### About Solving Radical Inequalities Part 2:

In order to solve a radical inequality, we will first consider any domain restrictions. Then we will convert our inequality into an equality and solve the resulting equation. The solution will give us the boundaries or critical values. From there, we will test values on each side of the boundary. We must restrict any value from our solution that violates the domain.

Test Objectives

- Demonstrate the ability to solve a radical equation
- Demonstrate the ability to solve a radical inequality
- Demonstrate the ability to write an inequality solution in interval notation
- Demonstrate the ability to graph an interval

#1:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}\large{\sqrt[4]{3x - 1}< 2}$$

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#2:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}\large{x - 2 ≤ \sqrt{2x + 20}}$$

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#3:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}\large{\sqrt{10 - x}> x - 8}$$

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#4:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}\large{\sqrt{x^2 - 8x - 33}≤ x + 10}$$

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#5:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}\sqrt[3]{3x + 2}≥ 3x - 4$$

Note:

$$9x^3 - 36x^2 + 47x - 22=(x - 2)(9x^2 - 18x + 11)$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\frac{1}{3}≤ x < \frac{17}{3}$$ $$\left[\frac{1}{3}, \frac{17}{3}\right)$$

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#2:

Solutions:

$$a)\hspace{.2em}{-}10 ≤ x ≤ 8$$ $$[-10, 8]$$

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#3:

Solutions:

$$a)\hspace{.2em}x < 9$$ $$(-\infty, 9)$$

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#4:

Solutions:

$$a)\hspace{.2em}{-}\frac{19}{4}≤ x ≤ -3 \hspace{.1em}\text{or}\hspace{.1em}x ≥ 11$$ $$\left[-\frac{19}{4}, -3\right] ∪ [11, ∞)$$

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#5:

Solutions:

$$a)\hspace{.2em}x ≤ 2$$ $$(-∞, 2]$$