To determine the solutions of the equation
[tex]\[
\left|\frac{1}{5} x + 2\right| - 6 = 2
\][/tex]
we need to isolate the absolute value expression. Follow these steps:
1. Isolate the absolute value expression:
[tex]\[
\left|\frac{1}{5} x + 2\right| - 6 = 2
\][/tex]
Add 6 to both sides to isolate the absolute value:
[tex]\[
\left|\frac{1}{5} x + 2\right| = 8
\][/tex]
2. Remove the absolute value:
The absolute value equation [tex]\(\left|A\right| = B\)[/tex] has two solutions: [tex]\(A = B\)[/tex] and [tex]\(A = -B\)[/tex]. Therefore:
[tex]\[
\frac{1}{5} x + 2 = 8 \quad \text{and} \quad \frac{1}{5} x + 2 = -8
\][/tex]
3. Solve the equation [tex]\(\frac{1}{5} x + 2 = 8\)[/tex]:
[tex]\[
\frac{1}{5} x + 2 = 8
\][/tex]
Subtract 2 from both sides:
[tex]\[
\frac{1}{5} x = 6
\][/tex]
Multiply both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 30
\][/tex]
4. Solve the equation [tex]\(\frac{1}{5} x + 2 = -8\)[/tex]:
[tex]\[
\frac{1}{5} x + 2 = -8
\][/tex]
Subtract 2 from both sides:
[tex]\[
\frac{1}{5} x = -10
\][/tex]
Multiply both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -50
\][/tex]
Hence, the solutions to the equation [tex]\(\left|\frac{1}{5} x + 2\right| - 6 = 2\)[/tex] are [tex]\(x = 30\)[/tex] and [tex]\(x = -50\)[/tex].
Therefore, the correct answer is:
[tex]\[
x = -50 \text{ and } x = 30
\][/tex]
