The problem asks to solve the inequality $2|x - 7| - 4 < 14$ algebraically. Let's go through the steps to solve it.

Step 1: Isolate the absolute value term

Start by adding 4 to both sides of the inequality to isolate the absolute value expression: $2|x - 7| < 18$

Step 2: Divide by 2

Next, divide both sides by 2: $|x - 7| < 9$

Step 3: Remove the absolute value

To solve the inequality $|x - 7| < 9$, consider the definition of absolute value. The inequality $|A| < B$ implies $-B < A < B$. Therefore: $-9 < x - 7 < 9$

Step 4: Solve the resulting compound inequality

Add 7 to each part of the inequality: $-9 + 7 < x < 9 + 7$ $-2 < x < 16$

Final Answer:

The solution to the inequality is: $x \in (-2, 16)$

This means that $x$ can be any real number between $-2$ and $16$, but not including $-2$ and $16$.

Would you like any further details or have any questions?

Related Questions:

1. How would the solution change if the inequality were $2|x - 7| - 4 \leq 14$?
2. What is the graphical representation of the solution set on a number line?
3. How would you solve the inequality if it were $2|x - 7| - 4 > 14$?
4. What happens to the solution set if the absolute value expression had a different coefficient, such as $3|x - 7|$?
5. Can you solve the inequality if the constant term on the right side of the inequality was negative?

Tip:

When isolating an absolute value expression in an inequality, make sure to correctly distribute any multiplicative constants before removing the absolute value. This ensures that your solution is accurate.