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.