Answer :

Sure, let's solve the inequality [tex]\( |2x + 4| > 16 \)[/tex] step-by-step.

1. Understanding Absolute Value Inequality:
The given inequality is [tex]\( |2x + 4| > 16 \)[/tex]. Absolute value inequalities can be split into two linear inequalities. Specifically, [tex]\( |A| > B \)[/tex] is equivalent to [tex]\( A > B \)[/tex] or [tex]\( A < -B \)[/tex].

2. Setting Up the Inequalities:
Applying this to our inequality [tex]\( |2x + 4| > 16 \)[/tex], we split it into two separate inequalities:
[tex]\[ \begin{cases} 2x + 4 > 16 \\ 2x + 4 < -16 \end{cases} \][/tex]

3. Solving Each Inequality:

- First Inequality: [tex]\( 2x + 4 > 16 \)[/tex]
[tex]\[ 2x + 4 > 16 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x > 12 \][/tex]
Divide both sides by 2:
[tex]\[ x > 6 \][/tex]
So, one solution is [tex]\( x > 6 \)[/tex].

- Second Inequality: [tex]\( 2x + 4 < -16 \)[/tex]
[tex]\[ 2x + 4 < -16 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x < -20 \][/tex]
Divide both sides by 2:
[tex]\[ x < -10 \][/tex]
So, another solution is [tex]\( x < -10 \)[/tex].

4. Combining the Solutions:
The solutions to the inequality [tex]\( |2x + 4| > 16 \)[/tex] are the values of [tex]\( x \)[/tex] that satisfy either [tex]\( x > 6 \)[/tex] or [tex]\( x < -10 \)[/tex].

Therefore, the solution to the inequality [tex]\( |2x + 4| > 16 \)[/tex] can be expressed as:
[tex]\[ x > 6 \quad \text{or} \quad x < -10 \][/tex]

In interval notation, the solution is:
[tex]\[ (-\infty, -10) \cup (6, \infty) \][/tex]

This thorough, step-by-step process demonstrates how the solution to the inequality [tex]\( |2x + 4| > 16 \)[/tex] is determined.