Virginia SOL Math / Algebra II / Equations & Inequalities

A2.EI.4

Equations With Rational Expressions

What you'll learn

  • Solve equations containing rational expressions
  • Multiply both sides by the least common denominator
  • Check for and reject extraneous solutions

Words to know

Rational equation —
an equation with a variable in a denominator
Least common denominator —
the smallest common multiple of the denominators
Clear the denominators —
multiply through to remove fractions
Extraneous solution —
a value that does not satisfy the original equation
Excluded value —
a value that makes a denominator 0

Why it matters Rates, mixtures, and shared-work problems lead to rational equations, so solving them answers questions like how long a job takes together.

Worked examples

Model ItI do

Solve 1/x + 1/2 = 3/4.

The least common denominator is 4x. Multiply every term by 4x: 4 + 2x = 3x. Solve: 4 = x, so x = 4. Since x = 4 does not make any denominator 0, it is valid.

Try It TogetherWe do

Solve 3/x = 6.

Multiply both sides by x: 3 = 6x. Divide by 6: x = 1/2. Since 1/2 does not make the denominator 0, it is valid.

Now YouYou do

Solve 2/x + 1 = 3.

Multiply through by x to clear the fraction, then solve. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Solve 6/x = 12.

  2. 2.Solve 1/x = 1/9.

  3. 3.What value must be excluded in 5/(x + 4) = 2?

On Level

The core Quick Check from the book.

  1. 1.Solve 5/x = 10.

  2. 2.Solve 1/x = 1/4.

  3. 3.Solve 2/x = 8.

  4. 4.What value must be excluded in 4/(x - 2) = 1?

  5. 5.Solve x/3 = 6.

Challenge

Stretch your thinking.

  1. 1.Solve 1/(x - 2) + 1/(x + 2) = 4/(x^2 - 4).

  2. 2.Solve x/(x - 3) = 6/(x - 3) + 2.

  3. 3.Solve 2/x + 3 = 5/x.