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Wize High School Grade 12 Pre-Calculus Textbook > Inequalities

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.

Step 1.
It is already in factored form.

Step 2.
xx-intercepts: −4,2-4,2
Test points: −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}


Step 4.
The solution relies on the inequality in the question (x−2)(x+4)≥0(x-2)(x+4)\geq{}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).
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Example: Solving Polynomial Inequalities Using a Number Line

Solve x3+x2−14x−24≥0x^3+x^2-14x-24\geq{}0.

Step 1.
Factor using synthetic division.
Let 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}
The quotient becomes x2−x−12=(x−4)(x+3)x^2-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.

Step 2.
X-Intercepts:−3,−2,4-3,-2,4
Test points: −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} .


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

Practice: Solving Polynomial Inequalities Using a Number Line

Match the polynomial with its correct solution.
A.
4x2+19x+12≤04x^2+19x+12\leq{}0
B.
−6x2+x+1>0-6x^2+x+1>0
C.
−(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{}0
(−13,12)\Bigg(-\displaystyle\frac{1}{3},\displaystyle\frac{1}{2}\Bigg)
[−4,−34]\Bigg[-4,-\displaystyle\frac{3}{4}\Bigg]
[−53,0] ∪ [12,∞)\Bigg[-\displaystyle\frac{5}{3},0\Bigg]~\cup ~\Bigg[\displaystyle\frac{1}{2},\infin\Bigg)
(−1,2) ∪ (72,∞)(-1,2)~\cup~\Bigg(\displaystyle\frac{7}{2},\infin\Bigg)

Practice: Solving Polynomial Inequalities Using a Number Line

Solve x3−4x2+x+6<0x^3-4x^2+x+6<0.

Practice: Solving Polynomial Inequalities Using a Number Line

The graph of 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) , determine an inequality that describes f(x)f(x). (Express in factored form.)
Extra Practice
Solve Polynomial Inequalities Using a Number Line
Practice: Solving Polynomial Inequalities Using a Number Line
The graph of f(x) f(x)~ is shown below.
If the solution is (−∞,−5] ∪ [−4,1] ∪ [7,∞)(-\infin,-5]~\cup~[-4,1]~\cup~[7,\infin), determine an inequality that describes 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)~ is shown below.
If the solution is [−3,−1] ∪ [2,5][-3,-1]~\cup~[2,5] , determine an inequality that describes 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{}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{}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.