Solve the Absolute Value Inequality: |2x - 4| < 8

Given:

$\left|2x-4\right|<8$

Which of the following statements is necessarily true?

To solve the absolute value inequality $|2x - 4| < 8$, we begin by removing the absolute value expression. This gives us a compound inequality:

$-8 < 2x - 4 < 8$.

We will solve this compound inequality by handling each part separately:

• Start with the left inequality: $-8 < 2x - 4$.
• Add 4 to both sides to isolate the term with $x$: $-8 + 4 < 2x$.
• Simplify: $-4 < 2x$.
• Finally, divide both sides by 2: $-2 < x$.

• Now, solve the right inequality: $2x - 4 < 8$.
• Add 4 to both sides to isolate the term with $x$: $2x - 4 + 4 < 8 + 4$.
• Simplify: $2x < 12$.
• Finally, divide both sides by 2: $x < 6$.

Combining the two solutions from the parts, we find:

$-2 < x < 6$.

The solution indicates that $x$ must be greater than -2 and less than 6. This form matches answer choice 4. Therefore, the correct solution is:

$-2 < x < 6$.