To solve the inequality \( |x - 3| \leq 13 \), we will break it down into two cases, as is common with absolute value inequalities.

### Step 1: Rewrite the inequality
The inequality \( |x - 3| \leq 13 \) means that the expression inside the absolute value, \( x - 3 \), must lie between \( -13 \) and \( 13 \). This can be rewritten as:
\[
-13 \leq x - 3 \leq 13
\]

### Step 2: Solve for \( x \)
Now, solve for \( x \) by adding 3 to all parts of the inequality:
\[
-13 + 3 \leq x - 3 + 3 \leq 13 + 3
\]
\[
-10 \leq x \leq 16
\]

### Solution
The solution is:
\[
x \in [-10, 16]
\]

This means \( x \) is between \( -10 \) and \( 16 \), inclusive.

Would you like more details on any step or further explanation?

Here are 5 related questions for practice:
1. Solve \( |x + 5| \leq 8 \).
2. Solve \( |x - 7| \geq 9 \).
3. Solve \( |2x - 1| \leq 5 \).
4. Find the solution to \( |3x + 4| \geq 12 \).
5. Solve \( |x - 2| < 6 \).

**Tip:** When solving absolute value inequalities, always remember to split the inequality into two cases—one for the positive and one for the negative value inside the absolute value.

Learn how to solve the absolute value inequality ∣x−3∣≤13 with step-by-step instructions. This problem demonstrates the key algebraic concept of absolute value inequalities and uses properties of absolute value to find the solution, suitable for students in Grades 8-10.

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