To solve the inequality [tex]\( |4w + 2| - 6 > 8 \)[/tex], we need to follow several steps. Let's begin by isolating the absolute value expression.
1. Start with the given inequality:
[tex]\[
|4w + 2| - 6 > 8
\][/tex]
2. Add 6 to both sides to isolate the absolute value:
[tex]\[
|4w + 2| > 14
\][/tex]
Next, we need to isolate the value within the absolute value function. Recall that the inequality [tex]\( |x| > a \)[/tex] means that [tex]\( x > a \)[/tex] or [tex]\( x < -a \)[/tex].
3. Break the absolute value inequality into two separate inequalities:
[tex]\[
4w + 2 > 14 \quad \text{or} \quad 4w + 2 < -14
\][/tex]
4. Solve each inequality separately:
- For [tex]\(4w + 2 > 14\)[/tex]:
[tex]\[
4w + 2 > 14
\][/tex]
Subtract 2 from both sides:
[tex]\[
4w > 12
\][/tex]
Divide by 4:
[tex]\[
w > 3
\][/tex]
- For [tex]\(4w + 2 < -14\)[/tex]:
[tex]\[
4w + 2 < -14
\][/tex]
Subtract 2 from both sides:
[tex]\[
4w < -16
\][/tex]
Divide by 4:
[tex]\[
w < -4
\][/tex]
Putting it all together, the solution to the inequality [tex]\( |4w + 2| - 6 > 8 \)[/tex] is:
[tex]\[
w > 3 \quad \text{or} \quad w < -4
\][/tex]
So, the correct answer is:
[tex]\[
w > 3 \quad \text{or} \quad w < -4
\][/tex]
