Algebra 2 Review Worksheet with 20 Questions

Problem Analysis and Solution

The image contains a set of algebra problems. Below, I will solve each problem step by step.

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#### Problem 1: Graph the equation \( y = |x + 4| - 2 \)

The given equation is \( y = |x + 4| - 2 \). This is an absolute value function shifted horizontally and vertically.

1. Identify the vertex: The general form of an absolute value function is \( y = |x - h| + k \), where \((h, k)\) is the vertex. Here, the equation can be rewritten as \( y = |x - (-4)| - 2 \), so the vertex is at \((-4, -2)\).
2. Graph the function:
- For \( x \geq -4 \), \( y = (x + 4) - 2 = x + 2 \).
- For \( x < -4 \), \( y = -(x + 4) - 2 = -x - 6 \).
- The graph will be a V-shaped curve with the vertex at \((-4, -2)\).

From the options provided, the correct graph is:

- Correct Answer: B

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#### Problem 2: Solve the equation \( |3x - 4| = 8 \)

The absolute value equation \( |3x - 4| = 8 \) means that \( 3x - 4 \) can be either 8 or -8.

1. Case 1: \( 3x - 4 = 8 \)
\[
3x - 4 = 8 \implies 3x = 12 \implies x = 4
\]

2. Case 2: \( 3x - 4 = -8 \)
\[
3x - 4 = -8 \implies 3x = -4 \implies x = -\frac{4}{3}
\]

So, the solutions are \( x = 4 \) and \( x = -\frac{4}{3} \).

- Correct Answer: D

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#### Problem 3: Simplify the expression \( \sqrt{75a^5b^7} \)

To simplify \( \sqrt{75a^5b^7} \):

1. Factorize the radicand:
\[
75 = 25 \cdot 3 = 5^2 \cdot 3
\]
\[
a^5 = a^4 \cdot a = (a^2)^2 \cdot a
\]
\[
b^7 = b^6 \cdot b = (b^3)^2 \cdot b
\]

2. Take the square root:
\[
\sqrt{75a^5b^7} = \sqrt{5^2 \cdot 3 \cdot (a^2)^2 \cdot a \cdot (b^3)^2 \cdot b}
\]
\[
= \sqrt{5^2} \cdot \sqrt{(a^2)^2} \cdot \sqrt{(b^3)^2} \cdot \sqrt{3ab}
\]
\[
= 5 \cdot a^2 \cdot b^3 \cdot \sqrt{3ab}
\]

So, the simplified expression is \( 5a^2b^3\sqrt{3ab} \).

- Correct Answer: C

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#### Problem 4: Solve the equation \( \sqrt{x} = \frac{1}{6} \)

Given \( \sqrt{x} = \frac{1}{6} \):

1. Square both sides to eliminate the square root:
\[
(\sqrt{x})^2 = \left(\frac{1}{6}\right)^2
\]
\[
x = \frac{1}{36}
\]

So, the solution is \( x = \frac{1}{36} \).

- Correct Answer: A

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#### Problem 5: Solve the proportion \( \frac{n + 11}{m - 7} = \frac{15}{11} \)

Given the proportion \( \frac{n + 11}{m - 7} = \frac{15}{11} \):

1. Cross-multiply:
\[
(n + 11) \cdot 11 = (m - 7) \cdot 15
\]
\[
11(n + 11) = 15(m - 7)
\]
\[
11n + 121 = 15m - 105
\]

2. Rearrange to isolate terms involving \( n \) and \( m \):
\[
11n = 15m - 105 - 121
\]
\[
11n = 15m - 226
\]
\[
n = \frac{15m - 226}{11}
\]

So, the solution is \( n = \frac{15m - 226}{11} \).

- Correct Answer: C

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Final Answers:

1. B
2. D
3. C
4. A
5. C

\[
\boxed{B, D, C, A, C}
\]