Solve the inequality $9 - 2|4x + 1| > 3$.

1. Problem Description

Solve the inequality $9 - 2|4x + 1| > 3$.

2. Solution Steps

First, isolate the absolute value term:

$9 - 2|4x + 1| > 3$

Subtract 9 from both sides:

$-2|4x + 1| > 3 - 9$

$-2|4x + 1| > -6$

Divide both sides by -2 (and reverse the inequality sign):

$|4x + 1| < 3$

Now, consider the two cases for the absolute value:

Case 1: $4x + 1 < 3$

$4x < 3 - 1$

$4x < 2$

$x < \frac{2}{4}$

$x < \frac{1}{2}$

Case 2: $-(4x + 1) < 3$

$-4x - 1 < 3$

$-4x < 3 + 1$

$-4x < 4$

$x > -1$

Combining the two inequalities, we have $-1 < x < \frac{1}{2}$.

3. Final Answer

$-1 < x < \frac{1}{2}$