Which Quadrant?

[iikarthik]

In a rectangular Co-ordinate system,which Quadrant if any,contains no point (x,y) that satisfies the inequality 2x-3y<= -6?

A.I Quadrant
B.II Quadrant
C.III Quadrant
D.IV Quadrant
E.None

The OS is: [spoiler]D..I tried to substitute values from each quadrant in the equation eg: 1,1 ; 1,-1 but this didn't work.[/spoiler]
Thanks
karthik

[albatross86]

I feel the easiest way to solve this problem is to simply plot the border line of the inequality in a rough graph.

i.e. Draw 2x - 3y = -6 on a graph. This has the intercepts (-3,0) and (0,2)

To find which direction this region is pointing to, simply test the origin (0,0)

2x - 3y would then be = 0 which is GREATER than -6, and hence does not satisfy our inequality. Thus the direction is away from the origin, or in other words are region is the half that does not contain the origin.

Please see the attached image for how to plot it. You can do it rough once you have your intercepts.

Thus, we can see that Quadrant IV does not contain any point that would satisfy our inequality.

[ankurmit]

You can convert this equation in form of a line as:

-3y<= -2x-6

3y>= 2x+6

y>= (2/3)x + 2

this line has positive slope with y intercept as 2

so this line will never pass through 4th quadrant..

got it?

Figure will be same as shown above by abhay

[gmatsensei]

I feel the easiest way to solve this problem is to simply plot the border line of the inequality in a rough graph.

i.e. Draw 2x - 3y = -6 on a graph. This has the intercepts (-3,0) and (0,2)

To find which direction this region is pointing to, simply test the origin (0,0)

2x - 3y would then be = 0 which is GREATER than -6, and hence does not satisfy our inequality. Thus the direction is away from the origin, or in other words are region is the half that does not contain the origin.

Please see the attached image for how to plot it. You can do it rough once you have your intercepts.

Thus, we can see that Quadrant IV does not contain any point that would satisfy our inequality.

Fantastic work. I always recommend the 'graphical' approach for such problems. Obtaining the x and y intercepts will only take a few seconds. Using these, you can plot the line and bingo! The other upside to this method is that you won't have the nagging doubt as to whether you solved the inequality correctly