To solve the equation, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 1 - 2x ≥ 0
In this case, we can remove the absolute value signs and rewrite the equation as:
7 - 4√(2x - 1) = 3(1 - 2x)

Let's solve this equation:
7 - 4√(2x - 1) = 3 - 6x
4√(2x - 1) = 4x - 4
√(2x - 1) = x - 1

Now, we square both sides of the equation to eliminate the square root:
2x - 1 = (x - 1)^2
2x - 1 = x^2 - 2x + 1
x^2 - 4x + 2 = 0

Using the quadratic formula, we find:
x = (4 ± √(16 - 4(1)(2))) / 2
x = (4 ± √(16 - 8)) / 2
x = (4 ± √8) / 2
x = (4 ± 2√2) / 2
x = 2 ± √2

So, in this case, the solutions are x = 2 + √2 and x = 2 - √2.

Case 2: 1 - 2x < 0
In this case, we need to flip the sign inside the absolute value and rewrite the equation as:
7 - 4√(2x - 1) = 3(2x - 1)

Let's solve this equation:
7 - 4√(2x - 1) = 6x - 3
4√(2x - 1) = 6x - 10
√(2x - 1) = 1.5x - 2.5

Now, we square both sides of the equation to eliminate the square root:
2x - 1 = (1.5x - 2.5)^2
2x - 1 = 2.25x^2 - 7.5x + 6.25
2.25x^2 - 9.5x + 7.25 = 0

Using the quadratic formula, we find:
x = (9.5 ± √(9.5^2 - 4(2.25)(7.25))) / (2.25)
x = (9.5 ± √(90.25 - 65.25)) / 2.25
x = (9.5 ± √25) / 2.25
x = (9.5 ± 5) / 2.25

The solutions are x = 7/2 and x = 1/2.

Therefore, the solutions to the equation 7 - 4√(2x - 1) = 3|1 - 2x| are:
x = 2 + √2, x = 2 - √2, x = 7/2, and x = 1/2.