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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.
2x3+8x2+5x30
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 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 2x3+8x2+5x30 are AxB OR xC
When we find the values of A,B and C:A=3,B1.366 and C0.366, we can
complete the solution to the problem.
2x3+8x2+5x30
3x1.366 OR x0.366

Example
Solve the given inequality.
x42x35x2+8x+30
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:
x42x35x2+8x+30
2.034x0.320 OR 1.806x2.549
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 x3+5x2+5x+10
Determine the interval(s) for which x3+5x2+5x+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 y0, we can see that this corresponds to: 3.732x1 OR x0.268
For y<0, we can see that this corresponds to: x<3.732OR1<x<0.268

Exercises 2.3
1) Determine the interval(s) for which x34x2+2x+30
Determine the interval(s) for which x34x2+2x+3<0
2) Determine the interval(s) for which 4x34x219x+100
Determine the interval(s) for which 4x34x219x+10<0
3) Determine the interval(s) for which x32.5x27x1.50
Determine the interval(s) for which x32.5x27x1.5<0
4) Determine the interval(s) for which x33.5x2+0.5x+50
Determine the interval(s) for which x33.5x2+0.5x+5<0
5) Determine the interval(s) for which 6x413x3+2x24x+150
Determine the interval(s) for which 6x413x3+2x24x+15<0
6) Determine the interval(s) for which x4x3x2+3x50
Determine the interval(s) for which x4x3x2+3x5<0
7) Determine the interval(s) for which 3x4+3x314x2x+30
Determine the interval(s) for which 3x4+3x314x2x+3<0
8) Determine the interval(s) for which 4x44x37x2+4x+30
Determine the interval(s) for which 4x44x37x2+4x+3<0

Determine the interval(s) that satisfy each inequality.
9) x3+x25x+30
10) x37x+6>0
11) x313x+12>0
12) x410x2+9<0
13) 6x49x24x+120
14) x45x3+20x16>0
15) x32x27x+60
16) x46x3+2x25x+20
17) 2x4+3x32x24x+2>0
18) x5+5x44x3+3x220


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.

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