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# Find (x,y)..

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

Feb 3, 2019

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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

Feb 3, 2019
edited by CPhill  Feb 3, 2019
edited by CPhill  Feb 3, 2019
edited by CPhill  Feb 3, 2019