We are given the inequality:

$2 |x - 7| + 3 < 17$

Step 1: Isolate the absolute value expression.

First, subtract 3 from both sides to get:

$2 |x - 7| < 14$

Next, divide both sides by 2:

$|x - 7| < 7$

Step 2: Break the absolute value into two inequalities.

For $|x - 7| < 7$, we can rewrite it as a compound inequality:

$-7 < x - 7 < 7$

Step 3: Solve the inequality.

Now, add 7 to all parts of the inequality:

$0 < x < 14$

Solution:

The solution to the inequality is:

$0 < x < 14$

This means that $x$ must be between 0 and 14 (but not including 0 or 14).

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Would you like more details on any of the steps, or have any specific questions?

Here are 5 related questions to explore:

1. What happens if the inequality were $2 |x - 7| + 3 \geq 17$?
2. How would the solution change if the sign were $\leq$ instead of $<$?
3. Can you graph the solution on a number line?
4. How does the solution change if the absolute value is centered at a different point, e.g., $|x - 5|$?
5. What are the possible scenarios if the absolute value inequality includes a negative value?

Tip: Always start by isolating the absolute value expression when solving absolute value inequalities!