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

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

1 Answer
May 20, 2017

#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.