Rational expressions and equations

College Algebra. Also known as: math 101, math 1314, math 1111, math 110, mat 140, intermediate algebra, precalculus i, algebra for college students, math 1010, college algebra and trigonometry

Simplify, combine and solve fractions with variables, and find asymptotes and holes of rational functions.

Rational expressions

A rational expression is a fraction whose top and bottom are polynomials, such as (x² − 9)/(x² + x − 6). Treat it like a numeric fraction: factor first, then cancel common factors (never common terms).

Simplifying

(x² − 9)/(x² + x − 6) = (x − 3)(x + 3) / ((x + 3)(x − 2)) = (x − 3)/(x − 2).

The cancelled factor x + 3 was zero at x = −3, so the original expression is still undefined there. State the restrictions: x ≠ −3 and x ≠ 2.

Multiply, divide, add

  • Multiply: factor, cancel, multiply tops and bottoms. (2/x) · (x²/6) = 2x²/(6x) = x/3.
  • Divide: multiply by the reciprocal.
  • Add or subtract: use a common denominator. 1/x + 1/3 = 3/(3x) + x/(3x) = (x + 3)/(3x).

Solving rational equations

  1. Find the restrictions (values that make a denominator zero).
  2. Multiply both sides by the common denominator to clear fractions.
  3. Solve the resulting equation.
  4. Reject any answer that breaks a restriction. These are extraneous solutions.

Worked example

Solve 2/(x − 1) = 3/(x + 2).

  • Restrictions: x ≠ 1 and x ≠ −2.
  • Cross-multiply: 2(x + 2) = 3(x − 1).
  • Expand: 2x + 4 = 3x − 3.
  • Solve: 7 = x.
  • Check: left 2/6 = 1/3; right 3/9 = 1/3. ✓.

A trap: x/(x − 2) = 2/(x − 2). Multiplying by x − 2 gives x = 2, but 2 is a restricted value, so there is no solution.

Asymptotes and holes

For a rational function f(x) = p(x)/q(x) in lowest terms:

  • Vertical asymptotes occur where q(x) = 0 after cancelling. For (x + 1)/(x − 4), x = 4.
  • A factor that cancels leaves a hole, not an asymptote. (x − 3)(x + 2)/(x − 3) has a hole at x = 3.
  • Horizontal asymptote: compare degrees. If the bottom has the larger degree, y = 0. If the degrees are equal, y = ratio of leading coefficients (for (3x² + 1)/(x² − 5), y = 3). If the top has larger degree, there is no horizontal asymptote.

Work problems

If one pipe fills a tank in 4 hours and another in 12, their rates add: 1/4 + 1/12 = 4/12 = 1/3 of the tank per hour, so together they take 3 hours.

Common mistakes

  • Cancelling terms such as the x in (x + 2)/x.
  • Forgetting to check answers against restrictions.
  • Adding fractions by adding denominators.

Practice questions

  1. What is the vertical asymptote of f(x) = (x + 1)/(x − 4)?

  2. What is the horizontal asymptote of f(x) = (3x² + 1)/(x² − 5)?

  3. What is the horizontal asymptote of f(x) = (2x + 1)/(x² + 4)?

  4. Simplify (x² − 4)/(x − 2), for x ≠ 2.

  5. Solve 6/x = 3.

  6. Solve x/(x − 2) = 2/(x − 2).