Solving Rational Equations

Equations with rational expressions

Solving Rational Equations

Strategy

Find the LCD of all denominators
Multiply both sides by the LCD
Solve the resulting equation
Check for extraneous solutions

Extraneous Solutions

Solutions that make any denominator zero are extraneous and must be rejected.

Always check your answers!

Common Types

Proportion: $\frac{a}{b} = \frac{c}{d}$ → Cross multiply: $ad = bc$

Work Problems: $\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t_{together}}$

Rate Problems: $\text{Time} = \frac{\text{Distance}}{\text{Rate}}$

Example

Solve: $\frac{3}{x} + \frac{2}{x-1} = 5$

LCD: $x(x - 1)$

Multiply both sides: $3(x - 1) + 2x = 5x(x - 1)$ $3x - 3 + 2x = 5x^2 - 5x$ $5x - 3 = 5x^2 - 5x$ $0 = 5x^2 - 10x + 3$

Use quadratic formula to solve.

📚 Practice Problems

1Problem 1easy

❓ Question:

Solve: $\frac{x}{3} = \frac{5}{2}$

💡 Show Solution

Cross multiply: $2x = 3 \cdot 5$ $2x = 15$ $x = \frac{15}{2}$

Check: $\frac{15/2}{3} = \frac{15}{6} = \frac{5}{2}$ ✓

Answer: $x = \frac{15}{2}$ or $7.5$

2Problem 2medium

❓ Question:

Solve: $\frac{1}{x} + \frac{1}{x+2} = \frac{1}{2}$

💡 Show Solution

LCD: $2x(x + 2)$

Multiply both sides by LCD: $2x(x + 2) \cdot \frac{1}{x} + 2x(x + 2) \cdot \frac{1}{x+2} = 2x(x + 2) \cdot \frac{1}{2}$

$2(x + 2) + 2x = x(x + 2)$ $2x + 4 + 2x = x^2 + 2x$ $4x + 4 = x^2 + 2x$ $0 = x^2 - 2x - 4$

Use quadratic formula: $a = 1, b = -2, c = -4$ $x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5}$

Answer: $x = 1 + \sqrt{5}$ or $x = 1 - \sqrt{5}$

3Problem 3hard

❓ Question:

Solve: $\frac{6}{x-2} = \frac{x}{x-2} + 1$

💡 Show Solution

LCD: $x - 2$

Multiply both sides: $6 = x + (x - 2)$ $6 = x + x - 2$ $6 = 2x - 2$ $8 = 2x$ $x = 4$

Check: Does $x = 4$ make any denominator zero? $x - 2 = 4 - 2 = 2 \neq 0$ ✓

Verify: $\frac{6}{2} = \frac{4}{2} + 1$ → $3 = 2 + 1$ ✓

Answer: $x = 4$

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