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.

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

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 $x = 1 - \sqrt{5}$

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: $x = 4$