This section introduces inequalities and what they are (and aren’t!)
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We begin our deep dive into inequalities by first discussing what they are - and importantly - what they aren’t. To this end, we will start by discussing single variable inequalities, that is to say, inequalities with just one variable.
Inequalities are primarily used when we want to set bounds on something. For example, if you want to represent the statement “the board was at least three feet long” you could do this using the inequality: where is the length of the board in feet.
Before we get too far ahead of ourselves however, let’s review the different symbols used for inequalities. Although these are likely familiar to you, it is good to make sure we are all on the same page.
| Symbol | Meaning |
| is strictly less than | |
| is less than or equal to | |
| is strictly larger than | |
| is greater than or equal to | |
Notice that there is a kind of symmetric with the first two inequalities and the last two inequalities above. In our work, it is entirely likely that (for organizational reasons) we may switch between these symbols, thus we may have something like:
| Moved the 12 over to the right | |
| Flipped the inequality and locations of terms. | |
Sometimes, in our work, we may skip that middle step, and just go from the first to the third. If it looks like the inequality is the wrong way, or that terms moved without being subtracted/added to both sides, take a moment to see if this is what happened.
As we said at the start, inequalities are usually used when we have some kind of “bounding” value. In general however, inequalities are just a way of stating a relationship of inequality between two expressions. For example, instead of saying we could say . But when we stop to think about what “” actually means, it means that either is less than 2, or is greater than 2, which we can express as either or . So, if we know that two expressions are relateable in size, we can express this using inequalities.
For example, if we know that is “at least as big as” , then we can express this as . Notice that we include the equality part because if then it is “at least as big” as . In fact, we can even capture the idea of equality with inequalities. For example, if we want to express that , we can capture this (albeit somewhat awkwardly) with inequalities by saying that and . By claiming that the expression on the left is simultaneously “at least as big as” and “at most as big as” the same value, the only possibility is that they are, in fact, equal.