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Checkpoint 1: Assess Your Understanding, pages 545–548
7.1
1. Multiple Choice Which expression simplifies to
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x - 3
?
x + 4
A.
x2 - 9
, x Z -4, - 3
x + 7x + 12
B.
x2 - 9
, x Z 3, 4
x - 7x + 12
C.
x2 + 9
, x Z -4, - 3
x + 7x + 12
D.
x2 + 9
, x Z 3, 4
x - 7x + 12
2
2
DO NOT COPY.
2
2
Chapter 7: Rational Expressions and Equations—Checkpoint 1—Solutions
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2. Write two equivalent forms of each rational expression.
- 2(x - 6)
3x2
a) 5x
b)
x(x + 7)(x - 6)
The expression has x ⴝ 0
as a non-permissible value.
The expression has x ⴝ ⴚ7, 0, 6
as non-permissible values.
ⴚ 2(x ⴚ 6)
3x2
3x2 2x
ⴝ
5x
5x 2x
6x3
ⴝ
10x2
#
x(x ⴙ 7)(x ⴚ 6)
ⴝ
3x2
3x 2
ⴝ
5x
5x
3x
ⴝ
5
ⴝ
ⴚ 2(x ⴚ 6)
xⴙ7
x(x ⴙ 7)(x ⴚ 6) x ⴙ 7
#
ⴚ 2(x ⴚ 6)(x ⴙ 7)
x(x ⴙ 7)2(x ⴚ 6)
ⴚ 2(x ⴚ 6)
The equivalent expressions are:
6x3
3x
, x ⴝ 0 and , x ⴝ 0
2
5
10x
x(x ⴙ 7)(x ⴚ 6)
ⴝ
ⴝ
ⴚ 2 (x ⴚ 6)
x(x ⴙ 7) (x ⴚ 6)
ⴚ2
x(x ⴙ 7)
The equivalent expressions are:
ⴚ 2(x ⴚ 6)(x ⴙ 7)
x(x ⴙ 7)2(x ⴚ 6)
, x ⴝ ⴚ7, 0, 6 and
ⴚ2
, x ⴝ ⴚ 7, 0, 6
x(x ⴙ 7)
3. Write each rational expression in simplest form.
25x3
x2 + 7x - 18
a)
b)
2
10x
x2 - 81
The non-permissible value is:
xⴝ0
3
25x
ⴝ
10x2
ⴝ
c)
5
2
3
(x ⴙ 9)(x ⴚ 2)
ⴝ
(x ⴙ 9)(x ⴚ 9)
The non-permissible values are:
x ⴝ ⴚ9 and x ⴝ 9
25 x
10 x 2
5x
,xⴝ 0
2
ⴝ
(x ⴙ 9) (x ⴚ 2)
(x ⴙ 9) (x ⴚ 9)
xⴚ2
, x ⴝ ⴚ9, 9
ⴝ
xⴚ9
2(49 ⴚ x2)
98 - 2x2
ⴝ
4x - 24x - 28 4(x2 ⴚ 6x ⴚ 7)
2
ⴝ
2(7 ⴚ x)(7 ⴙ x)
4(x ⴚ 7)(x ⴙ 1)
The non-permissible values are: x ⴝ ⴚ1 and x ⴝ 7
ⴝ
ⴚ 2 (x ⴚ 7) (7 ⴙ x)
2
ⴝⴚ
14
4 (x ⴚ 7) (x ⴙ 1)
7ⴙx
, x ⴝ ⴚ1, 7
2(x ⴙ 1)
Chapter 7: Rational Expressions and Equations—Checkpoint 1—Solutions
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4. Here is a student’s solution for simplifying a rational expression.
Identify the error in the solution. Write a correct solution.
4(x - 2)
4x - 8
=
(x
+
4)(x - 2)
x + 2x - 8
4
=
, x Z -2, 4
x + 4
2
The student has made an error in the signs of the non-permissible values.
Each factor in the denominator should be set equal to 0 to get x ⴝ 2 and
x ⴝ ⴚ4 as non-permissible values.
Correct solution:
4 (x ⴚ 2)
4x ⴚ 8
ⴝ
(x
ⴙ
4) (x ⴚ 2)
x ⴙ 2x ⴚ 8
4
, x ⴝ ⴚ 4, 2
ⴝ
xⴙ4
2
7.2
5. Multiple Choice Which values of x are non-permissible for
x2 + 3x
x + 3
x - 2 , x + 5?
A. - 5, 2
B. -5, -3, 2
C. - 5, - 3, 0, 2
6. Simplify each expression.
14(a - 9)
5a
a)
2
7(a
- 9)
15a
2n2 - 128 6n - 10
n - 8
#
b) 3n - 5
Non-permissible values:
a ⴝ 0 and a ⴝ 9
2
ⴝ
14 (a ⴚ 9)
3
15 a 2
2
ⴝ , a ⴝ 0, 9
3a
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#
D. -3, 0, 2
#
Non-permissible values:
nⴝ
5 a
7 (a ⴚ 9)
ⴝ
5
and n ⴝ 8
3
2(n2 ⴚ 64) 2(3n ⴚ 5)
3n ⴚ 5
nⴚ8
#
2 (n ⴚ 8) (n ⴙ 8) 2 (3n ⴚ 5)
3n ⴚ 5
nⴚ8
5
ⴝ 4(n ⴙ 8), n ⴝ , 8
3
ⴝ
#
Chapter 7: Rational Expressions and Equations—Checkpoint 1—Solutions
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c)
12e
18e2
,
4(e + 2)
7(e + 2)
18 e 2
ⴝ
4 (e ⴙ 2)
3
#
2x2 + 10x - 28
5x2 - 11x + 2
,
x2 - 3x - 28
x2 + 11x + 28
d)
Non-permissible values:
e ⴝ ⴚ2 and e ⴝ 0
(5x ⴚ 1)(x ⴚ 2)
ⴝ
7 (e ⴙ 2)
2 12 e
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(x ⴚ 7)(x ⴙ 4)
(5x ⴚ 1)(x ⴚ 2)
ⴝ
21e
ⴝ
, e ⴝ ⴚ2, 0
8
(x ⴚ 7)(x ⴙ 4)
ⴜ
ⴜ
2(x2 ⴙ 5x ⴚ 14)
(x ⴙ 7)(x ⴙ 4)
2(x ⴙ 7)(x ⴚ 2)
(x ⴙ 7)(x ⴙ 4)
Non-permissible values:
x ⴝ ⴚ7, x ⴝ ⴚ 4, x ⴝ 2, and x ⴝ 7
ⴝ
ⴝ
(5x ⴚ 1) (x ⴚ 2)
(x ⴚ 7) (x ⴙ 4)
#
(x ⴙ 7) (x ⴙ 4)
2 (x ⴙ 7) (x ⴚ 2)
5x ⴚ 1
, x ⴝ ⴚ7, ⴚ4, 2, 7
2(x ⴚ 7)
7. Simplify each expression.
2a2b
4bc2 12a2c2
a)
3 ,
bc
9ab
3ab2
#
Non-permissible values: a ⴝ 0, b ⴝ 0, c ⴝ 0
ⴝ
3
2a2 b
2
9 a b3
#
3 a b2
4 b c2
#
3
12 a2 c 2
bc
2a4
ⴝ 2 , a ⴝ 0, b ⴝ 0, c ⴝ 0
bc
b)
3x - 27 x2 - 7x - 8
5x
, 2x - 2
2
2
x - 1
x - 81
#
ⴝ
3(x ⴚ 9)
# (x ⴚ 8)(x ⴙ 1)
(x ⴚ 1)(x ⴙ 1) (x ⴚ 9)(x ⴙ 9)
ⴜ
5x
2(x ⴚ 1)
Non-permissible values: x ⴝ ⴚ9, ⴚ 1, 0, 1, 9
ⴝ
ⴝ
16
3 (x ⴚ 9)
(x ⴚ 1) (x ⴙ 1)
6(x ⴚ 8)
5x(x ⴙ 9)
# (x ⴚ 8) (x ⴙ 1) # 2 (x ⴚ 1)
(x ⴚ 9) (x ⴙ 9)
5x
, x ⴝ ⴚ9, ⴚ1, 0, 1, 9
Chapter 7: Rational Expressions and Equations—Checkpoint 1—Solutions
DO NOT COPY.
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