Adding & Subtracting Rational Expressions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Add and subtract rational expressions using common denominators.
  • Find and use the least common denominator (LCD).
  • Combine numerators and simplify fully.
  • State all variable restrictions clearly.

Key Ideas

Adding or subtracting rational expressions works just like adding or subtracting fractions—you need a common denominator.

To add or subtract rational expressions:

  1. Find the LCD by factoring each denominator.
  2. Rewrite each expression using the LCD.
  3. Combine numerators (add or subtract).
  4. Factor & simplify the result.
  5. State restrictions based on all denominators.

Common Problem Types

1. Same Denominator

Simply combine numerators.

2. Different Denominators

Find the LCD, rewrite, then combine.

3. Complex Expressions

Factor denominators first to identify hidden common factors.

4. Simplifying After Adding/Subtracting

Always check whether the final expression can be simplified further.

Strategies

  • Factor denominators early—it helps you see the LCD more clearly.
  • When rewriting numerators, distribute carefully to avoid sign mistakes.
  • Collect restrictions before simplifying; canceled factors still matter.
  • Keep the combined numerator in parentheses if the expression is long.
  • Simplify only after combining.

Worked Examples

Example 1 — Same Denominator

Simplify: \[ \frac{5}{x + 2} + \frac{3}{x + 2} \]

Solution:
Same denominator → combine numerators:

\[ \frac{8}{x + 2} \]

Restriction:
\[ x \ne -2 \]

Example 2 — Different Denominators

Simplify: \[ \frac{1}{x} + \frac{1}{x + 1} \]

Solution:
LCD = \(x(x + 1)\).

Rewrite each fraction:

\[ \frac{1}{x} = \frac{x + 1}{x(x + 1)}, \qquad \frac{1}{x + 1} = \frac{x}{x(x + 1)} \]

Add:

\[ \frac{(x + 1) + x}{x(x + 1)} = \frac{2x + 1}{x(x + 1)} \]

Restrictions:
\[ x \ne 0, -1 \]

WarningCommon Mistakes
  • Adding denominators instead of finding an LCD.
  • Forgetting to distribute numerators when rewriting fractions.
  • Canceling terms instead of full factors.

Practice Problems

  1. \(\dfrac{2}{x} + \dfrac{3}{x}\)
  2. \(\dfrac{1}{y} - \dfrac{1}{y + 3}\)
  3. \(\dfrac{x}{x - 2} + \dfrac{2}{x - 2}\)
  4. \(\dfrac{3}{a} + \dfrac{1}{a + 1}\)
  5. \(\dfrac{5}{x + 4} - \dfrac{3}{x}\)

1.
Same denominator: \[ \frac{2}{x} + \frac{3}{x} = \frac{5}{x}, \quad x \ne 0 \]

2.
LCD = \(y(y + 3)\): \[ \frac{y + 3 - y}{y(y + 3)} = \frac{3}{y(y + 3)}, \quad y \ne 0, -3 \]

3.
Same denominator: \[ \frac{x + 2}{x - 2}, \quad x \ne 2 \]

4.
LCD = \(a(a + 1)\): \[ \frac{3(a + 1) + a}{a(a + 1)} = \frac{4a + 3}{a(a + 1)}, \quad a \ne 0, -1 \]

5.
LCD = \(x(x + 4)\): \[ \frac{5x - 3(x + 4)}{x(x + 4)} = \frac{5x - 3x - 12}{x(x + 4)} = \frac{2x - 12}{x(x + 4)}, \quad x \ne 0, -4 \]

Summary

  • To add or subtract rational expressions, find the LCD and rewrite each fraction.
  • Combine numerators only after rewriting over the same denominator.
  • Factor and simplify the final expression when possible.
  • Always list restrictions based on original denominators.
  • Proper factoring prevents incorrect cancellations.
  • Factor denominators first to see the LCD easily.
  • Always distribute when adjusting numerators.
  • Restrictions come from every original denominator.
  • After combining, look for factors to simplify the result.