Adding and Subtracting Rational Expressions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Adding and Subtracting Rational Expressions.

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

Concept Recap

Adding or subtracting rational expressions by finding a least common denominator (LCD), rewriting each fraction with the LCD, then combining the numerators over the common denominator.

Just like $\frac{1}{3} + \frac{1}{4}$ requires a common denominator of 12, adding $\frac{2}{x+1} + \frac{3}{x-2}$ requires the LCD $(x+1)(x-2)$ . Rewrite each fraction so both have the same denominator, then add the numerators. The process mirrors numeric fractions but with polynomial denominators.

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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: Build the LCD, rewrite each fraction over it, and add or subtract only the tops.

Common stuck point: The procedure for adding and subtracting rational expressions is the easy part; the trap is adding denominators too. Asking "Do the denominators match yet — and if not, what is the LCD I must rewrite both over?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the denominators match yet — and if not, what is the LCD I must rewrite both over?

Worked Examples

Example 1

medium

Add

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

Answer

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

First step

Step 1: LCD =

(x+1)(x-2)

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Example 2

hard

Subtract

\frac{x}{x+2} - \frac{3}{x^2 + 4x + 4}

Example 3

easy

Worked example: combine

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

for

x\ne 0

Example 4

medium

Worked example: combine

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

for

x\ne\pm 2

Example 5

hard

Worked example: simplify

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

for

x\ne\pm 1

Practice Problems

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

Example 1

easy

Add

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

Example 2

medium

Subtract

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

Example 3

easy

Add

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

x\neq0

Example 4

easy

Subtract

\frac{5}{x} - \frac{2}{x}

x\neq0

Example 5

easy

Add

\frac{1}{x} + \frac{1}{y}

x,y\neq0

Example 6

easy

Add

\frac{3}{x} + \frac{2}{x^2}

x\neq0

Example 7

easy

Subtract

\frac{7}{2x} - \frac{1}{2x}

x\neq0

Example 8

easy

Add

\frac{x}{4} + \frac{x}{6}

Example 9

easy

Add

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

x\neq-1

Example 10

easy

Subtract

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

x\neq-2

Example 11

medium

Add

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

x\neq-1,2

Example 12

medium

Subtract

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

x\neq\pm1

Example 13

medium

Add

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

x\neq0,2

Example 14

medium

Add

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

x\neq0

Example 15

medium

Subtract

\frac{x}{x - 3} - \frac{3}{x - 3}

and simplify,

x\neq3

Example 16

medium

Add

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

x\neq\pm1

Example 17

medium

Subtract

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

x\neq0,-1

Example 18

medium

Add

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

x\neq1,-3

Example 19

medium

Subtract

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

x\neq0,-1

Example 20

challenge

Add

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

x\neq\pm2

Example 21

challenge

Subtract

\frac{x}{x^2 - x - 6} - \frac{2}{x - 3}

x\neq3,-2

Example 22

challenge

Simplify

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

x\neq\pm1

Example 23

easy

Add

\frac{4}{x}+\frac{1}{x}

for

x\ne 0

Example 24

easy

Subtract

\frac{8}{y}-\frac{3}{y}

for

y\ne 0

Example 25

easy

Add

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

Example 26

easy

Add

\frac{a}{b}+\frac{c}{b}

for

b\ne 0

Example 27

medium

Add

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

for

x\ne 1,-2

Example 28

medium

Subtract

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

for

x\ne -3,1

Example 29

medium

Combine

\frac{1}{x}-\frac{1}{x+h}

for

x\ne 0,-h

Example 30

medium

Add

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

for

x\ne\pm 2

Example 31

medium

Subtract

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

for

x\ne\pm 2

Example 32

medium

Combine

\frac{2}{x^2}-\frac{3}{x^3}

for

x\ne 0

Example 33

medium

Add

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

for

x\ne\pm 1

Example 34

medium

Simplify

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

for

x\ne 0,-1

Example 35

hard

Combine

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

for

x\ne 0,\pm 1

Example 36

hard

Combine

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

for

x\ne\pm 3

Example 37

hard

Simplify

\frac{2x}{x^2-4}-\frac{1}{x-2}

for

x\ne\pm 2

Example 38

hard

Combine

\frac{x+1}{x^2-x-6}-\frac{1}{x-3}

for

x\ne 3,-2

Example 39

hard

Combine

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

for

x\ne 0,-1,-2

Example 40

challenge

Simplify

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

in closed form.

Example 41

challenge

Given

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

, find

A

and

B

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Related Concepts

Simplifying Rational Expressions Least Common Multiple Solving Rational Equations Partial Fraction Decomposition

Background Knowledge

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

simplifying rational expressionsleast common multiple