2.7 | Nonlinear Inequalities


Strategy for Solving Polynomial Inequalities:

  • Move everything to one side so that you have zero on the other side.
  • Let the non-zero side be \(p(x)\).
  • Graph \(p(x)\) and include the \(x\)-intercepts.
  • The solution set for \(p(x) \geq 0\) comprise of the intervals along the \(x\)-axis for which the graph of \(p(x)\) is above or on the \(x\)-axis. Similarly, the solution set for \(p(x) \leq 0\) comprise of the intervals along the \(x\)-axis for which the graph of \(p(x)\) is below or on the \(x\)-axis. \(p(x) < 0\), \(p(x) > 0\), and \(p(x) =0\) are defined in an analogous way.
Strategy for Solving Rational Inequalities:

  • Move everything to one side so that you have zero on the other side.
  • Let the non-zero side be \(r(x)\).
  • Graph \(r(x)\) and include the \(x\)-intercepts and asymptotes
  • The solution set for \(r(x) \geq 0\) comprise of the intervals along the \(x\)-axis for which the graph of \(r(x)\) is above or on the \(x\)-axis. Similarly, the solution set for \(r(x) \leq 0\) comprise of the intervals along the \(x\)-axis for which the graph of \(r(x)\) is below or on the \(x\)-axis. \(r(x) < 0\), \(r(x) > 0\), and \(r(x) =0\) are defined in an analogous way.

E 2.7 Exercises