Solving Rational Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Rational Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving equations that contain rational expressions by multiplying every term by the LCD to clear all denominators, solving the resulting polynomial equation, and checking for extraneous solutions.

Fractions make equations messy. Multiply every term by the LCD to 'clear' all the denominators at once, turning a rational equation into a simpler polynomial equation. But be careful—values that make any original denominator zero are excluded from the domain and must be rejected even if they appear as solutions.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Clear all denominators at once with the LCD, solve the resulting polynomial, and reject roots that were excluded.

Common stuck point: The procedure for solving rational equations is the easy part; the trap is multiplying only some terms by the LCD. Asking "Is this an EQUATION (has $=$ ) with the variable in a denominator that I clear by the LCD and then check?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this an EQUATION (has $=$ ) with the variable in a denominator that I clear by the LCD and then check?

Worked Examples

Example 1

medium

Solve

\frac{3}{x} + \frac{1}{2} = \frac{5}{x}

Answer

x = 4

First step

Step 1: LCD =

2x

. Multiply every term:

2x \cdot \frac{3}{x} + 2x \cdot \frac{1}{2} = 2x \cdot \frac{5}{x}

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Example 2

hard

Solve

\frac{x}{x-2} - \frac{1}{x+1} = \frac{3}{(x-2)(x+1)}

Example 3

easy

Solve

\frac{4}{x} = \frac{2}{3}

Example 4

medium

Solve

\frac{2}{x+1} + \frac{3}{x-1} = 1

Example 5

hard

Solve

\frac{x+2}{x-3} - \frac{x-1}{x+3} = \frac{4}{x^2-9}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Solve

\frac{10}{x} = 5

Example 2

medium

Solve

\frac{2}{x-1} = \frac{4}{x+3}

Example 3

easy

Solve

\frac{x}{3} = 4

Example 4

easy

Solve

\frac{6}{x} = 2

Example 5

easy

What value of

x

must be excluded from

\frac{5}{x-2}

Example 6

easy

Solve

\frac{x}{4} = \frac{3}{2}

Example 7

easy

Solve

\frac{2}{x} = \frac{1}{4}

Example 8

easy

What is the LCD of

\frac{1}{x}

and

\frac{1}{x+1}

Example 9

easy

Solve

\frac{x+1}{2} = 3

Example 10

easy

challenge

Solve

\frac{x}{x-3} + \frac{3}{x-3} = 2

Example 21

challenge

Solve

\frac{1}{x} + \frac{1}{x+2} = \frac{1}{2}

for the positive root.

Example 22

challenge

For what value of

k

does

\frac{x}{x-2} = k

have NO solution?

Example 23

easy

Solve

\frac{x}{5} = 7

Example 24

easy

Solve

\frac{12}{x} = 3

Example 25

easy

Solve

\frac{x-1}{3} = 4

Example 26

easy

Solve

\frac{3}{x+2} = 1

Example 27

easy

Find the LCD of

\frac{1}{x-2}

and

\frac{3}{x+5}

Example 28

medium

Solve

\frac{2}{x} + \frac{3}{x} = 5

Example 29

medium

Solve

\frac{1}{x} - \frac{1}{x+1} = \frac{1}{12}

for the positive root.

Example 30

medium

Solve

\frac{x+1}{x-2} = 3

Example 31

medium

Solve

\frac{6}{x} = \frac{x+1}{2}

Example 32

medium

Solve

\frac{x}{x+3} = \frac{4}{x+3}

Example 33

medium

Two pipes fill a tank together in

4

hours. Alone, pipe A takes

6

hours. How long does pipe B take?

Example 34

medium

Solve

\frac{x-3}{x+2} = \frac{1}{4}

Example 35

hard

Solve

\frac{x}{x-1} - \frac{1}{x+1} = \frac{2}{x^2-1}

Example 36

hard

Solve

\frac{2}{x-3} + \frac{1}{x+3} = \frac{18}{x^2-9}

Example 37

hard

A boat travels

30

km upstream and

30

km downstream in

4

hours total. The boat's still-water speed is

16

km/h. Find the current's speed

c

Example 38

hard

Solve

\frac{3}{x-2} - \frac{2}{x+1} = \frac{5}{(x-2)(x+1)}

Example 39

hard

Solve

\frac{6}{x-1} = x + 4

Example 40

hard

For what value of

k

does

\frac{2x+1}{x-3} = k

have NO solution?

Example 41

challenge

Solve

\frac{1}{x} + \frac{1}{x-1} + \frac{1}{x+1} = 0

Example 42

challenge

Solve

x + \frac{1}{x} = \frac{10}{3}

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Related Concepts

Adding and Subtracting Rational Expressions Solving Linear Equations Rational Functions

Background Knowledge

These ideas may be useful before you work through the harder examples.

adding subtracting rational expressionssolving linear equations