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.