# Higher Order Inequalities

*x*

^{2}+ 4 < 2 and

*x*

^{3}â€“ 9 > 0. The number line is often helpful in solving these types of inequalities.

^{2}> â€“ 6x â€“ 5 ?

First, replace the inequality symbol with an equal symbol: | x^{2} = â€“ 6x â€“ 5 |

Adding 6x and 5 to both sides yields |
x^{2} + 6x + 5 = 0 |

Factoring yields (see General Trinomials in the chapter Factoring) | (x + 5)(x + 1) = 0 |

Setting each factor to 0 yields | x + 5 =0 and x + 1 =0 |

Or | x = â€“5 and x = â€“1 |

When *x* = â€“6, *x*^{2} > â€“ 6*x* â€“ 5 becomes 36 > 31. This is true.

Hence, all numbers in Interval I satisfy the inequality. That is, *x* < â€“5. When *x* = â€“3, *x*^{2} > â€“ 6*x* â€“ 5 becomes 9 > 13. This is false.

Hence, no numbers in Interval II satisfy the inequality. When *x* = 0, *x*^{2} > â€“ 6*x* â€“ 5 becomes 0 > â€“5. This is true.

Hence, all numbers in Interval III satisfy the inequality. That is, *x* > â€“1. The graph of the solution follows:

**Note, **if the original inequality had included the greater-than-or-equal symbol, __>__, the solution set would have included both â€“5 and â€“1. On the graph, this would have been indicated by filling in the circles above â€“5 and â€“1. The open circles indicate that â€“5 and â€“1 are not part of the solution.

**Summary of steps for solving higher order inequalities:**

- Replace the inequality symbol with an equal symbol.
- Move all terms to one side of the equation (usually the left side).
- Factor the equation.
- Set the factors equal to 0 to find zeros.
- Choose test points on either side of the zeros.
- If a test point satisfies the original inequality, then all numbers in that interval satisfy the inequality. Similarly, if a test point does not satisfy the inequality, then no numbers in that interval satisfy the inequality.