We are given the inequality:

\[
| \frac{4}{x} - 1 | < 2
\]

To solve this, we need to break the absolute value inequality into two separate inequalities:

\[
-2 < \frac{4}{x} - 1 < 2
\]

### Step 1: Break it into two inequalities

1. **Left side:**
\[
-2 < \frac{4}{x} - 1
\]

2. **Right side:**
\[
\frac{4}{x} - 1 < 2
\]

### Step 2: Solve each inequality separately

#### 1. Solving the left side inequality:
\[
-2 < \frac{4}{x} - 1
\]
Add \(1\) to both sides:
\[
-2 + 1 < \frac{4}{x}
\]
\[
-1 < \frac{4}{x}
\]
Multiply both sides by \(x\) (noting that if \(x\) is negative, the inequality direction changes):
\[
x > -\frac{4}{1}
\]
This gives us \(x < -4\) or \(x > 0\), but it also implies we should check for cases with \(x = 

Learn how to solve the absolute value inequality |4/x - 1| < 2 step by step. This detailed guide covers how to break the inequality into two cases, solve rational expressions, and manage inequalities. Suitable for students in Grades 9-12.

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