This section discusses how to algebraically manipulate inequalities.
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Thus far in this course we’ve mostly assumed the student knows the basic rules of algebra - how to add, subtract, multiply, and divide both sides of an equality to maintain a valid chain of steps to solve for a specific variable. With inequalities however, the rules of algebraic manipulation have a few important exceptions.
In general, we can still do all the same operations, but we have to be careful when we do so, that we preserve the inequality relationship. Most (but definitely not all!) of the previous rules still work exactly the same:
- and
In other words the following actions preserve the inequality (are allowed):
- Multiplying or Dividing by a positive number on both sides.
- Adding or Subtracting a (positive or negative) number on both sides.
- Multiplying by a negative number on both sides flips the direction of the inequality.
The above really covers the vast majority of our typical operations that we use with equalities - but there are some things we do with equalities without realizing that we need them to be equalities (not inequalities). Consider the following example: if then we can square both sides to get . So we can square both sides of an equality. This isn’t necessarily true for an inequality however.
| This is clearly true! | |||
| Square both sides. | |||
| Clearly however. | |||
As we can see, squaring both sides clearly doesn’t preserve inequalities. The reason is that one of the values is negative and the other one is not. If they are both the same sign, then we could predict the inequality relationship correctly. If they are both positive, then the inequality will stay the same, but if they are both negative, then the inequality would flip direction (because you are multiplying both sides by a negative number).
Another thing to consider, is how one might merge inequalities. The rules for merging inequalities tend to be easier to remember through intuition than memorization. If you have and and you want to merge them while maintaining the inequalities, then you can:
- Add smaller parts together and bigger parts together to get .
- Multiply the smaller ones together and larger ones together, so long as (meaning that both and are positive) to get: .
We can combine these two results along with the previous ones which tells us that, if then and , to get our full list of ways to combine inequalities: