Does taking the reciporcal of both sides flip an inequality

Solution:

There is one very important exception to the rule that multiplying or dividing an inequality is the same as multiplying or dividing an equation. Whenever we multiply or divide an inequality by a negative number, you must flip the inequality sign.

But in general that we can take the reciprocal of both sides, unfortunately, we can only take the reciprocal of a single number or a single fraction, NOT a sum or difference of fractions. The reciprocal of a sum is not the sum of the reciprocals.