Let's consider what multiplication by -1 means geometrically. Imagine a number. You are at the number 5. Multiply by -1. You are now at -5 and geometrically we see that multiplication by -1 means to reflect about 0.
Now reflecting about 0 preserves the distance from zero. This is why |-5| and
|5| are the same number. Reflection about zero is a distance preserving action.
But inequality doesn't measure distance from zero. Inequality measure which number is further to the right on the number line. This last fact must be kept in mind. Now, when I take two points on the number line, say A and B, and the reflect about zero what happens? The point that is furthest right reflects to a point that is to the left of the reflection of the point that was furthest left.
So when solving inequalities one must reverse the inequality when multiplying by a negative number. Multiplying by a negative number reverses the order of the numbers.