We are given the inequality:

\[
2|x - 5| + 7 < 11
\]

### Step 1: Isolate the absolute value term
First, subtract 7 from both sides to isolate the absolute value term:

\[
2|x - 5| < 11 - 7
\]
\[
2|x - 5| < 4
\]

### Step 2: Divide both sides by 2
Now, divide both sides of the inequality by 2:

\[
|x - 5| < \frac{4}{2}
\]
\[
|x - 5| < 2
\]

### Step 3: Break the absolute value into two inequalities
The absolute value inequality \(|x - 5| < 2\) means that the expression inside the absolute value lies between \(-2\) and \(2\). Therefore, we can write:

\[
-2 < x - 5 < 2
\]

### Step 4: Solve the inequality
Now, solve for \(x\) by adding 5 to all parts of the inequality:

\[
-2 + 5 < x < 2 + 5
\]
\[
3 < x < 7
\]

### Final answer:
The solution is:

\[
3 < x < 7
\]

This means \(x\) lies in the interval \( (3, 7) \).

Do you want any further details or have any questions?

Here are 5 related questions for practice:
1. Solve the inequality \(3|x - 4| - 5 < 7\).
2. Solve \(|x + 2| \leq 5\).
3. What is the solution to \(5|x - 1| > 15\)?
4. Solve for \(x\) in \(4|x + 3| + 2 \geq 10\).
5. Solve the equation \( |2x - 3| = 7 \).

**Tip:** When solving absolute value inequalities, remember to break them into two separate inequalities.

Learn how to solve the absolute value inequality 2|x − 5| + 7 < 11 step-by-step. This solution covers important algebraic concepts like absolute value and inequalities. Suitable for students in Grades 9-11.

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