A **quadratic inequality** is a polynomial inequality with a degree (highest exponent) of 2.

Consider the graph of the quadratic function y=-x^{2}+4 and the solutions of the corresponding inequalities on the number lines below.

Solution to -x^{2}+4\leq 0

Solution to -x^{2}+4\geq 0

- What relationships do you notice between the graph of the function and the solutions of the corresponding inequalities?
- How can we use the graph to find the solutions -x^{2}+4 \lt c where c is any real number?

The **solution set of a quadratic inequality** are the values that make the inequality true. Similar to quadratic equations, we can visualize where these solution sets come from by considering the graph of the corresponding quadratic function.

\displaystyle x^{2}\gt4

\bm{y=x^{2}}

Corresponding function

Notice that the end points at -2 and 2 are unfilled. This is because those values do not satisfy the inequality: \left(-2\right)^{2}\ngtr 4 and \left(2\right)^{2} \ngtr 4.

If we change the inequality sign, the solution set will change as well. Consider the inequality x^{2}-4\leq0. Now, the blue region of the graph above will be the solution set because those are the values that are below zero.

This time, the end points are filled because the solutions do satisfy the inequality.

\displaystyle \left(-2\right)^{2}-4 \leq 0 \text{ and }\left(2\right)^{2}-4 | \displaystyle \leq | \displaystyle 0 |

\displaystyle 4-4\leq 0\text{ and }4-4 | \displaystyle \leq | \displaystyle 0 |

\displaystyle 0\leq0\text{ and }0 | \displaystyle \leq | \displaystyle 0 |

Solve the inequality {x^{2}}-2x-8\lt7 and graph the solution set on a number line.

Worked Solution

Graph the corresponding quadratic function and solve each of the inequalities.

a

x^{2} \lt 9

Worked Solution

b

-x^{2}+16 \leq 0

Worked Solution

c

0 \leq \left(x-4\right)\left(x-1\right)

Worked Solution

Write a corresponding quadratic inequality for the given solution set on the number line.

Worked Solution

A company that produces children's toys makes a total profit, P in hundreds of dollars, given by the function P\left(x\right)=-5x^{2}+80x-315, where x is the number of toys produced in hundreds.

a

Write an inequality whose solution represents when the company will make a profit.

Worked Solution

b

Use the table to determine when the company would make a profit.

x | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|

P\left(x\right) | -40 | -15 | 0 | 5 | 0 | -15 | -40 |

Worked Solution

Idea summary

The solutions to a quadratic inequality are any values that make the inequality true.

When using a graph to solve a quadratic inequality with a number on one side, we look for where the y-values are equal to that number. If the inequality symbol is \lt or \leq, the solution is where the graph is below that y-value. If the inequality is \gt or \geq, the solution is where the graph is above that y-value.