Practice Adding and Subtracting Rational Expressions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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 13+14\frac{1}{3} + \frac{1}{4} requires a common denominator of 12, adding 2x+1+3xโˆ’2\frac{2}{x+1} + \frac{3}{x-2} requires the LCD (x+1)(xโˆ’2)(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.

Showing a random 20 of 50 problems.

Example 1

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Add 5x+3xโˆ’2\frac{5}{x} + \frac{3}{x - 2}, xโ‰ 0,2x\neq0,2.

Example 2

hard
Combine xx2โˆ’9+1xโˆ’3โˆ’1x+3\frac{x}{x^2-9}+\frac{1}{x-3}-\frac{1}{x+3} for xโ‰ ยฑ3x\ne\pm 3.

Example 3

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Add 2x+1+3xโˆ’2\frac{2}{x + 1} + \frac{3}{x - 2}, xโ‰ โˆ’1,2x\neq-1,2.

Example 4

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Subtract x+1xโˆ’xโˆ’1x+1\frac{x + 1}{x} - \frac{x - 1}{x + 1}, xโ‰ 0,โˆ’1x\neq0,-1.

Example 5

challenge
Subtract xx2โˆ’xโˆ’6โˆ’2xโˆ’3\frac{x}{x^2 - x - 6} - \frac{2}{x - 3}, xโ‰ 3,โˆ’2x\neq3,-2.

Example 6

challenge
Add 3xโˆ’2+2x+2+1x2โˆ’4\frac{3}{x - 2} + \frac{2}{x + 2} + \frac{1}{x^2 - 4}, xโ‰ ยฑ2x\neq\pm2.

Example 7

medium
For what value(s) of xx is 1xโˆ’3+1x+3\frac{1}{x-3}+\frac{1}{x+3} undefined?

Example 8

easy
Add 3x+2x2\frac{3}{x} + \frac{2}{x^2}, xโ‰ 0x\neq0.

Example 9

hard
Find the value of 1xโˆ’2+1x+2\frac{1}{x-2}+\frac{1}{x+2} at x=3x=3.

Example 10

medium
Subtract 1x+2โˆ’1xโˆ’2\frac{1}{x+2}-\frac{1}{x-2} for xโ‰ ยฑ2x\ne\pm 2.

Example 11

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Add 2xโˆ’1+3x+2\frac{2}{x-1}+\frac{3}{x+2} for xโ‰ 1,โˆ’2x\ne 1,-2.

Example 12

easy
Simplify 2xx+1โˆ’xโˆ’1x+1\frac{2x}{x+1}-\frac{x-1}{x+1} for xโ‰ โˆ’1x\ne -1.

Example 13

hard
Combine 1x+1x+1+1x+2\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2} for xโ‰ 0,โˆ’1,โˆ’2x\ne 0,-1,-2.

Example 14

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Add 1x+2x2+3x3\frac{1}{x} + \frac{2}{x^2} + \frac{3}{x^3}, xโ‰ 0x\neq0.

Example 15

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Combine 2x2โˆ’3x3\frac{2}{x^2}-\frac{3}{x^3} for xโ‰ 0x\ne 0.

Example 16

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Add 2x+1+3xโˆ’2\frac{2}{x+1} + \frac{3}{x-2}.

Example 17

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Worked example: combine 3x2โˆ’4+1xโˆ’2\frac{3}{x^2-4}+\frac{1}{x-2} for xโ‰ ยฑ2x\ne\pm 2.

Example 18

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Subtract xxโˆ’3โˆ’3xโˆ’3\frac{x}{x - 3} - \frac{3}{x - 3} and simplify, xโ‰ 3x\neq3.

Example 19

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Subtract 5xโˆ’3โˆ’2x+1\frac{5}{x-3} - \frac{2}{x+1}.

Example 20

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Add xxโˆ’2+2x+2\frac{x}{x-2}+\frac{2}{x+2} for xโ‰ ยฑ2x\ne\pm 2.
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