To solve the inequality [tex]\( 20|30x - 2| \leq 180 \)[/tex], follow these steps:
1. Isolate the absolute value:
[tex]\[
20|30x - 2| \leq 180
\][/tex]
Divide both sides by 20 to simplify:
[tex]\[
|30x - 2| \leq 9
\][/tex]
2. Remove the absolute value by considering the definition of absolute value:
[tex]\[
-9 \leq 30x - 2 \leq 9
\][/tex]
3. Solve the two inequalities separately:
First inequality:
[tex]\[
30x - 2 \leq 9
\][/tex]
Add 2 to both sides:
[tex]\[
30x \leq 11
\][/tex]
Divide by 30:
[tex]\[
x \leq \frac{11}{30}
\][/tex]
Approximating [tex]\(\frac{11}{30}\)[/tex]:
[tex]\[
x \leq 0.3667
\][/tex]
Second inequality:
[tex]\[
30x - 2 \geq -9
\][/tex]
Add 2 to both sides:
[tex]\[
30x \geq -7
\][/tex]
Divide by 30:
[tex]\[
x \geq \frac{-7}{30}
\][/tex]
Approximating [tex]\(\frac{-7}{30}\)[/tex]:
[tex]\[
x \geq -0.2333
\][/tex]
4. Combine the results from the two inequalities:
[tex]\[
-0.2333 \leq x \leq 0.3667
\][/tex]
5. Determine which interval from the options fits this result:
The correct interval is:
[tex]\[
[-0.23, 0.36]
\][/tex]
Thus, the correct choice is:
[tex]\[
[-0.23, 0.36]
\][/tex]
