4x - 7 / |3x - 5| = 4 - 3 / |4x - 7|

Let $$y = |4x - 7|$$. The equation becomes $$4x - 7/y = 4 - 3/y$$. Multiplying by $$y$$ (assuming $$y
eq 0$$), we get $$y(4x - 7) - 3 = 4y$$. Substituting back $$y = |4x - 7|$$, we have $$|4x - 7|^2 - 3 = 4|4x - 7|$$. Let $$z = |4x - 7|$$. Then $$z^2 - 4z - 3 = 0$$. Using the quadratic formula, $$z = \frac{4 \pm \sqrt{16 - 4(1)(-3)}}{2} = \frac{4 \pm \sqrt{28}}{2} = 2 \pm \sqrt{7}$$. Since $$z = |4x - 7| \ge 0$$, we must have $$z = 2 + \sqrt{7}$$. Thus, $$|4x - 7| = 2 + \sqrt{7}$$. This gives two possibilities: $$4x - 7 = 2 + \sqrt{7}$$ or $$4x - 7 = -(2 + \sqrt{7})$$. Case 1: $$4x = 9 + \sqrt{7}$$, so $$x = \frac{9 + \sqrt{7}}{4}$$. Case 2: $$4x = 7 - (2 + \sqrt{7}) = 5 - \sqrt{7}$$, so $$x = \frac{5 - \sqrt{7}}{4}$$. We must also ensure that $$|3x - 5|
eq 0$$, which means $$x
eq 5/3$$. For $$x = \frac{9 + \sqrt{7}}{4}$$, $$3x - 5 = 3(\frac{9 + \sqrt{7}}{4}) - 5 = \frac{27 + 3\sqrt{7} - 20}{4} = \frac{7 + 3\sqrt{7}}{4}
eq 0$$. For $$x = \frac{5 - \sqrt{7}}{4}$$, $$3x - 5 = 3(\frac{5 - \sqrt{7}}{4}) - 5 = \frac{15 - 3\sqrt{7} - 20}{4} = \frac{-5 - 3\sqrt{7}}{4}
eq 0$$. Therefore, the solutions are $$x = \frac{9 + \sqrt{7}}{4}$$ and $$x = \frac{5 - \sqrt{7}}{4}$$.