Question

Explain what needs to happen to the inequities sign when dividing or multiplying by a negative number

Answer 1

Inequality symbols need to be reversed when multiplying or dividing by a negative number, because that reverses the ordering of values.
2 > 1
-2 < -1 . . . . . multiplied or divided by -1

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A more "interesting" case is when a function is applied that has a negative slope.
0 < a < b < π
cos(0) > cos(a) > cos(b) > cos(π)