To find the value of x in the given equation, we first need to simplify the equation by combining like terms and then solve for x. Let's break down the equation step by step.

Given equation: (1-2x) + (1+2x) ÷ (4x+1)(x-3) = 1/2
First, simplify the expression inside the parenthesis:
1-2x + 1+2x = 2
Now, simplify the entire equation:
2 ÷ (4x+1)(x-3) = 1/2

To eliminate the fraction, cross multiply:
2 * 2 = (4x+1)(x-3)
4 = 4x^2 - 12x + x - 3

Combine like terms:
4 = 4x^2 - 11x - 3

Rearrange the equation to set it equal to zero:
4x^2 - 11x - 7 = 0

Now, we need to solve the quadratic equation:
x = [-(-11) ± √((-11)^2 - 4*4*(-7))] / 2*4
x = [11 ± √(121 + 112)] / 8
x = [11 ± √233] / 8
Therefore, the solutions for x are:
x = (11 + √233) / 8 and x = (11 - √233) / 8