Adding and subtracting rational expressions

About this lesson

Precalculus › Polynomial and rational functions

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Adding and subtracting rational expressions

Warm-up

Three quick questions from earlier lessons.

Discover

What is 1/2 + 1/3?

  1. 5/6
  2. 2/5

Key idea

Add rational expressions over a common denominator.

Factor the denominators, find the least common denominator, rewrite each fraction over it, then add or subtract the numerators only. Simplify at the end.

Worked example

Write 2/x + 3/(x + 1) as one fraction.

x(x + 1)
LCD.
2(x + 1)/(x(x + 1)) + 3x/(x(x + 1))
Rewrite.
(5x + 2)/(x(x + 1))
Add.

Your turn, guided

Your turn: the value of 1/x + 1/x at x = 4.

12\tfrac12
2/x at x = 4.

Practice ★

  1. Simplify 6/(x − 1) − 9/(x − 1).
  2. Simplify 9/(x − 1) − 2/(x − 1).
  1. What is the least common denominator of 1/(x² − 16) and 1/(x + 4)?

Watch out!

Practice ★★

  1. Write 1/x + 1/(x + 8) as a single fraction.
  2. Write 1/x + 1/(x + 1) as a single fraction.
  1. Simplify 6/(x − 1) − 9/(x − 1).

Exit ticket

Three questions to check you have got it.

  1. Write 1/x + 1/(x + 8) as a single fraction.
  2. Write 1/x + 1/(x + 1) as a single fraction.
  3. Write 1/x + 1/(x + 4) as a single fraction.

Summary card

Adding rational expressions

Common denominator, then add the numerators.

Watch out: never add the denominators.

End of the lesson

Next lesson: Rational functions: vertical and horizontal asymptotes Back to the chapter

Answers: Adding and subtracting rational expressions

A

  1. −3x−1- \frac{3}{x - 1}
  2. 7x−1\frac{7}{x - 1}

B

  1. x2−16x^{2} - 16

C

  1. 2x+8x(x+8)\frac{2 x + 8}{x \left(x + 8\right)}
  2. 2x+1x(x+1)\frac{2 x + 1}{x \left(x + 1\right)}

D

  1. −3x−1- \frac{3}{x - 1}

E

  1. 2x+8x(x+8)\frac{2 x + 8}{x \left(x + 8\right)}
  2. 2x+1x(x+1)\frac{2 x + 1}{x \left(x + 1\right)}
  3. 2x+4x(x+4)\frac{2 x + 4}{x \left(x + 4\right)}

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