You have followed the right path.
Case 1: here, you have strictly restricted $x$ to less-than-or-equal-to $2$. The solution you got was $x > 1/2$. Keeping the initial restriction in mind, you have $1/2<x \le 2$. Remember that you have ignored the numbers that follow $2$ because they don't follow the original restriction $x \le 2$.
Case 2: here, you have strictly restricted $x$ to greater than $2$. The solution you got was $x > -1$. Keeping the initial restrictions in mind, you get $x > 2$ as you will ignore all the numbers between $-1$ and $2$ (since they don't follow your original restriction).
So, finally, you get the solution $2 \ge x> 1/2$ and $x >2$. Here is where the number line comes into play. Represent your solutions on a number line, and you'll notice that it's the same as $x > 1/2$.