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Let's go through the problems step by step.

### Problem 4b (Compound Inequality):
You need to solve the compound inequality:

\[
2(x - 2) + 7 \geq -1 \quad \text{and} \quad 5 - 4x > 9
\]

**Step 1: Solve \(2(x - 2) + 7 \geq -1\):**
\[
2(x - 2) + 7 \geq -1
\]
\[
2x - 4 + 7 \geq -1
\]
\[
2x + 3 \geq -1
\]
\[
2x \geq -4
\]
\[
x \geq -2
\]

**Step 2: Solve \(5 - 4x > 9\):**
\[
5 - 4x > 9
\]
\[
-4x > 9 - 5
\]
\[
-4x > 4
\]
\[
x < -1
\]

**Final Solution:**
The solution is the intersection of these two inequalities:
\[
x \geq -2 \quad \text{and} \quad x < -1
\]
So, the solution is:
\[
-2 \leq x < -1
\]

### Problem 5 (Transformations of \( g(x) \)):

#### a. \( g(x) = 3|x + 2| - 3 \)

- The **vertical stretch** factor is 3 (since the coefficient outside the absolute value is 3).
- There is a **horizontal shift** to the left by 2 units (since \(x + 2\) means shifting left by 2).
- There is a **vertical shift** downward by 3 units (because of the \(-3\)).

#### b. \( g(x) = -3|x - 1| + 4 \)

- The negative sign indicates a **reflection** over the x-axis.
- The **vertical stretch** factor is 3 (since the coefficient is -3).
- There is a **horizontal shift** to the right by 1 unit (since \(x - 1\)).
- There is a **vertical shift** upward by 4 units (due to the \(+4\)).

### Problem 6 (Triangle Angles):
You are asked to find the unknown angles \(x\).

#### a. Triangle with angles \(56^\circ\), \(76^\circ\), and \(x^\circ\):
The sum of the interior angles of a triangle is always 180°.
\[
x + 56 + 76 = 180
\]
\[
x + 132 = 180
\]
\[
x = 48^\circ
\]

#### b. Triangle with angles \(23^\circ\), \(74^\circ\), and \(x^\circ\):
Similarly, for this triangle:
\[
x + 23 + 74 = 180
\]
\[
x + 97 = 180
\]
\[
x = 83^\circ
\]

### Problem 7 (Perpendicular Lines):
To determine if two lines are perpendicular, check their slopes. Two lines are perpendicular if the product of their slopes is \(-1\). In other words, if one line has slope \(m_1\) and the other line has slope \(m_2\), then:

\[
m_1 \times m_2 = -1
\]

Do you want more detailed explanations for any of these, or have further questions?

Here are 5 related questions to explore further:
1. How do you solve compound inequalities involving "or" instead of "and"?
2. What happens to the graph of an absolute value function when you change the sign of \(a\) or \(b\)?
3. How do you find the slopes of two lines if only their equations are given?
4. Can you solve a system of inequalities graphically?
5. How does reflecting a function over the x-axis change its graph?

**Tip:** When solving for the angles of a triangle, remember that the sum is always 180°, and you can apply this in any triangle geometry problem!

Solve compound inequalities, transformations of absolute value functions, calculate triangle angles, and determine if lines are perpendicular.

This guide covers solving compound inequalities, understanding transformations in absolute value functions, calculating unknown angles in triangles, and determining if two lines are perpendicular using slopes. Suitable for high school students in grades 9-11.

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