About Solving Polynomial Inequalities:
In this section, we will learn how to solve polynomial inequalities. This process will involve finding the critical values in order to set up intervals on the number line. Once this is done, we will test inside of each interval or use a sign chart to find the regions where the function is above or below the x-axis. We will use this information along with our inequality symbol to write our solution.
Test Objectives
- Demonstrate the ability to find the zeros of a polynomial function
- Demonstrate the ability to solve a polynomial inequality
#1:
Instructions: Solve each inequality.
$$a)\hspace{.2em}$$ $$(-x - 8)(3x - 4)(2x - 7) > 0$$
$$b)\hspace{.2em}$$ $$(x - 9)(x - 3)(x - 7)^3 > 0$$
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#2:
Instructions: Solve each inequality.
$$a)\hspace{.2em}$$ $$(x - 1)(x + 1)^3 ≤ 0$$
$$b)\hspace{.2em}$$ $$(3x - 2)(-3x + 2)(x - 3)^2 > 0$$
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#3:
Instructions: Solve each inequality.
$$a)\hspace{.2em}$$ $$\frac{1}{100}(2x + 5)(3x - 8)(x - 5)(-x - 9) ≥ 0$$
$$b)\hspace{.2em}$$ $$-\frac{1}{30}(2x - 7)(3x - 1)^2(x + 3)^2 < 0$$
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#4:
Instructions: Solve each inequality.
$$a)\hspace{.2em}$$ $$(x + 8)^2(x + 6)^2(3x + 11) ≥ 0$$
$$b)\hspace{.2em}$$ $$(2x - 9)(x - 4)^2(5x^2 + 3x + 20) < 0$$
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#5:
Instructions: Solve each inequality.
$$a)\hspace{.2em}$$ $$(x + 4)(x - 1)(-2x^2 - 5x - 9) ≤ 0$$
$$b)\hspace{.2em}$$ $$(3x^2 + x + 2)(-2x^2 - x - 1) ≤ 0$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$x < -8$$ $$\text{or}$$ $$\frac{4}{3} < x < \frac{7}{2}$$ Interval Notation: $$(-∞, -8) ∪ \left(\frac{4}{3}, \frac{7}{2}\right)$$ Graphing the Interval on the Number Line: 
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$$b)\hspace{.2em}$$ $$3 < x < 7$$ $$\text{or}$$ $$x > 9$$ Interval Notation: $$(3, 7) ∪ (9, ∞)$$ Graphing the Interval on the Number Line: 
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$-1 ≤ x ≤ 1$$ Interval Notation: $$[-1, 1]$$ Graphing the Interval on the Number Line: 
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$$b)\hspace{.2em}$$ $$\text{No Solution}$$ Demos Link for More Detail
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#3:
Solutions:
$$a)\hspace{.2em}$$ $$-9 ≤ x ≤ -\frac{5}{2}$$ $$\text{or}$$ $$\frac{8}{3} ≤ x ≤ 5$$ Interval Notation: $$\left[-9, -\frac{5}{2}\right] ∪ \left[\frac{8}{3}, 5\right]$$ Graphing the Interval on the Number Line: 
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$$b)\hspace{.2em}$$ $$x > \frac{7}{2}$$ Interval Notation: $$\left(\frac{7}{2}, ∞\right)$$ Graphing the Interval on the Number Line: 
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#4:
Solutions:
$$a)\hspace{.2em}$$ $$x = -8, -6$$ $$\text{or}$$ $$x ≥ -\frac{11}{3}$$ Interval Notation: $$[-8, -8] ∪ [-6, -6] ∪ \left[-\frac{11}{3}, ∞\right)$$ Graphing the Interval on the Number Line: 
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$$b)\hspace{.2em}$$ $$x < 4$$ $$\text{or}$$ $$4 < x < \frac{9}{2}$$ Interval Notation: $$(-∞, 4) ∪ \left(4, \frac{9}{2}\right)$$ Graphing the Interval on the Number Line: 
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#5:
Solutions:
$$a)\hspace{.2em}$$ $$x ≤ -4 \, \text{or} \, x ≥ 1$$ Interval Notation: $$(-∞, -4] ∪ [1, ∞)$$ Graphing the Interval on the Number Line: 
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$$b)\hspace{.2em}$$ All real numbers
Interval Notation: $$(-∞, ∞)$$ Graphing the Interval on the Number Line: 
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