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Lesson 7.5 Exercises, pages 583–590
A
3. Identify the non-permissible values of the variable in each equation.
2
3x
6
a) 5 = x
b) x - 1 = x
x ⴝ 0 and x ⴝ 1
xⴝ0
2
3
c) 6 - x =
x + 5
d)
x ⴝ ⴚ5 and x ⴝ 6
4. Solve each equation.
8
4
a) 5 =
d
x ⴝ ⴚ2 and x ⴝ 2
e
12
b) 3 = e
Non-permissible value: d ⴝ 0
Common denominator: 5d
Non-permissible value: e ⴝ 0
Common denominator: 3e
5 da b ⴝ 5 d a b
3 ea b ⴝ 3 e a e b
3
4
5
4d ⴝ 40
d ⴝ 10
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-1
2x
= x - 2
3x + 6
DO NOT COPY.
8
d
e
12
e2 ⴝ 36
e ⴝ 6 or e ⴝ ⴚ6
Solutions are e ⴝ ⴚ6 and
e ⴝ 6.
7.5 Solving Rational Equations—Solutions
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5. Solve each equation.
3
4
1
a) 2z = 3z - 2
4
6z a
3
4
2
b ⴝ 6z a b ⴚ
2z
3z
3
Non-permissible value: x ⴝ 0
Common denominator: x2
6 za b
1
2
x2(2) ⴚ x 2 a x b ⴝ x2 a
4
6
b
x2
2x2 ⴚ 4x ⴝ 6
2x ⴚ 4x ⴚ 6 ⴝ 0
2(x2 ⴚ 2x ⴚ 3) ⴝ 0
2(x ⴚ 3)(x ⴙ 1) ⴝ 0
x ⴝ 3 or x ⴝ ⴚ1
Solutions are x ⴝ ⴚ1 and x ⴝ 3.
9 ⴝ 8 ⴚ 3z
3z ⴝ ⴚ1
z ⴝⴚ
6
b) 2 - x = 2
x
Non-permissible value: z ⴝ 0
Common denominator: 6z
3
Quit
2
1
3
B
6. Solve each equation.
5
3
a) q - 2 =
q + 4
Non-permissible values: q ⴝ 2 and q ⴝ ⴚ4
Common denominator: (q ⴚ 2)(q ⴙ 4)
(q ⴚ 2) (q ⴙ 4) a
3
5
b ⴝ (q ⴚ 2) (q ⴙ 4) a
b
qⴚ2
qⴙ4
3q ⴙ 12 ⴝ 5q ⴚ 10
22 ⴝ 2q
q ⴝ 11
3
3
b) 2s - 4 = s - 2
Non-permissible value: s ⴝ 2
The numerators are the same, so equate the denominators.
2s ⴚ 4 ⴝ s ⴚ 2
sⴝ2
s ⴝ 2 is a non-permissible value, so the equation has no solution.
7. Solve each equation.
a - 1
a + 1
a) a - 3 = a - 4
Non-permissible values: a ⴝ 3 and a ⴝ 4
Common denominator: (a ⴚ 3)(a ⴚ 4)
(a ⴚ 3) (a ⴚ 4) a
aⴚ1
aⴙ1
b ⴝ (a ⴚ 3) (a ⴚ 4) a
b
aⴚ3
aⴚ4
(a ⴚ 4)(a ⴚ 1) ⴝ (a ⴚ 3)(a ⴙ 1)
a2 ⴚ 5a ⴙ 4 ⴝ a2 ⴚ 2a ⴚ 3
ⴚ3a ⴝ ⴚ7
aⴝ
36
7
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4w - 3
2w + 1
b) w - 4 =
2w + 1
Non-permissible values: w ⴝ 4 and w ⴝ ⴚ
1
2
Common denominator: (w ⴚ 4)(2w ⴙ 1)
(w ⴚ 4) (2w ⴙ 1) a
2w ⴙ 1
4w ⴚ 3
b ⴝ (w ⴚ 4) (2w ⴙ 1) a
b
wⴚ4
2w ⴙ 1
(2w ⴙ 1)(2w ⴙ 1) ⴝ (w ⴚ 4)(4w ⴚ 3)
4w2 ⴙ 4w ⴙ 1 ⴝ 4w2 ⴚ 19w ⴙ 12
23w ⴝ 11
wⴝ
c)
11
23
x - 2
6
=
x + 1
2x2 + 2x
6
xⴚ2
ⴝ
xⴙ1
2x(x ⴙ 1)
Non-permissible values: x ⴝ ⴚ1 and x ⴝ 0
Common denominator: 2x(x ⴙ 1)
2x(x ⴙ 1) a
6
xⴚ2
b ⴝ 2x (x ⴙ 1) a
b
xⴙ1
2x(x ⴙ 1)
6 ⴝ 2x2 ⴚ 4x
2x ⴚ 4x ⴚ 6 ⴝ 0
2(x2 ⴚ 2x ⴚ 3) ⴝ 0
2(x ⴚ 3)(x ⴙ 1) ⴝ 0
x ⴝ 3 or x ⴝ ⴚ1
x ⴝ ⴚ1 is a non-permissible value.
So, the only solution is x ⴝ 3.
2
8. Here is a student’s solution for solving a rational equation.
Identify the error in the solution. Write a correct solution.
1
2
= 1 + x
8
1
3
= x
8
1
3
8 x a b = 8 x ax b
8
x = 24
2
There is an error in line 2: the student added 1 ⴙ x without using a
common denominator. Correct solution:
Non-permissible value: x ⴝ 0
Common denominator: 8x
1
2
ⴝ1ⴙx
8
1
xⴙ2
ⴝ x
8
xⴙ2
8 xa b ⴝ 8 x a x b
8
1
x ⴝ 8x ⴙ 16
7x ⴝ ⴚ16
xⴝⴚ
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7.5 Solving Rational Equations—Solutions
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9. Solve each equation.
1
x - 2
1
a) x + x - 3 = x - 3
Non-permissible values: x ⴝ 0 and x ⴝ 3
1
1
xⴚ2
x ⴝxⴚ3ⴚxⴚ3
1
xⴚ3
x ⴝxⴚ3
1
x ⴝ1
xⴝ1
1
u - 1
u + 4
b) u - 2 + 2
=
u + 2
u - 4
1
uⴚ1
uⴙ4
ⴙ
ⴝ
uⴚ2
uⴙ2
(u ⴚ 2)(u ⴙ 2)
Non-permissible values: u ⴝ ⴚ2 and u ⴝ 2
Common denominator: (u ⴚ 2)(u ⴙ 2)
(u ⴚ 2) (u ⴙ 2) a
1
uⴚ1
b ⴙ (u ⴚ 2)(u ⴙ 2) a
b
uⴚ2
(u ⴚ 2)(u ⴙ 2)
ⴝ (u ⴚ 2) (u ⴙ 2) a
uⴙ4
b
uⴙ2
u ⴙ 2 ⴙ u ⴚ 1 ⴝ (u ⴚ 2)(u ⴙ 4)
2u ⴙ 1 ⴝ u2 ⴙ 2u ⴚ 8
u2 ⴚ 9 ⴝ 0
u ⴝ 3 or u ⴝ ⴚ3
Solutions are u ⴝ ⴚ3 and u ⴝ 3.
c)
6b2
4b
b
+
=
5 - b
b + 5
b2 - 25
6b2
4b
b
ⴚ
ⴝ
(b ⴚ 5)(b ⴙ 5)
bⴚ5
bⴙ5
Non-permissible values: b ⴝ ⴚ5 and b ⴝ 5
Common denominator: (b ⴚ 5)(b ⴙ 5)
(b ⴚ 5)(b ⴙ 5) a
4b
6b2
b ⴚ (b ⴚ 5) (b ⴙ 5)a
b
(b ⴚ 5)(b ⴙ 5)
bⴚ5
ⴝ (b ⴚ 5) (b ⴙ 5) a
b
b
bⴙ5
6b2 ⴚ 4b(b ⴙ 5) ⴝ b2 ⴚ 5b
6b2 ⴚ 4b2 ⴚ 20b ⴝ b2 ⴚ 5b
b2 ⴚ 15b ⴝ 0
b(b ⴚ 15) ⴝ 0
b ⴝ 0 or b ⴝ 15
Solutions are b ⴝ 0 and b ⴝ 15.
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7.5 Solving Rational Equations—Solutions
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d)
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1
2z - 1
z + 1
3z - 1
+ 6 =
+
2z + 1
2z + 1
z + 3
3z ⴚ 1
2z ⴚ 1
1
z
ⴚ
ⴙ ⴝ
2z ⴙ 1
2z ⴙ 1
6
z
z
1
z
ⴙ ⴝ
2z ⴙ 1
6
z
ⴙ
ⴙ
ⴙ
ⴙ
1
3
1
3
1
2
Non-permissible values: z ⴝ ⴚ and z ⴝ ⴚ3
Common denominator: 6(2z ⴙ 1)(z ⴙ 3)
6 (2z ⴙ 1) (z ⴙ 3) a
1
z
b ⴙ 6 (2z ⴙ 1)(z ⴙ 3)a b
2z ⴙ 1
6
ⴝ 6(2z ⴙ 1) (z ⴙ 3) a
zⴙ1
b
zⴙ3
6z(z ⴙ 3) ⴙ (2z ⴙ 1)(z ⴙ 3) ⴝ (12z ⴙ 6)(z ⴙ 1)
6z2 ⴙ 18z ⴙ 2z2 ⴙ 7z ⴙ 3 ⴝ 12z2 ⴙ 18z ⴙ 6
4z2 ⴚ 7z ⴙ 3 ⴝ 0
(4z ⴚ3)(z ⴚ 1) ⴝ 0
zⴝ
Solutions are z ⴝ
3
or z ⴝ 1
4
3
and z ⴝ 1.
4
10. Solve each equation.
b
2
a) 2
= 2
b - 4
b - b - 6
b
2
ⴝ
(b ⴚ 2)(b ⴙ 2)
(b ⴚ 3)(b ⴙ 2)
Non-permissible values: b ⴝ 2, b ⴝ ⴚ2, and b ⴝ 3
Common denominator: (b ⴚ 2)(b ⴙ 2)(b ⴚ 3)
(b ⴚ 2) (b ⴙ 2) (bⴚ 3) a
b
b
(b ⴚ 2) (b ⴙ 2)
ⴝ (b ⴚ 2) (b ⴙ 2) (bⴚ3) a
2
b
(b ⴚ 3) (b ⴙ 2)
b2 ⴚ 3b ⴝ 2b ⴚ 4
b ⴚ 5b ⴙ 4 ⴝ 0
(b ⴚ 4)(b ⴚ 1) ⴝ 0
2
b ⴝ 4 or b ⴝ 1
Solutions are b ⴝ 4 and b ⴝ 1.
b)
16
6
= 2
2g2 + 2g - 12
g - 9
6
16
ⴝ
(g ⴚ 3)(g ⴙ 3)
2(g2 ⴙ g ⴚ 6)
16
6
ⴝ
2(g ⴙ 3)(g ⴚ 2)
(g ⴚ 3)(g ⴙ 3)
Non-permissible values: g ⴝ ⴚ3, g ⴝ 2, and g ⴝ 3
Common denominator: 2(g ⴙ 3)(g ⴚ 3)(gⴚ 2)
2 (g ⴙ 3) (g ⴚ 3) (g ⴚ 2) a
16
b
2 (g ⴙ 3) (g ⴚ 2)
ⴝ 2 (g ⴙ 3) (g ⴚ 3) (g ⴚ 2) a
6
b
(g ⴚ 3) (g ⴙ 3)
16g ⴚ 48 ⴝ 12g ⴚ 24
4g ⴝ 24
gⴝ6
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7.5 Solving Rational Equations—Solutions
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c)
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3n + 5
2
n
+ 2
=
n + 1
n
+
3
n + 4n + 3
n
3n ⴙ 5
2
ⴙ
ⴝ
nⴙ1
nⴙ3
(n ⴙ 1)(n ⴙ 3)
Non-permissible values: n ⴝ ⴚ1 and n ⴝ ⴚ3
Common denominator: (n ⴙ 1)(n ⴙ 3)
(n ⴙ 1) (n ⴙ 3) a
n
3n ⴙ 5
b ⴙ (n ⴙ 1)(n ⴙ 3) a
b
nⴙ1
(n ⴙ 1)(n ⴙ 3)
ⴝ (n ⴙ 1) (n ⴙ 3) a
2
b
nⴙ3
n2 ⴙ 3n ⴙ 3n ⴙ 5 ⴝ 2n ⴙ 2
n2 ⴙ 4n ⴙ 3 ⴝ 0
(n ⴙ 1)(n ⴙ 3) ⴝ 0
n ⴝ ⴚ1 or n ⴝ ⴚ3
n ⴝ ⴚ1 and n ⴝ ⴚ3 are non-permissible values.
So, the equation has no solution.
11. The solutions of the equation 4 = x +
8
are x = 3 and
x + m
x = -4. Determine the value of m. Show how you can verify your
answer.
Common denominator: x ⴙ m
(x ⴙ m)4 ⴝ (x ⴙ m)x ⴙ (x ⴙ m) a x ⴙ m b
8
4x ⴙ 4m ⴝ x2 ⴙ mx ⴙ 8
x ⴙ mx ⴚ 4x ⴙ 8 ⴚ 4m ⴝ 0
x2 ⴙ x(m ⴚ 4) ⴙ (8 ⴚ 4m) ⴝ 0
2
An equation with roots x ⴝ 3 and x ⴝ ⴚ4 is:
(x ⴚ 3)(x ⴙ 4) ⴝ 0
x2 ⴙ x ⴚ 12 ⴝ 0
Compare the equations: x2 ⴙ x(m ⴚ 4) ⴙ (8 ⴚ 4m) ⴝ 0
x2 ⴙ x ⴚ 12 ⴝ 0
So, m ⴚ 4 ⴝ 1 and 8 ⴚ 4m ⴝ ⴚ12
mⴝ5
mⴝ5
The value of m is 5.
To verify the answer, substitute m ⴝ 5 in the original equation, then solve
the equation.
12. The measure, d degrees, of each angle in a regular polygon with
360
n sides is given by the equation d = 180 - n .
a) What is the measure of each angle in a regular polygon with
15 sides?
Substitute n ⴝ 15:
d ⴝ 180 ⴚ
360
15
d ⴝ 180 ⴚ 24
d ⴝ 156
Each angle measures 156°.
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b) When each angle in a regular polygon is 162°, how many sides
does the polygon have?
Substitute d ⴝ 162:
360
162 ⴝ 180 ⴚ n
360
18 ⴝ n
n(18) ⴝ n a n b
360
18n ⴝ 360
n ⴝ 20
The polygon has 20 sides.
13. Without solving the equation, how do you know that the equation
12
4
4x - 4 = x - 1 has no solution?
Simplify the expression on the left.
3
12
4
ⴝ
xⴚ1
4 (x ⴚ 1)
3
4
ⴝ
xⴚ1
xⴚ1
Since the expressions have the same denominator but different
numerators, I know the equation has no solution.
C
14. a) Write a rational equation that has 4 as a solution.
Work backward.
xⴝ4
xⴝ7ⴚ3
xⴙ3ⴝ7
xⴙ3
7
ⴝ
xⴙ3
xⴙ3
7
, x ⴝ ⴚ3
1ⴝ
xⴙ3
Write 4 as a difference of 2 numbers.
Take 3 to the left side.
Divide each side by x ⴙ 3.
Simplify.
b) Write a rational equation that has -3 as a solution and has 3 as an
extraneous root.
1
has x ⴝ 3 as a non-permissible value.
xⴚ3
Let both rational expressions in the equation have denominator x ⴚ 3.
xⴚ3
ⴝ
xⴚ3
Write an equation that has x ⴝ 3 and x ⴝ ⴚ3 as roots:
x2 ⴝ 9
So, the rational equation becomes:
x2
9
ⴝ
xⴚ3
xⴚ3
This equation has x ⴝ 3 and x ⴝ ⴚ3 as roots. Since x ⴝ 3 is a nonpermissible value, x ⴝ 3 is an extraneous root. The only solution is
x ⴝ ⴚ3.
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15. Solve each equation.
2x
x + 1
a) x - 3 =
x + 2
Non-permissible values: x ⴝ 3 and x ⴝ ⴚ2
Common denominator: (x ⴚ 3)(x ⴙ 2)
(x ⴚ 3) (x ⴙ 2) a
x2 ⴙ 3x ⴙ 2 ⴝ 2x2 ⴚ 6x
x2 ⴚ 9x ⴚ 2 ⴝ 0
Use the quadratic formula.
√
xⴝ
9_ ( ⴚ 9)2 ⴚ 4(1)( ⴚ 2)
2(1)
√
xⴝ
xⴙ1
2x
b ⴝ (x ⴚ 3) (x ⴙ 2) a
b
xⴚ3
xⴙ2
9_ 89
2
Solutions are x ⴝ
b)
√
9ⴙ
89
2
and x ⴝ
√
9ⴚ
89
2
.
w2
5
= 3
w + 4
Non-permissible value: w ⴝ ⴚ4
Common denominator: 3(w ⴙ 4)
3 (w ⴙ 4) a
w2
5
b ⴝ 3 (w ⴙ 4) a b
wⴙ4
3
3w2 ⴝ 5w ⴙ 20
3w ⴚ 5w ⴚ 20 ⴝ 0 Use the quadratic formula.
2
wⴝ
√
5_ ( ⴚ 5)2 ⴚ 4(3)( ⴚ 20)
2(3)
√
wⴝ
5_ 265
6
Solutions are w ⴝ
1
√
5ⴙ
265
6
and w ⴝ
√
5ⴚ
265
6
.
1
c) v + v - 4 = 2
Non-permissible values: v ⴝ 0 and v ⴝ 4
Common denominator: v(v ⴚ 4)
v (v ⴚ 4) a v b ⴙ v (v ⴚ 4) a
b ⴝ v (v ⴚ 4)(2)
vⴚ4
1
1
v ⴚ 4 ⴙ v ⴝ 2v2 ⴚ 8v
2v ⴚ 10v ⴙ 4 ⴝ 0
Use the quadratic formula.
v2 ⴚ 5v ⴙ 2 ⴝ 0
2
√
vⴝ
5_ ( ⴚ 5)2 ⴚ 4(1)(2)
√
vⴝ
2(1)
5_ 17
2
Solutions are v ⴝ
42
5ⴙ
√
17
2
and v ⴝ
5ⴚ
√
17
2
7.5 Solving Rational Equations—Solutions
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