To solve the inequality \(|x - 3| < 3x - 4\), we consider two cases based on the definition of absolute value.
Case 1: Assume \(x - 3 \, \geq \, 0\), which implies \(x \, \geq \, 3\).
In this case, the inequality becomes:
\(x - 3 < 3x - 4\)
Solving for \(x\):
\(x - 3 < 3x - 4\)
\(x - 3x < -4 + 3\)
\(-2x < -1\)
\(x > \frac{1}{2}\)
Since \(x \, \geq \, 3\), the solution for this case is \(x > 3\).
Case 2: Assume \(x - 3 \, < \, 0\), which implies \(x \, < \, 3\).
In this case, the inequality becomes:
\(-(x - 3) < 3x - 4\)
\(-x + 3 < 3x - 4\)
Solving for \(x\):
\(-x + 3 < 3x - 4\)
\(-x - 3x < -4 - 3\)
\(-4x < -7\)
\(x > \frac{7}{4}\)
Since \(x \, < \, 3\), the solution for this case is \(\frac{7}{4} < x < 3\).
Combining both cases, the solution to the inequality is \(x > \frac{7}{4}\).
