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

Guest Feb 3, 2019

#1**+1 **

We can simplify this as

x / [ x - 10 ] > [ y - 5] / y

Since 0 < x, y < 10.....the left side will always be negative

And the right side will be negative when y < 5

So when x = 1

The possible integer values for y are 1, 2, 3 and 4

When x = 2

The possible integer values are 1, 2, 3

When x = 3

The possible integer values of y are 1,2,3

When x = 4

The possible integer values for y are 1, 2

When x = 5

The possible integer values for y are 1,2

When x = 6

The possible values for y are 1

When x = 7

The possible values for y are 1

When x = 8...there is no value of y that makes the inequality true

So....there are 16 pairs of (x, y) that make the inequality true

CPhill Feb 3, 2019