Solving Rational Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Rational Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving equations that contain rational expressions by multiplying every term by the LCD to clear all denominators, solving the resulting polynomial equation, and checking for extraneous solutions.

Fractions make equations messy. Multiply every term by the LCD to 'clear' all the denominators at once, turning a rational equation into a simpler polynomial equation. But be careful—values that make any original denominator zero are excluded from the domain and must be rejected even if they appear as solutions.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Clear denominators by multiplying by the LCD, solve the resulting equation, then check that no solution makes an original denominator zero.

Common stuck point: Extraneous solutions arise when a solution makes an original denominator zero. Always substitute back into the original equation to verify.

Sense of Study hint: Multiply every single term (including non-fraction terms) by the LCD, then solve and check for excluded values.

Worked Examples

Example 1

medium
Solve \frac{3}{x} + \frac{1}{2} = \frac{5}{x}.

Solution

  1. 1
    Step 1: LCD = 2x. Multiply every term: 2x \cdot \frac{3}{x} + 2x \cdot \frac{1}{2} = 2x \cdot \frac{5}{x}.
  2. 2
    Step 2: Simplify: 6 + x = 10.
  3. 3
    Step 3: Solve: x = 4.
  4. 4
    Check: \frac{3}{4} + \frac{1}{2} = \frac{5}{4} and \frac{5}{4} ✓

Answer

x = 4
To solve a rational equation, multiply every term by the LCD to clear all denominators. This converts the rational equation into a simpler polynomial equation.

Example 2

hard
Solve \frac{x}{x-2} - \frac{1}{x+1} = \frac{3}{(x-2)(x+1)}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve \frac{10}{x} = 5.

Example 2

medium
Solve \frac{2}{x-1} = \frac{4}{x+3}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

adding subtracting rational expressionssolving linear equations