Solving Rational Equations

A rational equation is an equation that contains at least one rational expression (a fraction with a variable in the denominator). To solve these equations, the goal is to eliminate the fractions by multiplying every term by the Least Common Denominator (LCD).

Steps to Solve Rational Equations

1. Find the LCD: Determine the least common denominator for all fractions in the equation.
2. State Restrictions: Identify any values of the variable that make a denominator zero. These are restricted values.
3. Clear Fractions: Multiply both sides of the equation by the LCD.
4. Solve: Solve the resulting polynomial equation.
5. Check for Extraneous Solutions: Ensure your answers do not match any of the restricted values from Step 2. If they do, they are "extraneous" and must be rejected.

Example 1: Algebraic Rational Equation

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

Step 1: Find the LCD and restrictions. The denominators are $(x - 2)$ and $(x + 1)$. The LCD is $(x - 2)(x + 1)$. Restrictions: $x \neq 2$ and $x \neq -1$ (since these make the denominator zero).

Step 2: Multiply both sides by the LCD. $(x - 2)(x + 1) \left( \frac{x}{x - 2} + \frac{2}{x + 1} \right) = (x - 2)(x + 1) \left( \frac{5}{(x - 2)(x + 1)} \right)$

This simplifies to: $x(x + 1) + 2(x - 2) = 5$

Step 3: Solve the resulting equation. $x^2 + x + 2x - 4 = 5$ $x^2 + 3x - 9 = 0$

Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-9)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 36}}{2} = \frac{-3 \pm 3\sqrt{5}}{2}$

Step 4: Check for extraneous solutions. Neither solution is $2$ or $-1$, so both are valid.

Example 2: Work Rate Word Problem

Problem: Two pipes can fill a pool in 6 and 8 hours respectively. How long will it take them to fill the pool if they work together?

Solution: Let $t$ be the time it takes to fill the pool together.

• Pipe 1's rate: $\frac{1}{6}$ of the pool per hour.
• Pipe 2's rate: $\frac{1}{8}$ of the pool per hour.
• Combined rate: $\frac{1}{t}$ of the pool per hour.

Set up the rational equation: $\frac{1}{6} + \frac{1}{8} = \frac{1}{t}$

The LCD for $6$, $8$, and $t$ is $24t$. Multiply the entire equation by $24t$: $24t \left( \frac{1}{6} \right) + 24t \left( \frac{1}{8} \right) = 24t \left( \frac{1}{t} \right)$ $4t + 3t = 24$ $7t = 24$ $t = \frac{24}{7} \approx 3.43 \text{ hours}$

Working together, the pipes will fill the pool in $\frac{24}{7}$ hours.