Integrated Math 3 / Unit 2: Rational Expressions and Functions
2.3 Adding and Subtracting Rational Expressions
Notes
Fractions need a common denominator before you add or subtract. Use the least common denominator (LCD): the product of each distinct factor, to its highest power.
- Same denominator: $\displaystyle \frac{3}{x} + \frac{2}{x} = \frac{5}{x}$.
- Different denominators: $\displaystyle \frac{1}{x - 1} + \frac{1}{x + 1} = \frac{(x + 1) + (x - 1)}{(x - 1)(x + 1)} = \frac{2x}{x^2 - 1}$.
- Distribute the minus sign across the whole numerator when subtracting.
Practice
Add: $\displaystyle \frac{1}{x - 1} + \frac{1}{x + 1}$.
The LCD is $(x - 1)(x + 1)$. The numerator is $(x + 1) + (x - 1) = 2x$.
What is the LCD of $\displaystyle \frac{1}{x^2 - 4}$ and $\displaystyle \frac{1}{x^2 + 4x + 4}$?
The denominators are $(x - 2)(x + 2)$ and $(x + 2)^2$. Use each factor to its highest power.
Subtract: $\displaystyle \frac{5}{x + 2} - \frac{3}{x - 1}$.
$5(x - 1) - 3(x + 2) = 5x - 5 - 3x - 6 = 2x - 11$.
Ready for more? Take the Unit 2 test.