Blog: What is a Compound Inequality?

This one is for my students ... should they ever come by here.

Everything in Arithmetic seemed to be about finding the answer. Let me add emphasis to that: THE Answer. Then along came Algebra, and, suddenly, THE Answer wasn't as important anymore. Don't get me wrong -- it was still important. However, how we got the answer and why it was correct seemed to matter more. Just writing, for example, "5" on the paper in that big, empty space wasn't good enough. Even if it was "obvious" that it was 5. Why? Because the next problem might not be so "obvious", so we still needed to know the rules so we could attack the next one, and the one after that, and so on, as they got more complicated.

But even as we showed our work and checked our answer, we knew one thing for sure: There are AN answer. One. Singular. The value that makes the equation True.

Until inequalities came along. Why did we even start that chapter? How could there be problems with not only more than one answer, but an infinite number of them, an entire range of values, shooting off into infinity. The answer isn't seven, it's greater than seven. Does that mean it's eight? Well, yes, but it's also nine, ten, eleven, twenty-seven, thirty-one and a half, the square root of 92, a googol (not a search engine). It's all those real numbers. The ones bigger than seven.

Okay, so equations have one answer (or maybe two?? -- what do you mean, "we'll talk about that?"), and inequalities have arrows that point to the left or the right and go on forever. That's it, right?

Weeeeeellllllll . . . .

Do you know how in English class (or ELA or whatever), you can have compound sentences, which are two sentences joined together by a conjunction. (Cue: Schoolhouse Rock's "Conjunction Junction".) Those conjunctions are "AND", "BUT", and "OR". In math, we can have compound inequalities, and they can be joined by "AND" or "OR". What about "BUT"? Here's a secret for you: "AND" and "BUT" mean the same thing:

I went to the store, and it was closed.

I went to the store, but it was closed.

So we don't use "BUT" in Algebra. ("What about when talking about someone's face?" "That's just wrong, Gordo." -- and now because it's rude, so much as that would be misspelled.) Likewise, "OR" isn't a conjunction in math. It's something called a disjunction, and we'll address that later. One topic per day, please.

If you wanted to get a "B" on your report card, you would need to score AT LEAST 80 and LESS THAN 90. (Exactly 90 or higher would be an "A", and while that would be great, let's be realistic here: A's are tough to get. But aim high.)

This is an example of a compound inequality. If we were to graph all the averages that result in a grade of B, there wouldn't be an arrow. Sure, x > 80 would have a closed circle above 80 and an arrow pointing to the right. And x < 90 would have an open circle and an arrow pointing to the left. However, the "AND" in the condition tells us that we only want the points that make both true, the area that they overlap. So we wind up with a line segment with one closed endpoint and one open endpoint, representing a range of numbers -- still infinite! -- that are solutions to the inequality x > 80 AND x < 90.

One last thing. Compound inequalities with AND can be written without the AND. Look at the endpoints. The minimum number (80) has to be less than or equal to the actual average (the variable) and that has to be less than the maximum score (90), so we could write it as 80 < x < 90.

Confused yet? Don't worry, you'll get the hang of it.