Wize High School Algebra II Textbook (Common Core) > Polynomial Functions

Solving Polynomial Inequalities (Number Line)

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Solving Polynomial Inequalities Using a Number Line
A polynomial inequality can be solved in similar ways the polynomial equations are solved. 

Step 1. 
Factor the polynomial. 

Step 2.
Identify the x-intercepts and your test points around the x-intercepts to determine where the polynomial exists.

Step 3.
Draw a table & a number line and use the test points around each x-intercept to see if the function is positive or negative. 

Step 4.
Express the answer in an appropriate format.

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Example
Let's look at the polynomial inequality (x−2)(x+4)≥0(x-2)(x+4)\geq{}0(x−2)(x+4)≥0. 

Step 1. 
It is already in factored form.

Step 2.
xxx-intercepts: −4,2-4,2−4,2
Test points: −5,0,3-5,0,3−5,0,3

Step 3.

−503(x+4)−++(x−2)−−+(x−2)(x+4)+−+\begin{array}{|c|c|c|c|}\hline
&-5&0&3\\\\\hline\\
(x+4)&-&+&+\\\\\hline\\
(x-2)&-&-&+\\\\\hline\\
(x-2)(x+4)&+&-&+\\\\\hline

\end{array}(x+4)(x−2)(x−2)(x+4)​−5−−+​0+−−​3+++​​    


Step 4.
The solution relies on the inequality in the question (x−2)(x+4)≥0(x-2)(x+4)\geq{}0(x−2)(x+4)≥0.

We are looking for the part of the polynomial that lies above 0 (or equals to). Therefore, we want our polynomial to be positive. 

The solution is (−∞,−4] ∪ [2,∞)(-\infin,-4]~\cup~[2,\infin)(−∞,−4] ∪ [2,∞).

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Example: Solving Polynomial Inequalities Using a Number Line
Solve x3+x2−14x−24≥0x^3+x^2-14x-24\geq{}0x3+x2−14x−24≥0. 

Step 1. 
Factor using synthetic division. 
Let x=−2.x=-2.x=−2.
11−14−24−2↓−22241−1−120\begin{array}{r|cccc}
&1&1&-14&-24\\
-2&\downarrow&-2&2&24\\\hline
&1&-1&-12&0
\end{array}−2​1↓1​1−2−1​−142−12​−24240​​
The quotient becomes x2−x−12=(x−4)(x+3)x^2-x-12=(x-4)(x+3)x2−x−12=(x−4)(x+3). 
Our polynomial inequality can be written as (x−4)(x+3)(x+2)≥0(x-4)(x+3)(x+2)\geq{}0(x−4)(x+3)(x+2)≥0. 

Step 2.
X-Intercepts:−3,−2,4-3,-2,4−3,−2,4
Test points: −4,−2.5,0,5-4,-2.5,0,5−4,−2.5,0,5

Step 3.

−4−2.505(x−4)−−−+(x+2)−−++(x+3)−+++(x3+x2−14x−24)−+−+\begin{array}{|c|c|c|c|c|}\hline
&-4&-2.5&0&5\\\hline\\
(x-4)&-&-&-&+\\\\\hline\\
(x+2)&-&-&+&+\\\\\hline\\
(x+3)&-&+&+&+\\\\\hline\\
(x^3+x^2-14x-24)&-&+&-&+\\\\\hline
\end{array}(x−4)(x+2)(x+3)(x3+x2−14x−24)​−4−−−−​−2.5−−++​0−++−​5++++​​    .     


Step 4.
The solution relies on the inequality in the question x3+x2−14x−24≥0x^3+x^2-14x-24\geq{}0x3+x2−14x−24≥0.

We are looking for the part of the polynomial that lies above 0 (or equals to). Therefore, we want our polynomial to be positive. 

The solution is [−3,−2] ∪ [4,∞)[-3,-2]~\cup~[4,\infin)[−3,−2] ∪ [4,∞).

Practice: Solving Polynomial Inequalities Using a Number Line
Match the polynomial with its correct solution.

A.

−6x2+x+1>0-6x^2+x+1>0−6x2+x+1>0  

B.

4x2+19x+12≤04x^2+19x+12\leq{}04x2+19x+12≤0   

C.

−(2x−7)(x−2)(x+1)<0-(2x-7)(x-2)(x+1)<0−(2x−7)(x−2)(x+1)<0  

D.

x(3x+5)(x−1)≥0x(3x+5)(x-1)\geq{}0x(3x+5)(x−1)≥0   

(−13,12)\Bigg(-\displaystyle\frac{1}{3},\displaystyle\frac{1}{2}\Bigg)(−31​,21​)  

[−4,−34]\Bigg[-4,-\displaystyle\frac{3}{4}\Bigg][−4,−43​] 

[−53,0] ∪ [12,∞)\Bigg[-\displaystyle\frac{5}{3},0\Bigg]~\cup
~\Bigg[\displaystyle\frac{1}{2},\infin\Bigg)[−35​,0] ∪ [21​,∞) 

(−1,2) ∪ (72,∞)(-1,2)~\cup~\Bigg(\displaystyle\frac{7}{2},\infin\Bigg)(−1,2) ∪ (27​,∞) 

I don't know

Practice: Solving Polynomial Inequalities Using a Number Line
Solve x3−4x2+x+6<0x^3-4x^2+x+6<0x3−4x2+x+6<0. 

A) (−∞,−1) ∪ (2,3)(-\infin,-1)~\cup~(2,3)(−∞,−1) ∪ (2,3) 

B) (−∞,−1] ∪ [2,3](-\infin,-1]~\cup~[2,3](−∞,−1] ∪ [2,3] 

C) (−1,2) ∪ (3,∞)(-1,2)~\cup~(3,\infin)(−1,2) ∪ (3,∞) 

D) [−1,2] ∪ [3,∞)[-1,2]~\cup~[3,\infin)[−1,2] ∪ [3,∞)

I don't know

Practice: Solving Polynomial Inequalities Using a Number Line
The graph of f(x) f(x)~f(x)  is shown below.

If the solution is (−52,−34) ∪ (12, 52)\Bigg(-\displaystyle\frac{5}{2},-\displaystyle\frac{3}{4}\Bigg)~\cup~\Bigg(\displaystyle\frac{1}{2},~\displaystyle\frac{5}{2}\Bigg)
(−25​,−43​) ∪ (21​, 25​), determine an inequality that describes f(x)f(x)f(x). (Express in factored form.)

f(x)=

I don't know

Extra Practice

Solve Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
The graph of f(x) f(x)~f(x)  is shown below.
If the solution is (−∞,−5] ∪ [−4,1] ∪ [7,∞)(-\infin,-5]~\cup~[-4,1]~\cup~[7,\infin)(−∞,−5] ∪ [−4,1] ∪ [7,∞), determine an inequality that describes f(x)f(x)f(x). (Express in factored form.)

Solve Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
The graph of f(x) f(x)~f(x)  is shown below.
If the solution is [−3,−1] ∪ [2,5][-3,-1]~\cup~[2,5]
[−3,−1] ∪ [2,5], determine an inequality that describes f(x)f(x)f(x). (Express in factored form.)

Solve Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
Solve x3+2x2−x−2≤0x^3+2x^2-x-2\leq{}0x3+2x2−x−2≤0. 

Solve Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
Solve 3x3−5x2−58x+40≥03x^3-5x^2-58x+40\geq{}03x3−5x2−58x+40≥0. 

Solving Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
Match the polynomial with its correct solution.

Solving Polynomial Inequalities Using a Number Line

Practice: Solving Polynomial Inequalities Using a Number Line
Match the polynomial with its correct solution.