Co-ordinate + Absolute

[hey_thr67]

In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|


OA is D

How should I approach absolute questions, I took lot of time to conclude that -y < |y| is only true for y>0. Also what values to put confuses me.

[sam2304]

In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|


OA is D

How should I approach absolute questions, I took lot of time to conclude that -y < |y| is only true for y>0. Also what values to put confuses me.

4 quadrats ++ +- -+ --
1.|xy| + x|y| + |x|y + xy > 0
++ -> easy go satisfies the inequality
+- -> Will result in two -xy and two +xy = 0. Does not satisfy the inequality
-+ -> Same as above
-- -> Middle two will be -xy and end values will +xy, sum = 0

So point lies in Quadrant I. SUFF

(2) -x < -y < |y|

Break the inequality. -y < |y|, this will get satisfied only when y > 0, since -ve values will have both sides equal.
-x < -y => x > y => Multiplied by -1.
Now y > 0, and x > y so both are +ve. SUFF

IMO D.

Inequalities take a lot of time if you substitute values and check. Try approaching with general +ve, -ve values i.e +ve will satisfy this, -ve values will not. I took lot of time to conclude that -y < |y| => Your query about this would also be easier if you had considered +ve or -ve value category of numbers. Test for values < 0, =0 and >0. Nothing more than that is needed. Do not substitute individual values and check. Always remember that when you have 0 in the inequality and no other numbers it tests for +ve -ve values as a whole and substituting individual numbers will slow you down.

[Anurag@Gurome]

In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|

Statement 1:
--> |xy| + x|y| + |x|y + xy > 0
--> |y|(|x|X + x) + y(|x| + x) > 0
--> (|x| + x)(|y| + y) > 0

Hence, the signs of (|x| + x) and (|y| + y) are same.
Now, if x and y are less than or equal to zero, then |x| = -x and |y| = -y, i.e. (|x| + x) = (|y| + y) = 0

Hence, neither x nor y is negative or zero.
Therefore, both x and y are positive, i.e. (x, y) lies in the first quadrant.

Sufficient

Statement 2: -x < -y < |y|
For y > 0, |y| = y > -y
For y < 0, |y| = -y

Hence, y is positive.

Now, -x < -y ---> x > y
Hence, x is positive too.

Therefore, (x, y) lies in the first quadrant.

Sufficient

The correct answer is D.

[dhonu121]

In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|

From 1: If we just mentally create a picture by taking either x negative or y negative or both negative, we see that in all the three cases, two elements are positive and two are negative , all of which can add up to either positive or negative.
hence x and y got to be both positive.

From 2: -x<-y which boils down to x>y. and -y<|y| boils down to y>0. and both thse make x>y>0.

Thus both 1 and 2 are suff.
Hence D.