Integrated Math 3 / Unit 2: Rational Expressions and Functions
2.5 Solving Rational Equations
Notes
Multiply every term by the LCD to clear the fractions, solve, then check each answer.
- Example: $\displaystyle \frac{3}{x} + \frac{1}{2} = 2 \Rightarrow 6 + x = 4x \Rightarrow x = 2$.
- A solution that makes a denominator 0 is extraneous and must be thrown out.
- For a proportion $\displaystyle \frac{a}{b} = \frac{c}{d}$ you can cross multiply.
Practice
Solve $\displaystyle \frac{2}{x + 1} = \frac{3}{x - 2}$.
$2(x - 2) = 3(x + 1) \Rightarrow 2x - 4 = 3x + 3 \Rightarrow x = -7$.
Solve $\displaystyle \frac{x^2}{x - 3} = \frac{9}{x - 3}$.
$x^2 = 9$ gives $x = \pm 3$, but $x = 3$ makes the denominator 0 so it is extraneous.
Solve $\displaystyle \frac{6}{x} - 1 = x$.
Multiply by $x$: $6 - x = x^2$, so $x^2 + x - 6 = (x + 3)(x - 2) = 0$. Both values check.
Ready for more? Take the Unit 2 test.