Inverse DTFT Problem
You can use partial fractions first to expand the $X(e^{j\omega})$ as

$$X(e^{j\omega})=\frac{\tfrac{7}{4}}{1+\frac{1}{2}e^{-j\omega}}+\frac{\tfrac{5}{4}}{1-\frac{1}{2}e^{-j\omega}} $$.

Then taking the inverse DTFT you get $\tfrac{7}{4}(-\frac{1}{2})^{n}u(n)+\tfrac{5}{4}(\frac{1}{2})^{n}u(n)$