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

May 2, 2020
edited by littlemixfan  May 2, 2020
edited by littlemixfan  May 2, 2020
edited by littlemixfan  May 2, 2020
edited by littlemixfan  May 2, 2020

#1
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Can you try and do the LaTex again, please...

May 2, 2020
#2
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sorry that was my bad, the latex didn't save when i published it :(

littlemixfan  May 2, 2020
#3
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I'm very sorry I haven't learned this...

I asked some people do but they are eating dinner of AFK

LuckyDucky  May 2, 2020
#4
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oh, no problem! ill just wait and ask other people. thanks you your help though!

littlemixfan  May 2, 2020
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Hi Littlemixfan,

Find the number of pairs of integers (x,y) with x is less than 0, y is less than 10, that satisfy

$$\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}$$

First I want to look at formula restrictions.

$$y\ne0\\ x\ne0\\ x\ne10$$

We also have the question restriction of

$$x<0\qquad and \qquad y<10$$

So where can it be so far:

NOW

$$\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}\\ \frac{1}{\frac{x-10}{x}} > 1 - \frac{5}{y}\\ \frac{x}{x-10} > \frac{x-10}{x-10} - \frac{5}{y}\\ \frac{x-(x-10)}{x-10} > - \frac{5}{y}\\ \frac{10}{x-10} > - \frac{5}{y}\\ \frac{x-10} {10}< - \frac{y}{5}\\ - \frac{y}{5}>\frac{x-10} {10} \\ \frac{y}{5}<\frac{-x+10} {10} \\ \frac{y}{1}<\frac{-x+10} {2} \\ y<-0.5x+5$$

So that is underneath the line y=-0.5x+5

NOTE: Maybe I made an error but I do not think so. Maybe it was meant to be a   <   sign.

Also,:

I left the > sign in for all my working but it would have been easier just to solve for = and then test the regions afterwards.

So far I have these 2 regions.  But since there are 2 regions one where y<0 and one where y>0 I will test both.

Test  (-1,+1) and (-1,-1)

$$\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}\\test\;(-1,1)\\ \frac{1}{11}>-4\quad true \\ test\;(-1,-1)\\ \frac{1}{11}>6\quad false \\$$

So here is the region:

I think there is an infinite number of integer pair solutions.

LaTex:

\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}\\
\frac{1}{\frac{x-10}{x}} > 1 - \frac{5}{y}\\
\frac{x}{x-10} > \frac{x-10}{x-10} - \frac{5}{y}\\
\frac{x-(x-10)}{x-10} >  - \frac{5}{y}\\
\frac{10}{x-10} >  - \frac{5}{y}\\
\frac{x-10} {10}<  - \frac{y}{5}\\
- \frac{y}{5}>\frac{x-10} {10} \\

\frac{y}{5}<\frac{-x+10} {10} \\

\frac{y}{1}<\frac{-x+10} {2} \\

y<-0.5x+5

May 2, 2020
#6
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thank you so much melody!

littlemixfan  May 10, 2020