Let's break the equation in several step to make things more clear: first of all, the numerator of the left hand side simplifies to
[tex] \dfrac{2x}{3}-\dfrac{1}{2} = \dfrac{4x-3}{6} [/tex]
Similarly, the denominator becomes
[tex] \dfrac{3x}{4}-\dfrac{1}{2} = \dfrac{3x-2}{4} [/tex]
Dividing by a fraction means to multiply by the inverse of that fraction:
[tex] \dfrac{\frac{a}{b}}{\frac{c}{d}} = \dfrac{a}{b}\cdot\dfrac{d}{c} [/tex]
So, in your case, we have
[tex]\dfrac{\frac{2x}{3}-\frac{1}{2}}{\frac{3x}{4}-\frac{1}{2}} = \dfrac{\frac{4x-3}{6}}{\frac{3x-2}{4}} = \dfrac{4x-3}{6}\cdot \dfrac{4}{3x-2} = \dfrac{2(4x-3)}{3(3x-2)} = \dfrac{8x-6}{9x-6}[/tex]
So, the equation has become
[tex]\dfrac{8x-6}{9x-6} = \dfrac{4}{3}[/tex]
Multiply both sides by 3(9x-6):
[tex] 3(8x-6) = 4(9x-6) \iff 24x-18 = 36x-24 [/tex]
Subtract 24x from both sides, and add 18 to both sides:
[tex] 36x-24x = 24-18 \iff 12x=6 \iff x = \dfrac{1}{2} [/tex]
