Section 4.4: Operations with Rational Expressions

This section covers adding, subtracting, multiplying, and dividing rational expressions, including finding common denominators and simplifying results.

Example 1: Adding Rational Expressions

Simplify \( \frac{2}{x} + \frac{3}{x} \).

Step 1: Common denominator is x

Step 2: Add numerators: \( \frac{2+3}{x} = \frac{5}{x} \)

Example 2: Subtracting Rational Expressions

Simplify \( \frac{x}{x+1} - \frac{2}{x+1} \).

Step 1: Common denominator is \( x+1 \)

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

Example 3: Multiplying Rational Expressions

Simplify \( \frac{x}{x+2} \cdot \frac{3}{x} \).

Step 1: Multiply numerators and denominators: \( \frac{3x}{x(x+2)} \)

Step 2: Cancel common factor x: \( \frac{3}{x+2} \)

Example 4: Dividing Rational Expressions

Simplify \( \frac{x^2-1}{x+2} \div \frac{x-1}{x} \).

Step 1: Factor numerator: \( x^2-1 = (x-1)(x+1) \)

Step 2: Rewrite division as multiplication: \( \frac{(x-1)(x+1)}{x+2} \cdot \frac{x}{x-1} \)

Step 3: Cancel common factor: \( \frac{(x+1)x}{x+2} \)

Practice Problems

  1. Simplify \( \frac{3}{x+1} + \frac{2}{x+1} \)
  2. Simplify \( \frac{x+2}{x} - \frac{1}{x} \)
  3. Simplify \( \frac{2x}{x+3} \cdot \frac{5}{x} \)
  4. Simplify \( \frac{x^2-9}{x-3} \div \frac{x+3}{1} \)
  5. Simplify \( \frac{4}{x} + \frac{3}{x+2} \) (find common denominator)
← Previous Next →