To solve the inequality \(|2x - 3| > 4|x + 1|\), we first consider the non-modular inequality:
\((2x - 3)^2 > 4^2(x + 1)^2\)
This simplifies to:
\((2x - 3)^2 > 16(x + 1)^2\)
Expanding both sides, we get:
\(4x^2 - 12x + 9 > 16(x^2 + 2x + 1)\)
\(4x^2 - 12x + 9 > 16x^2 + 32x + 16\)
Rearranging terms gives:
\(4x^2 - 12x + 9 - 16x^2 - 32x - 16 < 0\)
\(-12x^2 - 44x - 7 < 0\)
Solving the quadratic inequality \(12x^2 + 44x + 7 < 0\) gives the critical values:
\(x = -\frac{7}{2}\) and \(x = -\frac{1}{6}\)
The solution to the inequality is:
\(\frac{7}{2} < x < -\frac{1}{6}\)
