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Rational Equations
MATH 017
Intermediate Algebra
S. Rook
1
Overview
• Section 6.6 in the textbook
– Solving rational equations
– Extraneous solutions
– More examples
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Solving Rational Equations
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Solving Rational Equations
• So far, we have been working with rational
expressions
– No =
• Find the LCD of all factored denominators for
both rational expressions and rational equations
• For rational expressions
– Simplify by combining over one denominator
• Section 6.2
• For rational equations
– Multiply both sides of the equation by the LCD to
clear out the denominators
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Solving Rational Equations
(Example)
Ex 1: Solve
5x 1 x
 
6 2 3
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Solving Rational Equations
(Example)
3x  6 7


x
2x
2
Ex 2: Solve 5
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Extraneous Solutions
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Extraneous Solutions
• Extraneous solution: a solution to a rational
equation which causes at least one of the
denominators in the original equation to equal
zero
Ex: x = 1 is an extraneous solution to 1 / (x – 1) = 1
• Must EXCLUDE an extraneous solution from a
solution set
• What If all proposed solutions are extraneous?
– The answer is the empty set { }
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Extraneous Solutions (Example)
2x  1

x 3
x 3
Ex 3: Solve x  5
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Extraneous Solutions (Example)
Ex 4: Solve
3x
1
1
 
3x  1 x 3x  1
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Additional Examples
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Additional Examples
Ex 5: Solve
2 x 10  5 x
x
 2

x  4 x 16 x  4
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Additional Examples
Ex 6: Solve
3z
1
4
  2
2
3z  7 z  6 z 3z  2 z
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Additional Examples
Ex 7: Solve x
5
 3
4 x
14
Additional Examples
Ex 8: Solve
64
2a
1 
2
a  16
a4
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Summary
• After studying these slides, you should
know how to do the following:
– Solve rational equations
– Identify extraneous solutions and exclude
them from solution sets
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