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

Let's solve the inequality \left|3x - 1\right| < 5x :

1. Remove the absolute value by considering two cases:

   1. 3x - 1 < 5x
      Solve for $x$:

   2. 3x - 1 < 5x

      Subtract $3x$ from both sides:
      -1 < 2x

      Divide both sides by $2$:
      -\frac{1}{2} < x

      Rewriting this inequality yields: x > -\frac{1}{2}

   3. -(3x - 1) < 5x
      Simplifying yields:

   4. -3x + 1 < 5x

      Add $3x$ to both sides:
      1 < 8x

      Divide both sides by $8$:
      \frac{1}{8} < x

      This implies x > \frac{1}{8}

2. Combine results:

3. Both conditions imply x > \frac{1}{8} . Thus, the solution is x > \frac{1}{8} .