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Mom's Basement University

Solving Rational Equations

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to solve rational equations by clearing denominators and checking for extraneous solutions.

The Concept

Section titled “The Concept”

A rational equation is an equation that contains one or more rational expressions (fractions with polynomials).

To solve:

  1. Find the least common denominator (LCD) of all fractions.
  2. Multiply every term on both sides by the LCD to clear the denominators.
  3. Solve the resulting polynomial equation.
  4. Check your solutions in the original equation. Some may be extraneous (they make a denominator zero).

The checking step is critical. Multiplying by the LCD can introduce solutions that don’t actually work in the original equation. If a solution makes any original denominator equal to zero, throw it out.

Worked Example

Section titled “Worked Example”

Solve:

2x−3+3x+2=1\frac{2}{x - 3} + \frac{3}{x + 2} = 1

1. Find the LCD

The denominators are (x − 3) and (x + 2), so the LCD is (x − 3)(x + 2).

2. Multiply every term by the LCD

2(x+2)+3(x−3)=1⋅(x−3)(x+2)2(x + 2) + 3(x - 3) = 1 \cdot (x - 3)(x + 2)

3. Expand both sides

2x+4+3x−9=x2−x−65x−5=x2−x−6\begin{aligned} 2x + 4 + 3x - 9 &= x^2 - x - 6 \\[1em] 5x - 5 &= x^2 - x - 6 \end{aligned}

4. Bring all terms to one side

0=x2−x−6−5x+5=x2−6x−10 = x^2 - x - 6 - 5x + 5 = x^2 - 6x - 1

5. Solve the quadratic using the quadratic formula

Here a = 1, b = −6, c = −1:

x=6±36+42=6±402=6±2102=3±10x = \frac{6 \pm \sqrt{36 + 4}}{2} = \frac{6 \pm \sqrt{40}}{2} = \frac{6 \pm 2\sqrt{10}}{2} = 3 \pm \sqrt{10}

6. Check both solutions

x = 3 + √10 ≈ 6.16. Neither denominator is zero. Valid.

x = 3 − √10 ≈ −0.16. Neither denominator is zero. Valid.

Solutions: x = 3 + √10 and x = 3 − √10.

Real-World Application

Section titled “Real-World Application”

Rational equations appear in:

  • Combined work-rate problems (“How long until two people finish the job together?”)
  • Distance-rate-time problems with different speeds
  • Mixture concentration problems
  • Electrical circuits (parallel resistance)
  • Average cost calculations

Example: if Pipe A fills a tank in x hours and Pipe B fills it in x + 4 hours, their combined rate leads to a rational equation that must be solved.

Quiz

Section titled “Quiz”
When solving a rational equation, the most important final step is:
To clear denominators in a rational equation, multiply both sides by:
Why do we check solutions after solving a rational equation?
A solution that makes a denominator zero is called:
Solve: $\frac{3}{x} + \frac{1}{2} = \frac{5}{x}$