# Mathematics 1010 online

## Solving Inequalities

An inequality consists of two algebraic expressions connected by one of the symbols , , , or . These symbols are explained on the page discussing the real number line. There might also be the symbol , but the other cases occur more frequently.

Suppose we have an inequality involving some variable . (It might be called something else, however.) To solve the inequality (for ) means to convert it to an equivalent inequality of the form

where is one of the above symbols and is an algebraic or arithmetic expression (often just a plain number) that does not contain .

For example, suppose we know that

Clearly this is true if and only if

which is an inequality of the desired form.

Formally speaking we converted the initial inequality by adding 3 on both sides. Why can we do this, and what else can we do?

Two fruitful ways of thinking about inequalities are in terms of the real number line, or in terms of an old fashioned scale. Having an inequality means that one side of the scale is heavier than the other. Adding the same weight on both sides of the scale, or multiplying with a positive factor (e.g., doubling) on both sides leaves the heavier side heavier. (One can contemplate more involved operations, for example adding more on the heavier side than on the lighter side, which also leaves the heavier side heavier. But we will only use the most basic techniques in this class.) In terms of the real number line, if we have an inequality then one point is to the left of the other, and shifting both points to the left or right by the same amount does not alter the relationship.

There is one subtlety, however. Multiplying on both sides with the same negative factor reverses the relationship. Suppose for simplicity that we multiply with . In terms of the real number line we reflect a point through the origin. In terms of the scale, a weight that previously pushed downwards now pulls upwards. If we have two points, rotating around the origin flips the points, the one that previously was to the right of the other is now to the left.

Here is a simple example. We know that

If we multiply with on both sides we get the numbers and , but

The multiplication with the negative factor reversed the sign!

So basically we process inequalities the same way as we process equalities, except that we take care to reverse the sign when we multiply on both sides with the same negative factor. There are some things (such as squaring) that we can do easily with equalities, but where we have to be careful with inequalities.

For the purpose of this class we will only consider the operations of adding and subtracting the same term on both sides, multiplying with the same positive factor on both sides, and multiplying with the same negative factor on both sides. The first two operations leave the inequality unchanged, the third reverses the sign.

Let's see these principle in action in some examples:

• Example 1. Suppose

We subtract on both sides which gives and then divide by which gives the answer:

• Example 2. Suppose

We subtract on both sides which gives . Then we divide by and reverse the sign which gives the answer:

• Example 3. Suppose

We subtract and from both sides which gives Then we divide by and reverse the sign to obtain