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Let's address the problems in the image step by step.

### Problem 4b: Solve the compound inequality
\[ 2(x - 2) + 7 \geq -1 \text{ and } 5 - 4x > 9 \]

#### Solving the first inequality:
\[ 2(x - 2) + 7 \geq -1 \]
1. Distribute the 2:
   \[ 2x - 4 + 7 \geq -1 \]
2. Combine like terms:
   \[ 2x + 3 \geq -1 \]
3. Subtract 3 from both sides:
   \[ 2x \geq -4 \]
4. Divide by 2:
   \[ x \geq -2 \]

#### Solving the second inequality:
\[ 5 - 4x > 9 \]
1. Subtract 5 from both sides:
   \[ -4x > 4 \]
2. Divide by -4 (remember to flip the inequality sign):
   \[ x < -1 \]

#### Compound solution:
We have:
\[ x \geq -2 \text{ and } x < -1 \]

The solution is:
\[ -2 \leq x < -1 \]

### Problem 5: List the transformations for the given functions
The base function is \( f(x) = |x| \). For each transformed function, we need to list the transformations.

#### 5a. \( g(x) = 0.5|x - 2| + 3 \)
1. **Horizontal shift**: The \( x - 2 \) inside the absolute value shifts the graph **2 units to the right**.
2. **Vertical stretch/compression**: The coefficient \( 0.5 \) compresses the graph vertically by a factor of 0.5 (half as steep).
3. **Vertical shift**: The \( +3 \) outside the absolute value shifts the graph **3 units upward**.

#### 5b. \( g(x) = -3|x| - 4 \)
1. **Reflection**: The negative sign reflects the graph **across the x-axis**.
2. **Vertical stretch**: The coefficient \( 3 \) stretches the graph vertically by a factor of 3.
3. **Vertical shift**: The \( -4 \) shifts the graph **4 units downward**.

Would you like any further explanation on these problems?

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Here are 5 related questions for practice:
1. Solve the compound inequality: \( 3x + 5 > 2 \text{ and } 2x - 3 \leq 7 \).
2. What are the transformations of \( g(x) = -2|x + 1| - 5 \)?
3. Solve the inequality: \( 4(x + 3) \leq 12 - x \).
4. How does multiplying the absolute value function by a negative number affect its graph?
5. Explain the difference between horizontal and vertical shifts in transformations.

**Tip**: Always remember to flip the inequality sign when multiplying or dividing by a negative number!

Solve the compound inequality 2(x - 2) + 7 >= -1 and 5 - 4x > 9, and list the transformations of the graph g(x) = |x| for g(x) = 0.5|x - 2| + 3 and g(x) = -3|x| - 4.

Learn how to solve the compound inequality 2(x - 2) + 7 >= -1 and 5 - 4x > 9, and explore the transformations for g(x) = 0.5|x - 2| + 3 and g(x) = -3|x| - 4. This covers key concepts in algebra and graph transformations. Suitable for students in Grades 8-10.

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