GMATCHPOINT wrote:From OG11
(picture of the coordinate system and its 4 quadrants I, II, III and IV)
In the rectangular coordinate system shown above, with quadrant, if any, contains no point (x, y) that satisfies the inequality 2x-3y<= -6?
A) None
B) I
C) II
D) III
E) IV

I can go up to y>=2/3x+2 with no problem. What should I do from here? Randomly substitute values of x into this inequality to determine coordinates for plotting the line?!? Is this the only way to solve it?

First, plot the line y = 2/3(x) + 2

This inequality is >=, so every point on or above the line satisfies the inequality.

As you can see from your diagram, the only sector without any points on or above the line is the bottom right (sector IV) - so that's the correct answer.

We also could have very quickly eliminated 3/5 answers.

Since it's a >= inequality, there will always be points in the top two sectors (I and II) that are on or above the line (since it's not a vertical line), so you can quickly eliminate (B) and (C).

The line has a positive slope, so it definitely passes through the bottom left sector (lines with positive slope must pass through sectors I and III; lines with negative slope must pass through sectors II and IV) - we can also eliminate (D).

So, the only 2 possible answers are "(A) none" and "(E) IV". All we have to really determine is whether the line passes through the bottom right sector (in this case it doesn't, so choose (E)).

SUPER!

thanks Stuart, this was excellent explanation