How negative numbers flip the sign of the inequality
Effect of negative numbers on inequalities
We solve inequalities the same way we solve equations, except that when we multiply or divide both sides of the inequality by a negative number, we have to do something special to it.
Anytime you multiply or divide both sides of the inequality, you must “flip” or change the direction of the inequality sign. This means that if you had a less than sign ???<???, it would become a greater than sign ???>???. Likewise, if you started with ???>???, it would become ???<???.
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If the sign is greater than or equal to ???\geq???, or less than or equal to ???\leq???, the “equals” part of the sign is unaffected; it stays the same. You only have to flip the greater than sign to a less than sign, or flip the less than sign to a greater than sign.
How to change the inequality when multiplying or dividing by a negative number
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Solving inequalities by clearing the negative values
Example
Solve the inequality.
???-x+3>12???
Subtract ???3??? from both sides.
???-x+3-3>12-3???
???-x>9???
Now we have to multiply both sides by ???-1???, so we have to remember to change the direction of the inequality when we do.
???(-1)(-x)<9(-1)???
???x<-9???
Let’s try another example of solving inequalities with negatives.
Anytime you multiply or divide both sides of the inequality, you must “flip” or change the direction of the inequality sign.
Example
Solve the inequality.
???-2x+4\geq-6???
Subtract ???4??? from both sides.
???-2x+4-4\geq-6-4???
???-2x\geq-10???
Now we have to divide both sides by ???-2???, so we have to remember to change the direction of the inequality when we do.
???\frac{-2x}{-2}\leq\frac{-10}{-2}???
???x\leq5???
