# Inequalities with Quadratic Functions

These are my notes on inequalities with quadratic functions.

Ankr Store on Amazon, keep your electronics charged by the best! If you buy something, I get a small commission and that makes it easier to keep on writing. Thank you in advance if you buy something.

In this section, we solve inequalities that involve quadratic functions. We will
accomplish this by using their graphs. For example, to solve the inequality
$$ax^2+bx+c>0$$
we graph the function $$f(x)=ax^2+bx+c$$ and from the graph, determine where it
is above the x-axis and where f(x)>0. To solve the inequality, we graph the
function and determine where the graph is below the x-axis. If the inequality is
not strict, we include the x-intercepts in the solution.

Example 1
Solving an Inequality
Solve the inequality $$x^2-4x-12\leq0$$ and graph the solution set.

Solution:
We graph the function $$f(x)=x^2-4x-12$$
The intercepts are: x=6 and x=-2
the y-intercept is -12 with the x-intercepts at 6 and -2.
The vertex is at:
$$x=-\frac{b}{2a}=-\frac{-4}{2}=2$$
Since f(2)=-16, the vertex is (2,-16).
The graph is below the x-axis for -2<x<6. Since the original inequality is not
strict, we include the x-intercepts. The solution set is $${x|-2\leqx\leq6} or [-2,6]. Example 2 Solving an Inequality Solve the inequality \(2x^2<x+10$$ and graph the solution set.

Solution:
We arrange the inequality so that 0 is on the right side.
$$2x^2<x+10$$
$$2x^2-x-10<0$$
This inequality is equivalent to the one that we wish to solve.
Next, we graph the function $$f(x)=2x^2-x-10$$. The intercepts are:
y=-10, x=-2, x=5/2
The vertex is at $$x=-\frac{b}{2a}=-\frac{-1}{4}=\frac{1}{4}$$
Since f(1/4)=-10.125, the vertex is at $$(\frac{1}{4},-10.125)$$
The graph is below the x-axis between x=-2 and x=5/2. Since the inequality is
strict, the solution set is $${x|-2<x<\frac{5}{2}}$$ or $$(-2,\frac{5}{2})$$.

This article was updated on August 12, 2024