Section 9.3: Adding & Subtracting Rational Expressions

To add or subtract rational expressions, we need a common denominator. Once denominators match, combine the numerators and simplify.

Example 1

Add: \( \frac{3}{x} + \frac{2}{x} \)

Denominators already match: \( \frac{3}{x} + \frac{2}{x} = \frac{3+2}{x} = \frac{5}{x} \), restriction: \( x \neq 0 \).

Subtract: \( \frac{2}{x} - \frac{3}{x+1} \)

LCD = \( x(x+1) \)

Rewrite: \( \frac{2(x+1)}{x(x+1)} - \frac{3x}{x(x+1)} \)

Combine: \( \frac{2x+2 - 3x}{x(x+1)} = \frac{-x+2}{x(x+1)} \), restrictions: \( x \neq 0, -1 \).

Add: \( \frac{x}{x-2} + \frac{3}{x+2} \)

LCD = \( (x-2)(x+2) \)

Rewrite: \( \frac{x(x+2)}{(x-2)(x+2)} + \frac{3(x-2)}{(x-2)(x+2)} \)

Combine: \( \frac{x^2+2x + 3x - 6}{(x-2)(x+2)} = \frac{x^2+5x-6}{(x-2)(x+2)} \)

Factor numerator: \( (x+6)(x-1) \), final \( \frac{(x+6)(x-1)}{(x-2)(x+2)} \), restrictions: \( x \neq \pm2 \).

\( \frac{5}{y} + \frac{7}{y} \)
\( \frac{3}{x+2} - \frac{4}{x} \)
\( \frac{2}{a} + \frac{5}{a-1} \)
\( \frac{m}{m+3} - \frac{2}{m-3} \)
\( \frac{4x}{x^2-1} + \frac{3}{x-1} \)