Student's completed worksheet on absolute value inequalities, featuring solved equations and corresponding 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, the image is blurry, especially in the lower portion, so I can only clearly see the first two problems.

Let’s go through Problems 1 and 2 in detail, which are visible and correctly solved in your image. Then I’ll explain how to solve absolute value inequalities in general, and if you need help with the rest of the problems, feel free to describe them or clarify.

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🔷 Problem 1:

Solve and graph:
$$
|6n| \leq 18
$$

#### ✔ Step-by-step Solution:

The absolute value inequality $ |A| \leq B $ (where $ B \geq 0 $) means:
$$
-A \leq x \leq A
$$
So for $ |6n| \leq 18 $, we rewrite it as:
$$
-18 \leq 6n \leq 18
$$

Now divide all parts by 6:
$$
\frac{-18}{6} \leq n \leq \frac{18}{6}
$$
$$
-3 \leq n \leq 3
$$

✔ Solution: $ \boxed{-3 \leq n \leq 3} $

#### 📊 Graph:
On a number line:
- Draw a solid dot at $ -3 $ and $ 3 $
- Shade the region between them (inclusive)

This matches what's shown in your image — correct!

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🔷 Problem 2:

Solve and graph:
$$
|p + 4| \leq 8
$$

#### ✔ Step-by-step Solution:

Again, use the rule:
$ |A| \leq B $ → $ -B \leq A \leq B $

So:
$$
-8 \leq p + 4 \leq 8
$$

Subtract 4 from all parts:
$$
-8 - 4 \leq p \leq 8 - 4
$$
$$
-12 \leq p \leq 4
$$

✔ Solution: $ \boxed{-12 \leq p \leq 4} $

#### 📊 Graph:
- Solid dots at $ -12 $ and $ 4 $
- Shade between them

Also matches your handwritten answer — correct!

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🧠 General Rule for Solving Absolute Value Inequalities:

For any expression $ |X| \leq a $, where $ a \geq 0 $:
$$
-a \leq X \leq a
$$

For $ |X| \geq a $:
$$
X \leq -a \quad \text{OR} \quad X \geq a
$$

> Note: If $ a < 0 $, then:
> - $ |X| \leq a $ has no solution (since absolute value is never negative)
> - $ |X| \geq a $ is always true (since absolute value is always ≥ 0)

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🔍 Let’s Try One More Example (if needed):

Suppose there was a problem like:
$$
|2x - 5| > 7
$$

Then:
$$
2x - 5 < -7 \quad \text{OR} \quad 2x - 5 > 7
$$

Solve each:
1. $ 2x < -2 $ → $ x < -1 $
2. $ 2x > 12 $ → $ x > 6 $

✔ Solution: $ x < -1 $ or $ x > 6 $

Graph: Open circles at $ -1 $ and $ 6 $, shade left and right.

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✔ Summary:

You’ve already solved Problems 1 and 2 correctly! Your answers:
1. $ -3 \leq n \leq 3 $
2. $ -12 \leq p \leq 4 $

are correct, and the graphs match.

If you’d like help with the other problems on the sheet (which are blurry), please:
- Describe them (e.g., “Problem 3: |3x – 6| > 9”)
- Or type out the expressions

I’ll walk you through each one step-by-step.

Let me know!