To solve the inequality [tex]\(\frac{4}{3} \left| \frac{1}{4} x + 3 \right| < 4\)[/tex], we will follow these detailed steps:
1. Isolate the Absolute Value Expression:
[tex]\[
\frac{4}{3} \left| \frac{1}{4} x + 3 \right| < 4
\][/tex]
Multiply both sides of the inequality by [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\left| \frac{1}{4} x + 3 \right| < 3
\][/tex]
2. Remove the Absolute Value:
Recall that [tex]\(|A| < B\)[/tex] if and only if [tex]\(-B < A < B\)[/tex]. So:
[tex]\[
-3 < \frac{1}{4} x + 3 < 3
\][/tex]
3. Solve the Compound Inequality:
Split the compound inequality into two parts and solve each part separately.
For the left part:
[tex]\[
-3 < \frac{1}{4} x + 3
\][/tex]
Subtract 3 from both sides:
[tex]\[
-6 < \frac{1}{4} x
\][/tex]
Multiply by 4:
[tex]\[
-24 < x
\][/tex]
This simplifies to:
[tex]\[
x > -24
\][/tex]
For the right part:
[tex]\[
\frac{1}{4} x + 3 < 3
\][/tex]
Subtract 3 from both sides:
[tex]\[
\frac{1}{4} x < 0
\][/tex]
Multiply by 4:
[tex]\[
x < 0
\][/tex]
4. Combine the Results:
Combine the two parts of the compound inequality:
[tex]\[
-24 < x < 0
\][/tex]
Which can be rewritten as:
[tex]\[
x > -24 \quad \text{and} \quad x < 0
\][/tex]
Therefore, the solution to the inequality [tex]\(\frac{4}{3} \left| \frac{1}{4} x + 3 \right| < 4\)[/tex] is:
[tex]\[
x > -24 \quad \text{and} \quad x < 0
\][/tex]
So the correct answer is:
[tex]\[
\boxed{x > -24 \text{ and } x < 0}
\][/tex]
