In the rectangular coordinate system which quadrant, if any, contains no point ( x, y ) that satisfies the inequality 2x âˆ’ 3y â‰¤ âˆ’ 6 ?
(A) None
(B) Î™
(C) Î™I
(D) Î™II
(E) IV
OA is E.
My Response to the ques:
Assuming x=0, then 3y<= 6 or y>= 2.
Assuming y=0, x<= 3.
Therefore a line with coordinates (X<= 3, Y>=2) should be located in quadrant 2 only.
Having done this much, I got stuck. I did not really comprehend what the question stem meant.
PS Coordinate Geometry
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2x  3y <= 6
x = 0==> 3y <= 6 ==> y>=2
y = 0==> 2x <= 6 ==> x <= 3
Since, any line drawn using the above points will result lines passing 1st, 2nd & 3rd quadrant.
So, Answer {E}
x = 0==> 3y <= 6 ==> y>=2
y = 0==> 2x <= 6 ==> x <= 3
Since, any line drawn using the above points will result lines passing 1st, 2nd & 3rd quadrant.
So, Answer {E}
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Let us first trace the region depicted by 2x âˆ’ 3y â‰¤ âˆ’ 6pareekbharat86 wrote:In the rectangular coordinate system which quadrant, if any, contains no point ( x, y ) that satisfies the inequality 2x âˆ’ 3y â‰¤ âˆ’ 6 ?
(A) None
(B) Î™
(C) Î™I
(D) Î™II
(E) IV
2x âˆ’ 3y â‰¤ âˆ’ 6
âˆ’3y â‰¤ 2x âˆ’ 6
3y â‰¥ 2x + 6 ... multiplying both sides by (1)
y â‰¥ (2/3)x + 2
Consider the line y = (2/3)x + 2
Its y intercept = 2 (put x = 0 in the above equation)
Its x intercept = 3 (put y = 0 in the above equation)
We need a region which is greater than y = (2/3)x + 2
(since the original equation states greater than equal to...)
Original question: which quadrant contains no point that satisfies the inequality ...
The points in the 4th quadrant will not be a part of the equation y â‰¥ (2/3)x + 2 and thus the answer is E
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For those who are unfamiliar with the location of the quadrants in the xy coordinate plane, they are shown herepareekbharat86 wrote:In the rectangular coordinate system which quadrant, if any, contains no point ( x, y ) that satisfies the inequality 2x âˆ’ 3y â‰¤ âˆ’6 ?
(A) None
(B) Î™
(C) Î™I
(D) Î™II
(E) IV
Another approach here is the recognize how the coordinates of points look in each quadrant.
Quadrant I: (positive, positive)
Quadrant II: (negative, positive)
Quadrant III: (negative, negative)
Quadrant IV: (positive, negative)
From here, we can check each quadrant.
For example, let's see if it's possible for a point in QUADRANT I (where the x and y coordinates are both positive) to satisfy the inequality 2x âˆ’ 3y â‰¤ âˆ’6
Well, how about x = 1 and y = 10.
When we plug these in, we get 2(1) âˆ’ 3(10) â‰¤ âˆ’6
Simplify to get 28 â‰¤ âˆ’6 [PERFECT it works]
ELIMINATE B
Is it possible for a point in QUADRANT II (where the x is negative and y is positive) to satisfy the inequality 2x âˆ’ 3y â‰¤ âˆ’6?
Well, how about x = 10 and y = 10.
When we plug these in, we get 2(10) âˆ’ 3(10) â‰¤ âˆ’6
Simplify to get 50 â‰¤ âˆ’6 [PERFECT it works]
ELIMINATE C
Is it possible for a point in QUADRANT III (where the x is negative and y is negative) to satisfy the inequality 2x âˆ’ 3y â‰¤ âˆ’6?
Well, how about x = 10 and y = 1.
When we plug these in, we get 2(10) âˆ’ 3(1) â‰¤ âˆ’6
Simplify to get 17 â‰¤ âˆ’6 [PERFECT it works]
ELIMINATE D
Is it possible for a point in QUADRANT IV (where the x is positive and y is negative) to satisfy the inequality 2x âˆ’ 3y â‰¤ âˆ’6?
NO.
When we plug these in, we get 2(positive) âˆ’ 3(negative) â‰¤ âˆ’6
Simplify to get positive  negative â‰¤ âˆ’6
Since a positive number minus a negative number will always be positive, we get . . .
POSITIVE â‰¤ âˆ’6
Since this is IMPOSSIBLE, the correct answer must be E
Cheers,
Brent