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Algebra II

Adding and Subtracting Rational Expressions

Lesson

Adding or subtracting rational expressions works like adding or subtracting numeric fractions: get a common denominator, then combine numerators.

Procedure:

  1. If the denominators are already the same, just add (or subtract) the numerators and keep the denominator.
  2. If different, find the least common denominator (LCD): factor each denominator, then take each distinct factor at its highest power.
  3. Rewrite each fraction with the LCD.
  4. Add or subtract the numerators.
  5. Simplify the result if possible.

Worked example 1 — same denominator

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

Worked example 2 — different denominators

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

LCD is x(x+1)x(x + 1). Rewrite each:

1⋅(x+1)x(x+1)+2xx(x+1)\frac{1 \cdot (x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)}
=x+1+2xx(x+1)=3x+1x(x+1)= \frac{x + 1 + 2x}{x(x + 1)} = \frac{3x + 1}{x(x + 1)}

How to type your answer

Write the result as a single fraction with parens around the numerator and denominator. Distribute the denominator’s factors as a product. Examples: (2x+3)/(x+1), (3x+1)/(x(x+1)), (x-2)/(x+5).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

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

Problem 2

5xx−2−3x−2\frac{5x}{x - 2} - \frac{3}{x - 2}

Problem 3

xx+4+4x+4\frac{x}{x + 4} + \frac{4}{x + 4}

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

4x−5−2x+3\frac{4}{x - 5} - \frac{2}{x + 3}

Problem 11

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

Problem 12

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

Problem 13

5x+23\frac{5}{x} + \frac{2}{3}

Problem 14

x2+1x\frac{x}{2} + \frac{1}{x}

Problem 15

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

Problem 16

xx+5+5x+5\frac{x}{x + 5} + \frac{5}{x + 5}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

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

Problem 18

2x+1−1x2−1\frac{2}{x + 1} - \frac{1}{x^2 - 1}

Problem 19

xx2−4+1x−2\frac{x}{x^2 - 4} + \frac{1}{x - 2}

Problem 20

3x−2x+1\frac{3}{x} - \frac{2}{x + 1}

Problem 21

1x−3+xx2−9\frac{1}{x - 3} + \frac{x}{x^2 - 9}

Problem 22

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

Practice

Standard problems matching the lesson.

Problem 23

Printer A: 1 page per x min. Printer B: 1 page per (x + 1) min. Combine 1/x + 1/(x + 1) as a single fraction.

Problem 24

Pipe rates 3/(x + 2) + 4/(x + 2). Single fraction?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Two workers: 1/(x − 1) + 2/(x + 1). Combine as a single fraction.

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