why does the inequality sign change when both sides are multiplied or divided by a negative number?
Picture a number line with zero at the center. Negative numbers run off to the left and positive numbers run off to the right. When you multiply a number by negative one, it’s like flipping it to the other side of the zero. It’s the same distance from zero but in the opposite direction. Consider this example: 5>3. Five is farther away from zero, and it’s farther to the right, so it’s greater. If you multiply each number by negative one, you get -5 and -3; the -5 is still farther away from zero, but because they’re both negative, that means that it’s farther to the left, i.e., smaller. So 5>3 becomes -5<-3. I started with multiplying by negative one because it's easiest to visualize. If instead you divide by negative two, that's like dividing by two and then multiplying by negative one, so the same thing happens.
