Math problems testing algebraic concepts including function evaluation, system of equations, and inequality solution representation.

Let's solve each problem step by step.

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Problem 11:

Given $ f(x) = 5x - 4 $, find the value of $ x $ if $ f(x) = 31 $.

We are given:
$$
f(x) = 5x - 4 = 31
$$

Solve for $ x $:

$$
5x - 4 = 31
$$
Add 4 to both sides:
$$
5x = 35
$$
Divide by 5:
$$
x = 7
$$

✔ Answer: A. 7

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Problem 12:

Which answer best describes the number of solutions for the following system of equations?

$$
\begin{align*}
(1)\quad & 4x + y = 5 \\
(2)\quad & 8x + 2y = -6
\end{align*}
$$

Let’s analyze the system.

We can simplify equation (2):
$$
8x + 2y = -6 \quad \Rightarrow \quad \text{Divide all terms by 2: } 4x + y = -3
$$

Now compare with equation (1):
- Equation (1): $ 4x + y = 5 $
- Simplified (2): $ 4x + y = -3 $

These two equations have the same left-hand side but different right-hand sides, so they represent parallel lines that never intersect.

Therefore, there is no solution.

✔ Answer: C. no solutions

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Problem 13:

Which graph best represents the solution set of:
$$
15 - 2(x + 3) < -7
$$

First, solve the inequality:

$$
15 - 2(x + 3) < -7
$$

Distribute the $-2$:
$$
15 - 2x - 6 < -7
$$

Simplify:
$$
9 - 2x < -7
$$

Subtract 9 from both sides:
$$
-2x < -16
$$

Divide by $-2$. Remember: When dividing by a negative number, reverse the inequality sign:
$$
x > 8
$$

So the solution is $ x > 8 $

Now, look at the graphs:

- We need a number line where values greater than 8 are shaded.
- Since it's strictly greater than, we use an open circle at 8.
- Shading goes to the right.

Check the options:

- A: Closed circle at 8, shading to the right → Incorrect (should be open)
- B: Open circle at 8, shading to the right → ✔ Correct
- C: Closed circle at 8, shading to the left → Wrong
- D: Open circle at 8, shading to the left → Wrong

✔ Answer: B

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✔ Final Answers:

11. A. 7
12. C. no solutions
13. B

Let me know if you'd like visual explanations or further breakdown!