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.