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!