To solve the inequality [tex]\(\left|\frac{1}{4}x - \frac{1}{3}\right| \leq \frac{1}{3}\)[/tex], we need to consider the definition of absolute value.
The absolute value inequality [tex]\(|A| \leq B\)[/tex] can be rewritten as [tex]\(-B \leq A \leq B\)[/tex]. Applying this to our problem:
[tex]\[
\left|\frac{1}{4}x - \frac{1}{3}\right| \leq \frac{1}{3}
\][/tex]
This translates to:
[tex]\[
-\frac{1}{3} \leq \frac{1}{4}x - \frac{1}{3} \leq \frac{1}{3}
\][/tex]
Now we solve these two inequalities separately.
### Solving the left inequality:
[tex]\[
-\frac{1}{3} \leq \frac{1}{4}x - \frac{1}{3}
\][/tex]
Add [tex]\(\frac{1}{3}\)[/tex] to both sides:
[tex]\[
-\frac{1}{3} + \frac{1}{3} \leq \frac{1}{4}x
\][/tex]
[tex]\[
0 \leq \frac{1}{4}x
\][/tex]
Multiply both sides by 4:
[tex]\[
0 \leq x
\][/tex]
### Solving the right inequality:
[tex]\[
\frac{1}{4}x - \frac{1}{3} \leq \frac{1}{3}
\][/tex]
Add [tex]\(\frac{1}{3}\)[/tex] to both sides:
[tex]\[
\frac{1}{4}x - \frac{1}{3} + \frac{1}{3} \leq \frac{1}{3} + \frac{1}{3}
\][/tex]
[tex]\[
\frac{1}{4}x \leq \frac{2}{3}
\][/tex]
Multiply both sides by 4:
[tex]\[
x \leq \frac{8}{3}
\][/tex]
So, the combined result from both inequalities is:
[tex]\[
0 \leq x \leq \frac{8}{3}
\][/tex]
In decimal form, [tex]\(\frac{8}{3}\)[/tex] is approximately [tex]\(2.6667\)[/tex].
Thus, the solution to the inequality [tex]\(\left|\frac{1}{4}x - \frac{1}{3}\right| \leq \frac{1}{3}\)[/tex] is:
[tex]\[
0 \leq x \leq 2.6667
\][/tex]
