College Algebra Lecture Notes Section 3.7Page 1 of 4

Section 3.7: Polynomial and Rational Inequalities

Big Idea:Nonlinear inequalities are solved by getting all terms on one side, factoring the polynomial or rational expression on the one side, and then identifying the intervals between the zeroes or vertical asymptotes that have output with the sign demanded by the inequality.

Big Skill:You should be able to solve nonlinear inequalities using this technique.

A. Quadratic Inequalities

To solve a quadratic inequality:

  • Get all terms on the left hand side.
  • Graph the quadratic function you obtain from that left hand side.
  • Factor the left hand side so you know the exact location of the zeroes, and thus where the y-values change sign.
  • State the solution as the interval(s) that have the desired y-values (positive or negative), as demanded by the inequality.

Practice:

  1. Solve the inequality.
  1. Solve the inequality.

B. Polynomial Inequalities

Solving Polynomial Inequalities

Given f(x) is a polynomial in standard form,

  • Write f in factored form.
  • Plot real zeroes on the x-axis, noting their multiplicity.
  • If the multiplicity is odd, the function will change sign.
  • If the multiplicity is even, the function will not change sign.
  • Use the end behavior to determine the sign of f in the outermost intervals, then label the other intervals as f > 0 or f < 0 by analyzing the multiplicity of neighboring zeroes.
  • State the solution in interval notation.

Practice:

  1. Solve the inequality.
  1. Solve the inequality.

C. Rational Inequalities

To solve a rational inequality:

  • Get all terms on the left hand side.
  • Combine all terms into one single rational expression.
  • Factor the rational expression so you know the exact location of the zeroes and vertical asymptotes, and thus where the y-values change sign.
  • State the solution as the interval(s) that have the desired y-values (positive or negative), as demanded by the inequality.

Practice:

  1. Solve the inequality.
  1. Solve the inequality.

D. Solving Function Inequalities Using Interval Tests

  • Get all terms on the left hand side.
  • Combine all terms into one single polynomial or rational expression.
  • Factor to find the exact location of the zeroes and vertical asymptotes, and thus where the y-values change sign.
  • Pick a test value for x in each interval; if the output of the left hand side satisfies the inequality, then all x values in that interval will satisfy the inequality.
  • State the solution as the interval(s) that satisfy the inequality.

Practice:

  1. Solve the inequality.

E. Applications of Inequalities