we have that and are integers for . furthermore, we are told that:2.) a and b are two integers, and a>b. The sum of a and b is 5, but their product is less than 1. What can you say about a and b?
What I found is:
a is greater than b
a+b=5 , but when the multiply, it is less than 1
I dont know what to do next...i tried forming numbers that is valid for the description above..but failed..can anyone help me..
what can we say about and ?
i am not exactly what you are looking for here, but notice that if both and are positive. then we could get their sum to be 5, but after that, getting their product to be less than 1 is impossible. after all, there are only a few possibilities with this scenario:
(1) a = 4, b = 1 ...........ab > 1
(2) a = 3, b = 2 ...........ab > 1
we cannot switch a and b since we must have a>b, and so, only these two possibilities work for a and b positive and in both cases, ab > 1. so this cannot be true
okay, can a and b both be negative then? obviously not, we can't add two negatives and get a positive, so that won't work.
what else is there? a and b are different signs. since a > b, a must be positive and b negative. can that work? yes! as long as a = |b| + 5, we can have the desired result.
example. a = 8, b = -3. then a + b = 8 - 3 = 5, and ab = -24 < 1.
Bingo! so this is a possible relationship between a and b. so we can say this about a and b, a can be positive, but if so, b must be negative, with a = |b| + 5
is there anything else we can say? sure, we didn't consider the integer zero. what if one is zero. to add and get 5 the other must be 5. obviously we must have a = 5 and b = 0 since a>b. and so we have a + b = 5 + 0 = 5, and ab = 5(0) = 0 < 1. Bingo!
so finally, here is what we can say about a and b. given these conditions, we must have one of two cases for all the conditions to be fulfilled.
(1) a = 5, b = 0
(2) a > 0, b < 0 with a = |b| + 5
and we're done
did you follow the thought process ok? of course you would not write all this out, but you should be thinking somewhere along these lines to answer the question