# 2.3: Solution of Polynomial Inequalities by Graphing

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In this section, we will combine the concepts of the previous two sections to solve polynomial inequalities. In Section $$2.2,$$ we solved equations by graphing and finding the $$x$$ -values which made $$y=0 .$$ In solving an inequality, we will be concerned with finding the range of $$x$$ values that make $$y$$ either greater than or less than $$0,$$ depending on the given problem.
Example
Solve the given inequality.
$$2 x^{3}+8 x^{2}+5 x-3 \geq 0$$
First, we graph the function:

Then we identify the intervals of $$x$$ -values that make the $$y$$ value greater than or equal to zero, as indicated in the problem.

The indicated roots of the function $$(A, B \text { and } C)$$ are the $$x$$ -values that make $$y$$ equal to zero. These points divide the graph between the regions where $$y$$ is greater than zero and the regions where $$y$$ is less than zero. The solution to the given inequlaity $$2 x^{3}+8 x^{2}+5 x-3 \geq 0$$ are $$\mathrm{A} \leq x \leq \mathrm{B}$$ OR $$x \geq \mathrm{C}$$
When we find the values of $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}: \mathrm{A}=-3, \mathrm{B} \approx-1.366$$ and $$\mathrm{C} \approx 0.366,$$ we can
complete the solution to the problem.
$$2 x^{3}+8 x^{2}+5 x-3 \geq 0$$
$$-3 \leq x \leq-1.366$$ OR $$x \geq 0.366$$

Example
Solve the given inequality.
$$x^{4}-2 x^{3}-5 x^{2}+8 x+3 \leq 0$$
First, we graph the function:

In this problem, we're looking for the intervals of $$x$$ values that make $$y$$ less than or equal to zero. First, we identify the roots of the function:

Next, we'll identify the intervals where the $$y$$ values are less than zero:

So, the solution to the original inequality is:
$$x^{4}-2 x^{3}-5 x^{2}+8 x+3 \leq 0$$
$$-2.034 \leq x \leq-0.320$$ OR $$1.806 \leq x \leq 2.549$$
In the next example we'll be looking to identify both the intervals where $$y$$ is greater than zero, and the intervals where $$y$$ is less than zero.

Example
Determine the interval(s) for which $$x^{3}+5 x^{2}+5 x+1 \geq 0$$
Determine the interval(s) for which $$x^{3}+5 x^{2}+5 x+1<0$$
Once again, we'll start by graphing the function to find the roots:

Now that we've indentified the roots, we can determine where the $$y$$ values are greater than zero and where they're less than zero.

For $$y \geq 0,$$ we can see that this corresponds to: $$-3.732 \leq x \leq-1$$ OR $$x \geq-0.268$$
For $$y<0,$$ we can see that this corresponds to: $$x<-3.732 \mathrm{OR}-1<x<-0.268$$

Exercises 2.3
1) Determine the interval(s) for which $$x^{3}-4 x^{2}+2 x+3 \geq 0$$
Determine the interval(s) for which $$x^{3}-4 x^{2}+2 x+3<0$$
2) Determine the interval(s) for which $$4 x^{3}-4 x^{2}-19 x+10 \geq 0$$
Determine the interval(s) for which $$4 x^{3}-4 x^{2}-19 x+10<0$$
3) Determine the interval(s) for which $$x^{3}-2.5 x^{2}-7 x-1.5 \geq 0$$
Determine the interval(s) for which $$x^{3}-2.5 x^{2}-7 x-1.5<0$$
4) Determine the interval(s) for which $$x^{3}-3.5 x^{2}+0.5 x+5 \geq 0$$
Determine the interval(s) for which $$x^{3}-3.5 x^{2}+0.5 x+5<0$$
5) Determine the interval(s) for which $$6 x^{4}-13 x^{3}+2 x^{2}-4 x+15 \geq 0$$
Determine the interval(s) for which $$6 x^{4}-13 x^{3}+2 x^{2}-4 x+15<0$$
6) Determine the interval(s) for which $$x^{4}-x^{3}-x^{2}+3 x-5 \geq 0$$
Determine the interval(s) for which $$x^{4}-x^{3}-x^{2}+3 x-5<0$$
7) Determine the interval(s) for which $$3 x^{4}+3 x^{3}-14 x^{2}-x+3 \geq 0$$
Determine the interval(s) for which $$3 x^{4}+3 x^{3}-14 x^{2}-x+3<0$$
8) Determine the interval(s) for which $$4 x^{4}-4 x^{3}-7 x^{2}+4 x+3 \geq 0$$
Determine the interval(s) for which $$4 x^{4}-4 x^{3}-7 x^{2}+4 x+3<0$$

Determine the interval(s) that satisfy each inequality.
9) $$\quad x^{3}+x^{2}-5 x+3 \leq 0$$
10) $$\quad x^{3}-7 x+6>0$$
11) $$\quad x^{3}-13 x+12>0$$
12) $$\quad x^{4}-10 x^{2}+9<0$$
13) $$\quad 6 x^{4}-9 x^{2}-4 x+12 \geq 0$$
14) $$\quad x^{4}-5 x^{3}+20 x-16>0$$
15) $$\quad x^{3}-2 x^{2}-7 x+6 \leq 0$$
16) $$\quad x^{4}-6 x^{3}+2 x^{2}-5 x+2 \leq 0$$
17) $$\quad 2 x^{4}+3 x^{3}-2 x^{2}-4 x+2>0$$
18) $$\quad x^{5}+5 x^{4}-4 x^{3}+3 x^{2}-2 \leq 0$$

This page titled 2.3: Solution of Polynomial Inequalities by Graphing is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard W. Beveridge.