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Solving Rational Equations

Master the art of solving rational equations through clear explanations and targeted practice. Learn techniques such as clearing denominators, cross-multiplication, and checking for extraneous solutions.

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Solving Rational Equations

Study Guide

Key Definition

A rational equation is an equation that sets two (or more) rational expressions—fractions whose numerators and denominators are polynomials—equal to one another.

Important Notes

A rational expression is a ratio of two polynomials; its denominator must never be zero.
Identify domain restrictions (values that make any denominator zero) before you start solving.
Always check your final answers against those domain restrictions to rule out extraneous solutions.
Cross-multiplication or clearing denominators converts the equation to a polynomial equation that is often easier to solve.
After clearing denominators, simplify completely before solving.

Mathematical Notation

$\\frac{a}{b}$ represents a fraction

$+$ represents addition

$-$ represents subtraction

$\\times$ or $*$ represents multiplication

$\\div$ or $/$ represents division

Write each step clearly and keep track of restrictions so you can reject extraneous roots later.

Why It Works

Multiplying both sides of a rational equation by the least common denominator removes fractions, turning the problem into a familiar polynomial equation. Checking solutions against the original denominators ensures those solutions are valid.

Remember

Solutions that make any original denominator zero are not valid, even if they satisfy the transformed (cleared-denominator) equation.

Quick Reference

Cross-Multiplication:$\\frac{a}{b} = \\frac{c}{d} \\rightarrow ad = bc$

Understanding Solving Rational Equations

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Start with an equation like $\\frac{x}{2}=\\frac{3}{4}$. Multiply both sides by 4 (the common denominator) to get $4\\left(\\frac{x}{2}\\right)=4\\left(\\frac{3}{4}\\right)$, which simplifies to $2x=3$, then $x=\\tfrac{3}{2}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Solve $\\frac{x}{2}=\\frac{3}{4}$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You and a friend each eat $\\tfrac{1}{4}$ of a pizza. How much of the pizza remains uneaten?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

For $\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$, manipulate the equation to express $z$ in terms of $x$ and $y$.

Challenge Quiz

Single Choice Quiz

Advanced

Solve $\\frac{x+2}{x-3}=\\frac{x-1}{x+4}$. (Remember to check for extraneous solutions.)

Recap

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Review key concepts and takeaways