Question

Find the number of pairs of integers (x,y)?

Find the number of integers (x,y) with `0<x, y<10` that satisfy $$\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}$$

Answer 1

`16`

Explanation:

Given:

`1/(1-10/x) > 1-5/y`

As is typical when solving inequalities, it may be helpful to look at the corresponding equation first:

`1/(1-10/x) = 1-5/y`

Adding `5/y - 1/(1-10/x)` to both sides we get:

`5/y = 1-1/(1-10/x)`

`color(white)(5/y) = ((color(red)(cancel(color(black)(1)))-10/x)-color(red)(cancel(color(black)(1))))/(1-10/x)`

`color(white)(5/y) = (-10/x)/(1-10/x)`

`color(white)(5/y) = (10/x)/(10/x-1)`

`color(white)(5/y) = 10/(10-x)`

Taking the reciprocal of both ends, we find:

`y/5 = (10-x)/10 = 1-x/10`

Multiplying both sides by `5` we get:

`y = 5-x/2`

Hence, in order to satisfy the original inequality, we need one (which one to be determined) of these:

`y < 5-x/2" "` or `" "y > 5-x/2`

To find out which one we need we could carefully analyse what has gone before, but it is probably easier to just try.

Consider `x=1` and look at `y=4` and `y=5`...

With `x=1`, we have:

`1/(1-10/x) = 1/(1-10) = -1/9`

With `y=4`, we have:

`1-5/y = 1-5/4 = -1/4`

With `y=5`, we have:

`1-5/y = 1-5/5 = 0`

So we need the condition that gives us `y=4`, namely:

`y < 5-x/2`

So for each value of `x` we find the possible values of `y` are as follows:

`x=1:" "y = 1,2,3,4`

`x=2, 3:" "y = 1,2,3`

`x=4, 5:" "y = 1,2`

`x=6, 7:" "y = 1`

giving a total of `4+6+4+2 = 16` possible pairs.