To solve the inequality \(2|2x - a| < |x + 3a|\), we first consider the non-modular inequality:
\(2^2(2x - a)^2 < (x + 3a)^2\).
This simplifies to solving the quadratic inequality:
\(4(2x - a)^2 < (x + 3a)^2\).
Expanding both sides, we have:
\(4(4x^2 - 4ax + a^2) < x^2 + 6ax + 9a^2\).
Simplifying further, we get:
\(16x^2 - 16ax + 4a^2 < x^2 + 6ax + 9a^2\).
Rearranging terms gives:
\(15x^2 - 22ax - 5a^2 < 0\).
Solving the quadratic equation \(15x^2 - 22ax - 5a^2 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find the critical values:
\(x = \frac{5}{3}a\) and \(x = -\frac{1}{5}a\).
The solution to the inequality is the interval between these critical values:
\(-\frac{1}{5}a < x < \frac{5}{3}a\).
