To solve the inequality \(|4x + 3| > |x|\), we consider two cases based on the definition of absolute value.

Case 1: \(4x + 3 > x\)

Simplify to get \(3x + 3 > 0\), which gives \(x > -1\).

Case 2: \(4x + 3 < -x\)

Simplify to get \(5x < -3\), which gives \(x < -\frac{3}{5}\).

Combining both cases, the solution is \(x < -1\) or \(x > -\frac{3}{5}\).