Knowing that 6<x<7 and 10<y<12 find all possible values of: y-x

Answer 1

Since the intervals for x and y are given as inequalities, we can expect the answer to also be an inequality. So we ask, what is the largest value it can be?

Wouldn't that be the smallest x value subtracted from the largest y value? And, perhaps for a moment, we can think about it as when y is 12 and x is 6 (even though we really know that y cannot be 12 and x cannot be 6; thinking of y = 12 and x = 6 allows us to find the upper boundary of y - x). So the upper boundary is 6.

And wouldn't the lowest value be when y is smallest and x is biggest, so 10 - 7 or 3.

So 3 < (y-x) < 6

Answer 2

Let's start with the maximum possible value. To get the largest possible number from subtraction, we need the 2 numbers to be the furthest away that is possible. So here we would want to take the largest possible y value and the smallest possible x value to get the biggest possible difference. The top limit for y is 12 and the bottom limit for x is 6. If we subtract 12-6, we get 6. This must be the upper limit of y-x.

Now let's deal wit the smallest possible value. Here we are looking for the numbers closest together. This would be the smallest possible y value and the largest possible x value. The bottom limit for y is 10 and the top limit for x is 7. If we subtract 10-7, we get 3. This must be the lower limit of y-x. This means y-x must be between 6 and 3. So now we can write these to limits as:

3<y-x<6