Solving Absolute Value Inequalities worksheet with a fun spider web coloring activity.

The image you've shared is a "Solving Absolute Value Inequalities" activity that combines math problem-solving with a color-by-number coloring sheet. This type of worksheet is designed to make learning engaging by connecting math solutions to visual art.

Let’s break it down and explain how to solve this kind of problem, using the visible examples from the worksheet.

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🔍 What is the Task?

Students are given absolute value inequalities to solve. After solving each inequality, they match their answer to a corresponding solution set on the coloring sheet. Each solution corresponds to a color code (e.g., number 1 = gray, 2 = red, etc.), and students color the sections labeled with that number accordingly.

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🧮 Example Problems from the Worksheet

Let’s work through two of the visible problems:

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#### Problem 2: |7x| − 11 > 45

Step 1: Isolate the absolute value expression.

$$
|7x| - 11 > 45
$$
Add 11 to both sides:
$$
|7x| > 56
$$

Step 2: Solve the absolute value inequality.

For $|A| > B$, where $B > 0$, the solution is:
$$
A > B \quad \text{or} \quad A < -B
$$

So:
$$
7x > 56 \quad \text{or} \quad 7x < -56
$$

Divide both sides by 7:
$$
x > 8 \quad \text{or} \quad x < -8
$$

✔ Final Answer: $ x < -8 $ or $ x > 8 $

Now, look at the answer key on the coloring sheet:
- The option "x < -8 or x > 8" is associated with color #2, which is red.

👉 So, any section labeled "2" on the spider web should be colored red.

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#### Problem 4: 2|x + 6| ≤ 28

Step 1: Divide both sides by 2:
$$
|x + 6| ≤ 14
$$

Step 2: Solve the absolute value inequality.

For $|A| ≤ B$, the solution is:
$$
-B ≤ A ≤ B
$$

So:
$$
-14 ≤ x + 6 ≤ 14
$$

Subtract 6 from all parts:
$$
-20 ≤ x ≤ 8
$$

✔ Final Answer: $ -20 \leq x \leq 8 $

Check the answer key:
- This matches color #1, which is gray.

👉 Color all sections labeled "1" in gray.

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#### Problem 8: 4|x − 9| − 2 ≥ 14

Step 1: Add 2 to both sides:
$$
4|x - 9| ≥ 16
$$

Step 2: Divide by 4:
$$
|x - 9| ≥ 4
$$

Step 3: Solve:
$$
x - 9 ≥ 4 \quad \text{or} \quad x - 9 ≤ -4
$$

$$
x ≥ 13 \quad \text{or} \quad x ≤ 5
$$

✔ Final Answer: $ x \leq 5 $ or $ x \geq 13 $

Look at the answer key:
- This matches color #5, which is orange (based on the legend shown).

👉 Color sections labeled "5" in orange.

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🎨 How to Use the Coloring Sheet

1. Solve each inequality.
2. Match your answer to one of the options in the #/Color Answer key.
3. Find the number (e.g., 1, 2, 3) corresponding to that answer.
4. Color all regions on the spider web labeled with that number using the assigned color.

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✔ Summary of Key Concepts

| Inequality Type | Rule |
|------------------|------|
| $ |A| > B $ | $ A > B $ or $ A < -B $ |
| $ |A| < B $ | $ -B < A < B $ |
| $ |A| ≥ B $ | $ A ≥ B $ or $ A ≤ -B $ |
| $ |A| ≤ B $ | $ -B ≤ A ≤ B $ |

> Remember: If $ B < 0 $, then:
> - $ |A| > B $ → Always true (since absolute value is always ≥ 0)
> - $ |A| < B $ → No solution

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🌟 Why This Activity Works

This "flexible color by number" design allows students to:
- Practice solving absolute value inequalities.
- Check their answers visually.
- Engage creatively through coloring.
- Reinforce learning with immediate feedback (if colors don't match, they made a mistake).

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🖼️ Final Notes

Even though we can’t see all the problems, the structure is clear:
- Students solve inequalities.
- Match answers to color codes.
- Color the spider web based on numbers.

It's a fun, interactive way to master absolute value inequalities!

If you’d like, I can help generate a full solution guide for all problems on the worksheet—just let me know!