Lesson 99: Solving Rational Equations

New Concept

A rational equation is an equation containing at least one rational expression.

What’s next

Next, you’ll master two key methods for solving these equations and learn how to spot tricky “extraneous” solutions.

Property

A rational equation is an equation containing at least one rational expression. If it's a proportion, you can solve it using cross products: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Explanation

Think of these as fraction puzzles where 'x' is hiding! When two fractions are equal, you can cross-multiply to get rid of the denominators and turn it into a regular equation. It’s a neat trick to un-complicate things and find the hidden value of x.

Examples

To solve $\frac{7}{x} = \frac{4}{x-3}$, cross-multiply to get $7(x-3) = 4x$, so $x=7$.
To solve $\frac{x}{5} = \frac{2}{x+3}$, cross-multiply to get $x(x+3) = 10$, so $x^2+3x-10=0$, which means $x=2$ or $x=-5$.

Property

An extraneous solution is a solution that is acquired through the solving process, but makes a denominator in the original equation equal to 0.

Explanation

Watch out for these impostor solutions! They seem to solve the equation, but they are mathematical outlaws. Plugging them back into the original equation makes a denominator zero, which breaks the fundamental rules of fractions. Always check your answers to expose these fakes!

Examples

Solve $\frac{x+3}{x-4} = \frac{2x+1}{x-4}$. Cross-multiplying gives $x+3=2x+1$, so $x=2$. This solution is valid.
Solve $\frac{x-5}{x-5} = \frac{2x-9}{x-5}$. This simplifies to $1 = 2x-9$, so $x=5$. But $x=5$ makes the denominator 0, so it's an extraneous solution.

Sometimes an equation has a 'disguised' trap, let's learn how to find it. This example demonstrates how to solve rational equations using cross products, a key idea from this lesson, and how to identify extraneous solutions.

Example Problem

Solve the equation $\frac{x+3}{x-5} = \frac{x+7}{3x-15}$.

Property

If a rational equation includes a sum or difference, find the least common denominator (LCD) of all the terms and multiply the entire equation by it to solve.

Explanation

Fractions in an equation can be messy. Multiplying every single term by the LCD is like using a magic wand to make all the denominators disappear! This leaves you with a much friendlier, fraction-free equation to solve. It’s the ultimate cleanup tool for rational equations.

Examples

Solve $\frac{5}{x} + \frac{1}{2} = 3$. The LCD is $2x$. Multiply by $2x$ to get $10 + x = 6x$, so $x=2$.
Solve $\frac{4}{x-1} - \frac{1}{x} = \frac{2}{x}$. The LCD is $x(x-1)$. This becomes $4x - (x-1) = 2(x-1)$, which simplifies to $x=-1$.