I see the image you uploaded. It contains algebraic expressions and equations that need to be evaluated or solved. Let's go through them step by step.

1. Evaluate each expression if $x = -4$ and $y = -9$:

1. $|x - 8|$: $|x - 8| = |-4 - 8| = |-12| = 12$

2. $2|7y|$: $2|7y| = 2|7(-9)| = 2|-63| = 2 \times 63 = 126$

3. $-3|y|$: $-3|y| = -3|-9| = -3 \times 9 = -27$

4. $-2|3x + 8| - 4$: $-2|3x + 8| - 4 = -2|3(-4) + 8| - 4 = -2|-12 + 8| - 4 = -2|-4| - 4 = -2 \times 4 - 4 = -8 - 4 = -12$

2. Problem Solving (Fish Tank Temperature):

a. Write an equation to determine the least and greatest optimal temperatures.

If the ideal water temperature is within 2°F of 78°F, the optimal temperature range can be modeled by: $|T - 78| \leq 2$

b. Solve the equation you wrote in part a.

To solve $|T - 78| \leq 2$: $-2 \leq T - 78 \leq 2$ Adding 78 to all sides: $76 \leq T \leq 80$ So, the optimal temperature range is 76°F to 80°F.

c. If your aquarium’s thermometer is accurate to within plus or minus 1°F, what should the temperature of the water be to ensure it reaches the minimum temperature? Explain.

To ensure the water reaches at least 76°F, considering the thermometer's accuracy: $T - 1 \geq 76 \quad \text{or} \quad T \geq 77$ Thus, the water should be set to 77°F to guarantee the minimum temperature is reached.

3. Solve each equation. Check your solutions:

6. $|x + 8| = 12$: $x + 8 = 12 \quad \text{or} \quad x + 8 = -12$ $x = 4 \quad \text{or} \quad x = -20$

7. $|4 - 7a| = 11$: $4 - 7a = 11 \quad \text{or} \quad 4 - 7a = -11$ $-7a = 7 \quad \text{or} \quad -7a = -15$ $a = -1 \quad \text{or} \quad a = \frac{15}{7}$

8. $|a - 5| + 4 = 9$: $|a - 5| = 5 \quad \text{so} \quad a - 5 = 5 \quad \text{or} \quad a - 5 = -5$ $a = 10 \quad \text{or} \quad a = 0$

9. $3|2x - 3| = 5x + 12$: Solve by dividing both sides by 3 first: $|2x - 3| = \frac{5x + 12}{3}$ You would need to set up two separate cases to solve this.

10. $|4 - 3a| = 6 - a$: Set up two cases and solve: $4 - 3a = 6 - a \quad \text{or} \quad 4 - 3a = -(6 - a)$

11. $|4 - 3a| = 3a - 6$: Similar to question 11, set up two cases and solve.

Would you like more detailed steps on how to solve any of the equations or further clarification on any topic?

Relative Questions:

1. How do you determine the solution to absolute value equations involving inequalities?
2. What are the steps to solve the equation $|ax + b| = c$?
3. How does the accuracy of a thermometer affect the setting of the optimal temperature in a real-world scenario?
4. What are the implications of having two different cases when solving absolute value equations?
5. How do you solve more complex absolute value equations, like those in problems 10, 11, and 12?

Tip: When solving absolute value equations, remember to consider both the positive and negative scenarios, as the absolute value represents a distance from zero and could result from either direction on the number line.