Trouble understanding only one way implication truth

For real numbers:

$$x > 1$$

implies that $$ x^2 > 1 $$

But $x^2 > 1$ does not imply that $x > 1$. For instance, $(-2)^2 = 4 > 1$, but $-2$ is not greater than $1$.

$x = 2 \implies x \ge 2$ is true.

$x \ge 2 \iff x = 2$ is false.

An easy one: all squares are rectangles, but not every rectangle is a square.

The fallacy of the converse is something lots of students are guilty of. Remember, to say P implies Q means that if you have P, you have Q. The presence of Q doesn't mean P holds.

Another example: differentiability implies continuity. Every differentiable function is continuous on its domain's interior, but a continuous function need not have a derivative anywhere.