Prove or disprove if x∈Z, then $x=⌈\frac{x}{2}⌉+⌊\frac{x}{2}⌋$

Question

Prove or disprove if x∈Z

$$x=⌈\frac{x}{2}⌉+⌊\frac{x}{2}⌋$$

I'm just unsure how to go about this question and can't find decent examples in my textbook. I'm assuming that the assumption is true based on plugging in arbitrary integers, but I'm unsure how to phrase an explanation.

I was going along this route but wasn't sure if I was getting warmer

⌈x⌉ = n if and only if $n - 1 < x <= n$

$⌈\frac{x}{2}⌉$ = n if and only if n-12

⌊x⌋ = nif and only if n <= x < n + 1

$⌊\frac{x}{2}⌋ = n$ if and only if $\frac{n}{2} <= \frac{x}{2} < \frac{n + 1}{2}$

Answer 1

If $x$ is even, we're done. If $x$ is odd, write $x=2k+1$ so the right-hand side is $k+1+k$.