Adding and Subtracting Rational Expressions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Adding and Subtracting Rational Expressions.

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

Concept Recap

Adding or subtracting rational expressions by finding a least common denominator (LCD), rewriting each fraction with the LCD, then combining the numerators over the common denominator.

Just like \frac{1}{3} + \frac{1}{4} requires a common denominator of 12, adding \frac{2}{x+1} + \frac{3}{x-2} requires the LCD (x+1)(x-2). Rewrite each fraction so both have the same denominator, then add the numerators. The process mirrors numeric fractions but with polynomial denominators.

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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: Find the LCD by factoring all denominators. Multiply each fraction's numerator and denominator by whatever factors are missing from its denominator.

Common stuck point: Finding the LCD when denominators are polynomials that need factoring. Factor each denominator first, then build the LCD from all unique factors at their highest powers.

Sense of Study hint: Factor each denominator, build the LCD from all unique factors, then rewrite each fraction with that LCD.

Worked Examples

Example 1

medium
Add \frac{2}{x+1} + \frac{3}{x-2}.

Solution

  1. 1
    Step 1: LCD = (x+1)(x-2).
  2. 2
    Step 2: \frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)}.
  3. 3
    Step 3: Combine: \frac{2x - 4 + 3x + 3}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)}.
  4. 4
    Check: At x = 3: \frac{2}{4} + \frac{3}{1} = \frac{7}{2} and \frac{14}{4} = \frac{7}{2} ✓

Answer

\frac{5x - 1}{(x+1)(x-2)}
To add rational expressions with different denominators, find the LCD, rewrite each fraction with the LCD, then combine numerators. This mirrors adding numeric fractions.

Example 2

hard
Subtract \frac{x}{x+2} - \frac{3}{x^2 + 4x + 4}.

Practice Problems

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

Example 1

easy
Add \frac{3}{x} + \frac{5}{x}.

Example 2

medium
Subtract \frac{5}{x-3} - \frac{2}{x+1}.
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Background Knowledge

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

simplifying rational expressionsleast common multiple