To add $\frac{1}{3} + \frac{1}{4}$ you rewrite both over 12. Rational expressions work the same way, with the least common denominator built from the factored denominators. The arithmetic with factors is what you practiced in Multiplying and Dividing Rational Expressions.
Key idea
Factor each denominator. The LCD uses each factor the greatest number of times it appears in any one denominator. Rewrite each fraction over the LCD, add or subtract the numerators (the whole numerator, in parentheses), keep the denominator, and simplify.
Why it works
Fractions can only be added when they count the same size pieces, which means a common denominator. Multiplying a fraction by $\frac{x + 2}{x + 2}$ is multiplying by 1, so it changes the look but not the value. Adding the denominators would be like saying $\frac{1}{2} + \frac{1}{2} = \frac{2}{4}$.
Worked examples
Worked example: Different linear denominators
Simplify $\frac{3}{x + 1} + \frac{2}{x - 2}$.
- LCD: $(x + 1)(x - 2)$.
- Rewrite: $\frac{3(x - 2)}{(x + 1)(x - 2)} + \frac{2(x + 1)}{(x + 1)(x - 2)}$.
- Add the numerators: $3x - 6 + 2x + 2 = 5x - 4$.
Answer: $\frac{5x - 4}{(x + 1)(x - 2)}$
Worked example: Subtracting, with a shared factor
Simplify $\frac{x}{x^2 - 9} - \frac{1}{x - 3}$.
- Factor: $x^2 - 9 = (x - 3)(x + 3)$, so the LCD is $(x - 3)(x + 3)$.
- Rewrite the second fraction: $\frac{x + 3}{(x - 3)(x + 3)}$.
- Subtract the whole numerator: $x - (x + 3) = -3$.
Answer: $\frac{-3}{(x - 3)(x + 3)}$
Common mistake
Two classic slips. Adding the denominators: $\frac{1}{x} + \frac{1}{x} = \frac{2}{2x}$ is wrong (it should be $\frac{2}{x}$). And subtracting only the first term of the second numerator: $x - x + 3$ instead of $x - (x + 3)$. Put the second numerator in parentheses before you subtract, and check your answer at a number such as $x = 1$.