### About Solving Radical Equations Part 1:

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 a boundary or critical value. 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 on the number line

#1:

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

$$a)\hspace{.2em}\sqrt{-1 - 6x}> \sqrt{3x + 8}$$

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#2:

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

$$a)\hspace{.2em}\sqrt{3x + 22}+ 2 > 4$$

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#3:

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

$$a)\hspace{.2em}\sqrt{10 - x}≤ \sqrt{3x + 7}- 3$$

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#4:

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

$$a)\hspace{.2em}\sqrt{8 - x}≥ \sqrt{2x - 7}- 3$$

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#5:

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

$$a)\hspace{.2em}1 + \sqrt{6x - 2}≤ \sqrt{9x - 2}$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}-\frac{8}{3}< x < -1$$ $$\left(-\frac{8}{3}, -1 \right)$$

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#2:

Solutions:

$$a)\hspace{.2em}x > -6$$ $$(-6, \infty)$$

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#3:

Solutions:

$$a)\hspace{.2em}6 ≤ x ≤ 10$$ $$[6, 10]$$

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#4:

Solutions:

$$a)\hspace{.2em}\frac{7}{2}≤ x ≤ 8$$ $$\left[\frac{7}{2}, 8\right]$$

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#5:

Solutions:

$$a)\hspace{.2em}x=\frac{1}{3}\hspace{.2em}or \hspace{.2em}x ≥ 3$$ $$\left[\frac{1}{3}, \frac{1}{3}\right] ∪ [3, \infty)$$