Chapter 7 Rational Functions - College of the Redwoods

Section 7.7 Solving Rational Equations 711
7.7 Solving Rational Equations
When simplifying complex fractions in the previous section, we saw that multiplying
both numerator and denominator by the appropriate expression could “clear” all fractions
from the numerator and denominator, greatly simplifying the rational expression.
In this section, a similar technique is used.
Clear the Fractions from a Rational Equation. If your equation has rational
expressions, multiply both sides of the equation by the least common denominator
to clear the equation of rational expressions.
Let’s look at an example.
◮ Example 1. Solve the following equation for x.
x
2 − 2 3 = 3 4
To clear this equation of fractions, we will multiply both sides by the common
denominator for 2, 3, and 4, which is 12. Distribute 12 in the second step.
( x
12
2 3)
− 2 ( 3
= 12
4)
( x
) ( ( 2 3
12 − 12 = 12
2 3)
4)
Multiply.
6x − 8 = 9
We’ve succeeded in clearing the rational expressions from the equation by multiplying
through by the common denominator. We now have a simple linear equation which
can be solved by first adding 8 to both sides of the equation, followed by dividing both
sides of the equation by 6.
6x = 17
x = 17 6
We’ll leave it to our readers to check this solution.
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Version: Fall 2007