https://file.aiquickdraw.com/mathbot/file/1726754403907c1jv3n2q.jpg

The inequality shown in the image is:

\[
|x - 3| \leq 12
\]

To solve this absolute value inequality, we use the fact that:

\[
|A| \leq B \quad \text{implies} \quad -B \leq A \leq B
\]

Here, \( A = x - 3 \) and \( B = 12 \). Applying this to the inequality, we have:

\[
-12 \leq x - 3 \leq 12
\]

Now, solve for \( x \) by adding 3 to all parts of the inequality:

\[
-12 + 3 \leq x \leq 12 + 3
\]

\[
-9 \leq x \leq 15
\]

Thus, the solution to the inequality is:

\[
x \in [-9, 15]
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you solve absolute value inequalities with "greater than" signs?
2. Can you explain what happens when the absolute value inequality has a "strict" inequality, like \( |x - 3| < 12 \)?
3. What are some real-world applications of absolute value inequalities?
4. How would you graph the solution to this inequality on a number line?
5. How do we solve absolute value equations, such as \( |x - 5| = 10 \)?

**Tip:** When solving absolute value inequalities, always check whether the inequality is "less than" or "greater than" to use the correct approach!

Learn how to solve the absolute value inequality |x - 3| ≤ 12. This tutorial explains key algebraic concepts, including the property of absolute value inequalities and step-by-step solution methods. Suitable for students in Grades 9-10.

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