Let's start by solving the questions step by step.

1. Solving the equation:

$\frac{1}{3}x + 4 = 3$

Step 1: Subtract 4 from both sides to isolate the term with $x$.

$\frac{1}{3}x = 3 - 4$ $\frac{1}{3}x = -1$

Step 2: Multiply both sides by 3 to solve for $x$.

$x = -1 \times 3 = -3$

So, $x = -3$ is one solution. Now, let's check the answer choices:

• Option C: $x = -3$ is a correct answer.

Since the problem asks to select all that apply, we know that other choices might also work. Let's verify:

• Option B: $x = -\frac{22}{3}$
• Option D: $x = -\frac{4}{3}$
• Option E: $x = 3$
• Option F: $x = \frac{22}{3}$

None of these additional values satisfy the equation, so only C: $x = -3$ is correct.

2. Isolate the absolute value expression in the equation:

$-3|x - 3| + 3 = 6$

Step 1: Subtract 3 from both sides.

$-3|x - 3| = 6 - 3$ $-3|x - 3| = 3$

Step 2: Divide by -3 to isolate the absolute value.

$|x - 3| = -1$

Since the absolute value cannot equal a negative number, there is no solution to this equation.

3. Isolate the absolute value expression in the equation:

$5|2x + 4| - 3 = 7$

Step 1: Add 3 to both sides.

$5|2x + 4| = 7 + 3$ $5|2x + 4| = 10$

Step 2: Divide by 5 to isolate the absolute value.

$|2x + 4| = 2$

Step 3: Solve for $x$ by setting up two cases, since the absolute value can be positive or negative.

• Case 1: $2x + 4 = 2$ $2x = 2 - 4$ $2x = -2 \quad \Rightarrow \quad x = -1$

• Case 2: $2x + 4 = -2$ $2x = -2 - 4$ $2x = -6 \quad \Rightarrow \quad x = -3$

So, the solutions to the equation are $x = -1$ and $x = -3$.

Summary of Solutions:

1. $x = -3$ (Option C) is the only solution for the first equation.
2. No solution for the second equation.
3. The solutions for the third equation are $x = -1$ and $x = -3$.

Would you like more details on any of these steps? Here are five follow-up questions you can explore:

1. How do we handle equations involving fractions?
2. Why can't absolute value expressions equal negative numbers?
3. What happens if an absolute value equation has no solution?
4. How do you check if your solutions are correct in multi-step problems?
5. Can we always have more than one solution for absolute value equations?

Tip: When solving equations involving absolute values, always remember to set up two cases—one for the positive and one for the negative scenario.