Find the number of pairs of integers (x,y)?
Find the number of integers (x,y) with #0<x, y<10# that satisfy $$\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}$$
Find the number of integers (x,y) with
1 Answer
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/(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
#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
With
#1/(1-10/x) = 1/(1-10) = -1/9#
With
#1-5/y = 1-5/4 = -1/4#
With
#1-5/y = 1-5/5 = 0#
So we need the condition that gives us
#y < 5-x/2#
So for each value of
#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