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 3x+12=5x\frac{3}{x} + \frac{1}{2} = \frac{5}{x}.

Answer

x=4x = 4

First step

1
Step 1: LCD = 2x2x. Multiply every term: 2x⋅3x+2x⋅12=2x⋅5x2x \cdot \frac{3}{x} + 2x \cdot \frac{1}{2} = 2x \cdot \frac{5}{x}.

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Example 2

hard
Solve xx−2−1x+1=3(x−2)(x+1)\frac{x}{x-2} - \frac{1}{x+1} = \frac{3}{(x-2)(x+1)}.

Example 3

easy
Solve 4x=23\frac{4}{x} = \frac{2}{3}.

Example 4

medium
Solve 2x+1+3x−1=1\frac{2}{x+1} + \frac{3}{x-1} = 1.

Example 5

hard
Solve x+2x−3−x−1x+3=4x2−9\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 10x=5\frac{10}{x} = 5.

Example 2

medium
Solve 2x−1=4x+3\frac{2}{x-1} = \frac{4}{x+3}.

Example 3

easy
Solve x3=4\frac{x}{3} = 4.

Example 4

easy
Solve 6x=2\frac{6}{x} = 2.

Example 5

easy
What value of xx must be excluded from 5x−2\frac{5}{x-2}?

Example 6

easy
Solve x4=32\frac{x}{4} = \frac{3}{2}.

Example 7

easy
Solve 2x=14\frac{2}{x} = \frac{1}{4}.

Example 8

easy
What is the LCD of 1x\frac{1}{x} and 1x+1\frac{1}{x+1}?

Example 9

easy
Solve x+12=3\frac{x+1}{2} = 3.

Example 10

easy
Is x=0x = 0 a valid solution to 1x=1x\frac{1}{x} = \frac{1}{x}?

Example 11

medium
Solve 1x+12=1\frac{1}{x} + \frac{1}{2} = 1.

Example 12

medium
Solve 3x=x3\frac{3}{x} = \frac{x}{3}.

Example 13

medium
Solve xx−1=2\frac{x}{x-1} = 2.

Example 14

medium
Solve 2x−3=4x\frac{2}{x-3} = \frac{4}{x}.

Example 15

medium
Solve xx−2=2x−2\frac{x}{x-2} = \frac{2}{x-2}.

Example 16

medium
Solve 1x−1+1x+1=2x2−1\frac{1}{x-1} + \frac{1}{x+1} = \frac{2}{x^2-1}.

Example 17

medium
Solve x+2x=32\frac{x+2}{x} = \frac{3}{2}.

Example 18

medium
Solve 5x−3x=1\frac{5}{x} - \frac{3}{x} = 1.

Example 19

medium
Solve x2+x3=5\frac{x}{2} + \frac{x}{3} = 5.

Example 20

challenge
Solve xx−3+3x−3=2\frac{x}{x-3} + \frac{3}{x-3} = 2.

Example 21

challenge
Solve 1x+1x+2=12\frac{1}{x} + \frac{1}{x+2} = \frac{1}{2} for the positive root.

Example 22

challenge
For what value of kk does xx−2=k\frac{x}{x-2} = k have NO solution?

Example 23

easy
Solve x5=7\frac{x}{5} = 7.

Example 24

easy
Solve 12x=3\frac{12}{x} = 3.

Example 25

easy
Solve x−13=4\frac{x-1}{3} = 4.

Example 26

easy
Solve 3x+2=1\frac{3}{x+2} = 1.

Example 27

easy
Find the LCD of 1x−2\frac{1}{x-2} and 3x+5\frac{3}{x+5}.

Example 28

medium
Solve 2x+3x=5\frac{2}{x} + \frac{3}{x} = 5.

Example 29

medium
Solve 1x−1x+1=112\frac{1}{x} - \frac{1}{x+1} = \frac{1}{12} for the positive root.

Example 30

medium
Solve x+1x−2=3\frac{x+1}{x-2} = 3.

Example 31

medium
Solve 6x=x+12\frac{6}{x} = \frac{x+1}{2}.

Example 32

medium
Solve xx+3=4x+3\frac{x}{x+3} = \frac{4}{x+3}.

Example 33

medium
Two pipes fill a tank together in 44 hours. Alone, pipe A takes 66 hours. How long does pipe B take?

Example 34

medium
Solve x−3x+2=14\frac{x-3}{x+2} = \frac{1}{4}.

Example 35

hard
Solve xx−1−1x+1=2x2−1\frac{x}{x-1} - \frac{1}{x+1} = \frac{2}{x^2-1}.

Example 36

hard
Solve 2x−3+1x+3=18x2−9\frac{2}{x-3} + \frac{1}{x+3} = \frac{18}{x^2-9}.

Example 37

hard
A boat travels 3030 km upstream and 3030 km downstream in 44 hours total. The boat's still-water speed is 1616 km/h. Find the current's speed cc.

Example 38

hard
Solve 3x−2−2x+1=5(x−2)(x+1)\frac{3}{x-2} - \frac{2}{x+1} = \frac{5}{(x-2)(x+1)}.

Example 39

hard
Solve 6x−1=x+4\frac{6}{x-1} = x + 4.

Example 40

hard
For what value of kk does 2x+1x−3=k\frac{2x+1}{x-3} = k have NO solution?

Example 41

challenge
Solve 1x+1x−1+1x+1=0\frac{1}{x} + \frac{1}{x-1} + \frac{1}{x+1} = 0.

Example 42

challenge
Solve x+1x=103x + \frac{1}{x} = \frac{10}{3}.
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Background Knowledge

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

adding subtracting rational expressionssolving linear equations