Analyzing structure with linear inequalities: fruits | High School Math | Khan Academy

Shantanu bought more apples than bananas, and he bought more bananas than cantaloupes. Let A represent the number of apples Shantanu bought, let B represent the number of bananas, and let C represent the number of cantaloupes.

Let's compare the expressions B plus C and A. Which statement is correct? Is B plus C greater than A? Is it less than A, or are these two quantities equal? Or is there not enough information to tell? So, like always, pause this video and see if you can work through it on your own. Now, I will work through it with you.

All right, so let's just write down the information that they gave us. They say let A represent the number—oh, well that's more straightforward: A for apple, B for banana, C for cantaloupe. Here we have more apples than bananas, so A is greater than B. Then they also tell us he bought more bananas than cantaloupes, so B is greater than C. We could rewrite that as A is greater than B, which is greater than C. Or we could write that as C is less than B, which is less than A. This is essentially the information that they give us.

So, let's see which of these is going to be true: Is B plus C greater than A? Is B plus C less than A? One thing that we can try is to plug in some values, some numbers, to see if we can get combinations that are consistently in one of these buckets, or if they fall into multiple of these choices. Then we would say, "Hey, there's not enough information to tell."

In general, this is a good strategy for things like this, where we're dealing with very abstract quantities. So, let's make a little table here: A, B, C, and then I can also figure out what B plus C is. This is going to be A, B, C, and this is B plus C, and we can compare that to A.

So, let's see a situation where—we'll see if we can make B plus C greater than A. They both have to be less than A. Let's see if C is 5 and B is 6, and let's make A 11. In this situation, B plus C is going to be equal to 11. So, we're able to find a situation where if B plus C are close enough to A, then B plus C is going to be greater than A.

Now, let's see if we can figure out a scenario where B plus C is less than A. We could do the same: B plus C equals 6 plus 5. We can make A bigger than 6 plus 5; we can make A 12. And now this is a situation.

So, in the first situation, we have B plus C is greater than A. In the second situation, we have B plus C less than A. Depending on what your A, B, and C are that meet these constraints, notice both of these situations meet all the constraints where A is greater than B, which is greater than C. But it could be either one of these, so that immediately tells us that there is not enough information to tell.

Now, one thing that we—yeah, there's just not enough information to tell. I can even come up with a scenario where B plus C is equal to A. If it's 6, 5, and 11, then B plus C equals A. So based on the information they gave us, any of these are actually possible; thus, there's not enough information to tell.