When do inequalities flip

Learn more. Why do inequalities flip signs? Asked 3 years ago. Active 3 years ago. Viewed 6k times. Improve this question. Community Bot 1. Lenny Lenny 4 4 silver badges 12 12 bronze badges.

After all, as mathematics teachers we ought to teach mathematics, not blind mimicking. Add a comment. Active Oldest Votes. Improve this answer. Benjamin Dickman Benjamin Dickman I'm just curious how to show it if some student ever asks. I hope those who are satisfied by this explanation will read my past answers, as well as answers from other users, which have generally involved significantly more work! Multiplying by a negative number flips numbers around 0.

Thus, "left of" becomes "right of", or "greater than". Jasper Jasper 1, 11 11 silver badges 19 19 bronze badges. That's harder to show on a blackboard, but makes for a nice visual if you can find a way to demo it.

It might be worth breaking down into cases, e. I would draw graphics to illustrate Anyway] I'm not sure truly: unsure! It is an interesting perspective, though, thanks for your comment! Performing multiplication or division with an inequality is nearly identical to multiplying or dividing parts of traditional equations with one exception, covered below.

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 you multiply or divide an inequality by a negative number, you must flip the inequality sign. One very important implication of this rule is: You cannot divide by an unknown i. The same rule would apply if you're multiplying both sides by a fraction. Multiplying and dividing are inverses of the same process, kind of like adding and subtracting, so the same rules apply to both.

You also need to think about flipping the inequality sign when you're dealing with absolute value problems. Then first of all you want to isolate the absolute value expression on the left side of the inequality it makes life easier.

Subtract 6 from both sides to get:. Now, you need to rewrite this expression as a compound inequality. The output of an absolute value expression is always positive, but the " x " inside the absolute value signs might be negative, so we need to consider the case when x is negative.

That gives us our two inequalities or our "compound inequality". We can easily solve both of them. These kinds of problems take some practice, so don't worry if you aren't getting it at first!

Keep at it and it will eventually become second nature. You also often need to flip the inequality sign when solving inequalities with absolute values.