We are asked to solve the equation $4|2x-3| + 1 = 21$ for $x$. The equation involves an absolute value.

1. Problem Description

We are asked to solve the equation $4|2x-3| + 1 = 21$ for $x$. The equation involves an absolute value.

2. Solution Steps

We want to isolate the absolute value term.

$4|2x-3| + 1 = 21$

Subtract 1 from both sides:

$4|2x-3| = 21 - 1$

$4|2x-3| = 20$

Divide both sides by 4:

$|2x-3| = \frac{20}{4}$

$|2x-3| = 5$

The expression inside the absolute value can be either 5 or -

5. We have two cases:

Case 1: $2x-3 = 5$

Add 3 to both sides:

$2x = 5 + 3$

$2x = 8$

Divide both sides by 2:

$x = \frac{8}{2}$

$x = 4$

Case 2: $2x-3 = -5$

Add 3 to both sides:

$2x = -5 + 3$

$2x = -2$

Divide both sides by 2:

$x = \frac{-2}{2}$

$x = -1$

3. Final Answer

The solutions are $x = 4$ and $x = -1$.