w + (15 + 7)m
= 57 w + 22m
The expression 57 w + 22m represents the income from
w women and m men getting a haircut and a shampoo.
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Worked-Out Solutions
91
Chapter 3
18. To match the world record, the total time difference is 0.
Let x be the time difference for lap 4.
−1.23 + 0.45 + 0.18 + x =
0
−0.6 + x =
0
+ 0.6
The runner needs a time difference of 0.6 for the fourth
lap.
5. F;
penalty.
Variable: Let x be your score before the penalty.
Equation: 9.124 =
−
x
9.124 = x − 0.3
+ 0.3
3
1
3
− x + = −
4
8
8
3
1 1
3 1
− x + − = − −
4
8 8
8 8
3
3 1
− x = − −
4
8 8
0.3
4
6. 0.07 0.28 → 7 28
− 28
0
+ 0.3
So, − 0.28 ÷ ( − 0.07) = 4.
9.424 = x
Your score before the penalty is 9.424.
20. 5 x + 15 + 17 = 45
7. D; −96 = −6( x − 15)
−96 = − 6 x + 90
5 x + 32 = 45
P = 2l + 2 w
8. F;
5 x + 32 − 32 = 45 − 32
400 = 2(4 j ) + 2(60)
5 x = 13
x =
400 = 8 j + 120
13
3
, or 2
5
5
400 − 120 = 8 j + 120 − 120
280 = 8 j
3
So, x = 2 .
5
280
8j
=
8
8
35 = j
Chapter 3 Standards Assessment
1. B;
b
= −10
0.3
2. − 9;
9. B; 3 + x 2 ÷ y = 3 + ( − 2) ÷ 4
2
0( − 6) − ( − 6)
cd − d 2
=
4
4
0 − 36
=
4
− 36
=
4
= −9
3. G; − 38 − ( −14) = − 38 + 14
= − 24
92
7
The mean temperature of last week is 6°F.
0.6
Final is score
minus
score
before
penalty
− 2 + ( − 3) + ( −1) + 8 + 10 + 12 + 18
42
=
7
= 6
+ 0.6
x =
19. Words:
4. B; Mean =
Big Ideas Math Red
Worked-Out Solutions
= 3+ 4÷4
2
= 3+1
= 4
10. H; − 2
1  3
9
67
−  −8  = − +
4  8
4
8
18 67
= − +
8
8
49
1
=
, or 6
8
8
Copyright © Big Ideas Learning, LLC
All rights reserved.
Chapter 3
11. Part A:
The total
the amount
the amount
equals
plus
cost of the
you have
you plan to
bicycle
saved
save
times the number
of weeks.
Variable: Let w be the number of weeks.
Equation: 116 = 43.50 + 7.25 • w
116 = 43.50 + 7.25w
116 − 43.50 = 43.50 + 7.25w − 43.50
116 − 43.50 = 43.50 − 43.50 + 7.25w
72.5 = 7.25w
72.5
7.25w
=
7.25
7.25
10 = w
So, it will take 10 weeks.
Part B:
The total
minus coupon equals the amount
cost of the
you have saved
bicycle
plus the amount
you plan
to save
times the number
of weeks.
Variable: Let w be the number of weeks.
Equation: 116 − 20 = 43.50 + 8.75 • w
116 − 20 = 43.50 + 8.75w
96 = 43.50 + 8.75w
96 − 43.50 = 43.50 + 8.75w − 43.50
96 − 43.50 = 43.50 − 43.50 + 8.75w
52.50 = 8.75w
52.50
8.75w
=
8.75
8.75
6 = w
4 weeks sooner because in Part A it takes 10 weeks and
in Part B it takes 6 weeks. The difference is
10 − 6 = 4 weeks.
Copyright © Big Ideas Learning, LLC
All rights reserved.
Big Ideas Math Red
Worked-Out Solutions
93