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:
$clipboard_e3f8e47f913fd0e34106737c8e2343cf8.png$
Then we identify the intervals of $x$ -values that make the $y$ value greater than or equal to zero, as indicated in the problem.
$clipboard_e6b2909ccb26526cd094be44a912eed0f.png$
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:
$clipboard_e0d1f896a2f4849e3bffdbc8dfefdf341.png$
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:
$clipboard_e1ec85eef24e411dee8323fc1cd74e19d.png$
Next, we'll identify the intervals where the $y$ values are less than zero:
$clipboard_e23c9b391486aa6db2497514cd64fa7d8.png$
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:
$clipboard_e83e7f20222ec2c78613c50fea074f1cd.png$

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) $\quad x^{5}+5 x^{4}-4 x^{3}+3 x^{2}-2 \leq 0$

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