In order to check it you should try numbers less than $5/2$, not equal to. Once you plugged it in the original equation it was good that it wasn't a solution, or else something would have went horribly wrong. Try $x=2$.
The values of $x$ you have found are the ones less than two and half, not equal to. Why should it be $2x+4 \leq 1$? You don't need equality. Geometrically, you have the points belonging to the line $y=2x+4$ and below the line $y=1$, but you discount the intersection, which happens at the point $x= 5/2$.
Also, note that $5/2$ is not less than itself, thus it cannot be a solution! If it doesn't belong to the solution set, it cannot satisfy the given inequality.