Solve Absolute Value Inequality: |3x - 1| < 5x Analysis

The absolute value |3x - 1| means the distance from zero. This expression could be positive OR negative, so we must consider both possibilities to find all solutions.

You don't choose one case! You solve both cases and then find where they overlap. The final answer is where both conditions are satisfied.

That's normal! Case 1 gave us $x > -\frac{1}{2}$ and Case 2 gave us $x > \frac{1}{8}$. The solution is the more restrictive condition: $x > \frac{1}{8}$.

Great question! Since absolute values are always non-negative, if 5x < 0 (when x < 0), then there's no solution because we can't have |something| < negative number.

Pick a test value like $x = \frac{1}{4}$: $|3(\frac{1}{4}) - 1| = |\frac{1}{4}| = \frac{1}{4}$ and $5(\frac{1}{4}) = \frac{5}{4}$. Since $\frac{1}{4} < \frac{5}{4}$, it works!