Worksheet on solving absolute value inequalities with solutions and number line graphs.

It looks like you've uploaded a worksheet titled "Absolute Value Inequalities" from Kuta Software - Infinite Algebra 2, and you'd like me to solve the problems and explain the solutions. However, due to the image being blurry or low-resolution, only the first two problems are clearly visible and readable.

Let’s go through Problems 1 and 2, solve them step-by-step, and explain how to graph the solutions.

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🔹 Problem 1: $ |6n| \leq 18 $

#### Step 1: Understand the inequality
The absolute value inequality $ |6n| \leq 18 $ means that the expression inside the absolute value (i.e., $6n$) is less than or equal to 18 in magnitude.

This splits into two inequalities:
$$
6n \leq 18 \quad \text{AND} \quad 6n \geq -18
$$

#### Step 2: Solve each inequality
- $6n \leq 18$ → divide both sides by 6:
$$
n \leq 3
$$

- $6n \geq -18$ → divide both sides by 6:
$$
n \geq -3
$$

#### Step 3: Combine the results
So,
$$
-3 \leq n \leq 3
$$

✔ This matches the answer shown on the worksheet.

#### Step 4: Graph the solution
On a number line:
- Draw a closed circle at $-3$ and $3$ (since it's "≤", inclusive).
- Shade the line between them.

That’s what the graph shows: filled-in dots at $-3$ and $3$, with a shaded segment connecting them.

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🔹 Problem 2: $ |p + 4| \leq 8 $

#### Step 1: Break into two inequalities
The inequality $ |p + 4| \leq 8 $ means:
$$
p + 4 \leq 8 \quad \text{AND} \quad p + 4 \geq -8
$$

#### Step 2: Solve each
- $p + 4 \leq 8$ → subtract 4:
$$
p \leq 4
$$

- $p + 4 \geq -8$ → subtract 4:
$$
p \geq -12
$$

#### Step 3: Combine
$$
-12 \leq p \leq 4
$$

✔ Again, this matches the handwritten answer.

#### Step 4: Graph
- Closed circles at $-12$ and $4$
- Shade the region between them.

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✔ Summary of Solutions:

| Problem | Inequality | Solution | Graph |
|--------|------------|---------|-------|
| 1 | $ |6n| \leq 18 $ | $ -3 \leq n \leq 3 $ | Closed circles at $-3$ and $3$, shaded between |
| 2 | $ |p + 4| \leq 8 $ | $ -12 \leq p \leq 4 $ | Closed circles at $-12$ and $4$, shaded between |

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📌 General Rule for Absolute Value Inequalities:

For any expression $ |A| \leq B $, where $B > 0$:
$$
-A \leq A \leq B
$$

This applies when the inequality is "less than or equal to".

If it were greater than, like $ |A| > B $, then:
$$
A > B \quad \text{OR} \quad A < -B
$$

But since these are "≤", we use the "and" case (compound inequality).

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Let me know if you’d like help solving the rest of the problems — even though they're blurry, I can guide you through the method!