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).

---

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!

Discover how to solve the absolute value inequality 2|x - 7| + 3 < 17. This solution involves isolating the absolute value, breaking it into a compound inequality, and solving it step-by-step. Perfect for students in Grades 8-10.

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