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.
