To determine the solutions for the given equation:
[tex]\[ \left|\frac{2}{3} x + 2\right| - 8 = 0 \][/tex]
we'll solve it step-by-step.
1. Isolate the absolute value:
[tex]\[
\left|\frac{2}{3} x + 2\right| - 8 = 0
\][/tex]
Add 8 to both sides of the equation:
[tex]\[
\left|\frac{2}{3} x + 2\right| = 8
\][/tex]
2. Remove the absolute value definition:
The equation [tex]\(\left|A\right| = B\)[/tex] implies [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Therefore, we have two cases:
Case 1:
[tex]\[
\frac{2}{3} x + 2 = 8
\][/tex]
Case 2:
[tex]\[
\frac{2}{3} x + 2 = -8
\][/tex]
3. Solve for [tex]\( x \)[/tex] in both cases:
Case 1:
[tex]\[
\frac{2}{3} x + 2 = 8
\][/tex]
Subtract 2 from both sides:
[tex]\[
\frac{2}{3} x = 6
\][/tex]
Multiply both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x = 6 \cdot \frac{3}{2}
\][/tex]
[tex]\[
x = 9
\][/tex]
Case 2:
[tex]\[
\frac{2}{3} x + 2 = -8
\][/tex]
Subtract 2 from both sides:
[tex]\[
\frac{2}{3} x = -10
\][/tex]
Multiply both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x = -10 \cdot \frac{3}{2}
\][/tex]
[tex]\[
x = -15
\][/tex]
Thus, the solutions to the equation [tex]\(\left|\frac{2}{3} x + 2\right| - 8 = 0\)[/tex] are [tex]\(x = 9\)[/tex] and [tex]\(x = -15\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{x = -15 \text{ and } x = 9} \][/tex]
