Graphing inequality question (1 Viewer)

[Chris100]

Please help me approach this question

Graph the region 1/x>1/y

 

[HeroicPandas]

looks hard...

did u know that x,y =/= 0? (1/0 is undefined)

cross multiply: y>x

 

[QZP]

looks hard...

did u know that x,y =/= 0? (1/0 is undefined)

cross multiply: y>x

Not that simple Counter example: x = 1, y = -1 is a solution to 1/x > 1/y but not y > x.

 

[Chris100]

Hopefully someone will get it and share their working

 

[QZP]

I'd do it by considering cases but that seems rudimentary...

 

[braintic]

I believe the answer is (excluding the axes): all of the 4th quadrant, and those parts of the 1st and 3rd quadrants above y=x (and none of the 2nd quadrant).

Perhaps someone can confirm that - if anyone wants working, I'll do it after I've had dinner.

 

[Drongoski]

I just finished watching Rick Stein's curry programme from India. Worked on this question. Got the same results as you, braintic.

I wish I knew how to show the diagram. But, as you have indicated: excluding the x and y axes, and the line y=x, the triangular parts in quadrants 1 and 3, above the line, y = x, and the whole of quadrant 4, and none in quadrant 2.

 

[QZP]

Care to post solution?

 

[Drongoski]

(which holds if & only if both top & bottom have same sign)

For xy > 0(1st & 3rd quadrants, excluding the axes)

we need: y-x > 0, i.e. y > x which is the part of the half-plane above the line y = x. Therefore the regions here are the 2 triangular regions in quadrants 1 and 3 above the line y = x.

For xy < 0 (2nd and 4th quadrants, excluding the axes)

we need: y - x < 0, i.e. y < x, i.e. the half-plane under the line y = x. The region so defined is therefore the open 4th quadrant, but nothing in quadrant-2.

 

[braintic]

Solution:
View attachment Inequ.pdf

 

[Chris100]

^ thanks a lot guys

Btw drongoski, how do you type up the questions with the vincilum added and everything

 

[sleepyair]

Braintic, there's a typo in your solution. In 4th quad, x>0 and y<0

 

[Drongoski]

^ thanks a lot guys

Btw drongoski, how do you type up the questions with the vincilum added and everything

I used LaTeX. Many on this forum are very good with various software for mathematical typesetting, graphics, diagrams etc. I'm pretty hopeless in these.

 

[braintic]

Braintic, there's a typo in your solution. In 4th quad, x>0 and y<0

I probably didn't edit my copy and paste. But I don't really feel like fixing this up now.

 

[sleepyair]

Just pointing it out for others, that's all