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 =