Answer
$\{-13,1\}$
Work Step by Step
Step 1. Let $u=(2x-1)^{1/3}$, we have $u^2+2u-3=0$
Step 2. Factor and solve: $(u+3)(u-1)=0$ thus $u=-3, 1$
Step 3. For $u=-3$, we have $(2x-1)^{1/3}=-3$ or $2x-1=-27$ thus $x=-13$
Step 4. For $u=1$, we have $(2x-1)^{1/3}=1$ or $2x-1=1$ thus $x=1$
Step 5. Solution set $\{-13,1\}$