Analyzing structure with linear inequalities: fruits

- [Instructor] 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? So 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 and 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, 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, and then they also tell us he bought more bananas than cantaloupes, so B is greater than C, or we could rewrite that as A is greater than B is greater than C or we could write that as C is less than B 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? B plus C greater than A, B plus C greater than A, B plus C less than A. So one thing that we can try is let's try 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 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. So A, B, C and then I can also figure out what B plus C is. So 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 let's see if we can make B plus C greater than A. So they both have to be less than A. So let's see, if C is five and B is six and let's make A seven. So 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, that B plus C is going to be greater than A. So we're able to find this scenario. Let's see if we can figure out a scenario where B plus C is less than A. Well if, well we could do the same B plus C, six, five. We could make A bigger than six plus five. We can make A 12. And now this is a situation, so this first situation we have B plus C is greater than A. The second situation right over here, you have B plus C is less than A and so depending on what your A, B, and C's are that meet these constraints and notice, both of these situations I meet all the constraints where A is greater than B 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 six, five, and 11. Then B plus C is equal to A. So based on the information they gave us, any of these are actually possible. So there's not enough information to tell.