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# Reasoning about unknown variables

CCSS.Math:

## Video transcript

you go for a job interview and the first thing that your interviewer says like look you have great work experience you seem like a nice young person but what I really care about is your logical reasoning capabilities so what she says just sit down I'm gonna ask you a question about some math expressions and you say sure shoot away give me some questions about math expressions and she says all right so you have two integers integer a and integer B and she tells you that integer a is greater than 0 and integer B is less than 0 then she says that we also know that a over B is greater than is greater than a times B so then she says tell me some interesting things about a and B and a over B and a times B and you say ok I'll give my best shot and I encourage you to actually pause now and try to think about how these things might relate or how B might be constrained or a might be constrained and then unpause it so I assume you've you've unpause it so let's think about this a little bit so the first thing we know that a is positive B is negative so if I take a positive divided by a negative what am I going to get well this right over here a positive divided by negative is going to be a negative what happens if I take a positive times a negative well a B is also going to be a negative a positive times a negative is also a negative so really what we're saying is that we have two negative quantities and this one right over here is greater than that one so let's think let's visualize that on a number line let's visualize that on a number line so let's say that this right over here is zero this right over here is the positive direction this is the negative direction we know both of these are negative but a over B is larger so a over B is going to be to the right of a times B so we know that a over B a over B it has to be negative has to be to the left of 0 a over B is going to be to the right of a times B to the right of a a times B so one way to think about it is they're both negative but a over B is going to be less negative or another way of thinking about it it's going to have a smaller absolute value its distance from zero which is another way of thinking about absolute value its distance to the left of zero is less than a times B's distance to the left of zero but because they're both negative and we're talking about distances from the left of zero the one that has a smaller distance to the left of zero is less negative and is therefore a greater is a greater number so so far your interviewer seems impressed so that was pretty good you were able to get a lot of clues just out of this little bit of information that I gave you but tell me more tell me tell me more about what what B has to be and whether it's whether we can constrain it in some way and you say well okay got some clues here the fact that this absolute value right over here is going to be less than this absolute value we know that the absolute value of a over B is going to be less than a times B then the absolute value of a times B once again it's going to be a lot it's going to be not as far left of 0 as a times B is so that's how we can make that statement but let's actually manipulate algebraically manipulate this this inequality here we could multiply both sides times B so let's do that let's multiply both sides times B B is less than 0 B is less than 0 so if you're multiplying both sides of an inequality by something less than 0 it swaps the inequality so let me that's why I got rid of it I'm going to rewrite it so B times a over B is going to be we're going to it's negative we know it's negative so it's going to be less than a B times B if we just multiply it out these cancel out we get a is less than a times B squared well now if we want to simplify it we can divide both sides by a and since a is greater than 0 does not change the inequality so we divide both sides by a this becomes 1 and this just becomes B squared so we're left with 1 is less than is less than B squared or another way we could think about if we think we could say well if one is less than B squared that means that B that let's be careful here and that means that the absolute value of B is going to be greater than one so this means that the absolute value of B is greater than one you say wait Sal how did you how did you get from that to that right over there well just just think about this a little bit if I square something and if it's greater than one so that means that either B is less than negative one because if it was negative one if you squared it you would get one if it was greater than negative one or just directly greater than negative one if it was negative 0.99 then when you square it it would be also it would be something less than one so that wasn't work so B has to be less than negative one or B has to be greater greater than one and because for the same exact logic if it was exactly equal to one if you squared it this would be equal if it was 0.5 then of a B squared would be 0.25 which wouldn't be greater than so you know both of these things are true and this this this is another way of saying that the absolute value of B is greater than one now we have our other constraint that B is less than zero so since we know B is less than zero we can take this out of the picture and if the absolute value of B is greater than 1 and B is less than zero then we know that B has to be less than B has to be less than negative one so wherever negative one is B is going to be less than that and your interviewers very impressed and she says look you've done very good reasoning here I think you deserve the job