Download 6.5 Solving Equations Containing Fractions

Section 6.5 / Solving Equations Containing Fractions
6.5
Objective A
315
Solving Equations
Containing Fractions
To solve an equation containing fractions
Recall that to solve an equation containing fractions, clear denominators by
multiplying each side of the equation by the LCM of the denominators. Then
solve for the variable.
Solve:
HOW TO
3x 1
2
7
4
3
6
3x 1
2
7
4
3
6
3x 1 2
7
12
12 4
3
6
3x 1
2
7
12
12 12 4
3
6
3
12 3x 1
1
4
1
4
• The LCM is 12. To clear denominators, multiply
each side of the equation by the LCM.
• Simplify using the Distributive Property
and the Properties of Fractions.
2
12 2
12 7
1 3
1 6
1
1
9x 3 8 14
9x 5 14
9x 9
x1
• Solve for x.
1 checks as a solution. The solution is 1.
Occasionally, a value of the variable that appears to be a solution of an equation
will make one of the denominators zero. In this case, that value is not a solution
of the equation.
Solve:
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HOW TO
4
2x
1
x2
x2
2x
4
1
x2
x2
2x
4
x 2 1 x2
x2
2x
4
x 2
x 2 1 x 2
x2
x2
x 2
1
1
x 2
x 2
2x
4
x 2 1 1
x2
1
x2
1
• The LCM is x 2. Multiply each
side of the equation by the LCM.
• Simplify using the Distributive
Property and the Properties of
Fractions.
1
2x x 2 4
2x x 2
x2
When x is replaced by 2, the denominators of
Therefore, the equation has no solution.
• Solve for x.
2x
x2
and
4
x2
are zero.
316
Chapter 6 / Rational Expressions
Example 1
Solve:
You Try It 1
x
2
x4
x
Solve:
Solution
x
3
x6
x
Your solution
The LCM is xx 4.
x
2
x4
x
x
2
xx 4
xx 4
x4
x
1
• Multiply by
the LCM.
1
xx 4
x
xx 4 2
1
x4
1
x
1
• Divide by
common factors.
1
x2 x 42
x2 2x 8
• Simplify.
Solve the quadratic equation by factoring.
x2 2x 8 0
x 4x 2 0
x40
x20
x4
x 2
• Write in standard form.
• Factor.
• Principle of Zero Products
Both 4 and 2 check as solutions.
The solutions are 4 and 2.
Example 2
Solve:
You Try It 2
3x
12
5
x4
x4
Solve:
Solution
5x
10
3
x2
x2
Your solution
The LCM is x 4.
1
• Clear
denominators.
1
x 4
3x
x 4
12
x 45 1
x4
1
x4
1
3x x 45 12
3x 5x 20 12
3x 5x 8
2x 8
x4
1
• Solve for x.
4 does not check as a solution.
The equation has no solution.
Solutions on p. S17
Copyright © Houghton Mifflin Company. All rights reserved.
3x
12
5
x4
x4
3x
12
x 4
x 4 5 x4
x4
Section 6.5 / Solving Equations Containing Fractions
317
6.5 Exercises
Objective A
1.
To solve an equation containing fractions
Can 2 be a solution of the equation
6x
x1
x
x2
4? Explain your
answer.
2.
After multiplying each side of an equation by a variable expression,
why must we check the solution?
For Exercises 3 to 35, solve.
3.
2x
5
1
3
2
2
4.
x
1
1
3
4
12
5.
x
1
x
1
3
4
4
6
6.
2y
1
y
1
9
6
9
6
7.
2x 5
1
x
3
8
4
8
4
8.
3x 4
1
5x 2
1
12
3
12
2
9.
6
2
2a 1
10.
12
3
3x 2
12.
6
3
4 3x
13.
2
16.
3
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15. 1 9
4
x
11.
9
2
2x 5
5
7
x
14.
3
12
7
x
17.
2
59
y
8
5
n
18.
6
3 11
x
19.
3
4
x2
x
20.
5
3
x3
x1
21.
2
3
3x 1
4x 1
22.
5
3
3x 4
1 2x
23.
3
2
2x 5
x1
318
24.
Chapter 6 / Rational Expressions
4
2
5y 1
2y 1
27. 2 30.
3
a
a3
a3
x
1
x 12
x5
33. x 6
3x
x2
x2
25.
4x
5x
5
x4
x4
26.
2x
7x
5
x2
x2
28.
x
4
3
x4
x4
29.
x
8
x1
x2
31.
2x
3
x4
x1
32.
5
n
3n 8
n2
34.
x
35.
8
2
1
y
y2
6
2x
x3
x3
APPLYING THE CONCEPTS
36.
Explain the procedure for solving an equation containing fractions.
Include in your discussion how the LCM of the denominators is
used to eliminate fractions in the equation.
37.
3
1
2y 5
y 1 y 5
3
15
38.
3
1
a2
a 3 a 4
2
4
39.
b2
1
3
b
b 1
5
4
10
40.
x
3
3
2
2x x 1
x 1
2x 1
41.
x1
x2
3
2
x2 x 2
x 1
x2
42.
y2
y1
1
2
y2 y 2
y 4
y1
2
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For Exercises 37 to 42, solve.