Subtract, Then Flip

To solve $5-2x \ge 1$, subtract 5 from both sides. The sign stays the same, and you get $-2x \ge -4$. Now the coefficient of $x$ is -2. Divide both sides by -2 and flip the sign:

$$x \le 2$$

There are two steps, and one of them divides by a negative number, so the sign flips once. Check the answer with $x = 0$. It is at most 2, and $5-2\cdot 0 = 5$, which is at least 1.

The sign flips at the step where you divide by the negative coefficient, not before. Subtracting 5 is an ordinary subtraction, and it leaves the sign alone. Ask two separate questions at each step: what are you undoing, and are you multiplying or dividing by a negative number?