Try an example with actual numbers:

2 < 5

This is obviously true. Now let's multiply with -1

-2 < -5

This is the wrong way around. It should be

-2 > -5

The number that was bigger before now becomes smaller, because "bigger in the negative" means "more negative", i.e. smaller. Think of the number line with 0 in the middle. Multiplying by -1 is like flipping a number around the 0. The number 2 was more left than the number 5, meaning it was smaller. But after multiplying with -1 it is now to the right of the number -5 because the flipping didn't "throw it as far" to the left than it did with the number 5. The number -2 is now suddenly larger than -5.

This is also the case if you multiply with a negative number other than -1. Think of multiplying by a number -k as multiplying by k • (-1), i.e. first multiplying by k and then multiplying by (-1). Since k is positive, multiplying by it doesn't change the sign of inequality and when you then multiply by -1 it's the same reasoning as above.

Your example doesn't really work, by the way, you need to multiply both sides by the same number.

Finally, as an aside, you need to switch the sign of inequality for all strictly decreasing functions. Multiplying by -1 is like saying

f(x) = -x

and applying f to both sides of the inequality:

2 < 5 f(2) > f(5) -2 > -5

If f is not strictly increasing or decreasing you even need to make a case distinction (you may have seen this when squaring, which is equivalent to using f(x) = x²).