Is xy 0? (1) |3 - x|

Question

Is xy > 0?

(1) |3 - x| < x + 5
(2) |2 - 2y| < y - 1

Aparna1995 · Answer

Is xy > 0?

(1) |3 - x| < x + 5
(2) |2 - 2y| < y - 1

Considering statement (1)
|3 - x| < x + 5

For x<-5, statement (1) will be false.
For -5<x<3, statement (1) will be true.
For x>3, statement (1) again will be false.

So statement (1) is only valid for x between -5 & 3.

Not sure if xy>0, Hence option A & D are eliminated

Considering statement (2)
|2 - 2y| < y - 1
This statement can never be true.
Option B is eliminated

Combining we're still not able to find if xy>0.
Hence, Option E is the correct answer

shaurya98 · Answer

To find: Is xy>0?
OR is the product of x and y positive?
OR are BOTH x and y positive or negative numbers?

Constraint: None, i.e. x and y can be integers, fractions, decimals etc.

Statement 1: |3 - x| < x + 5

This is an absolute value inequality with variables on both sides. Hence, on removing the Modulus symbol, we get two inequalities: (i) 3-x < x+5 and (ii) x-3 < x+5

On further simplifying to find out the range of variable x, we get
(i) x>-1 and
(ii) -3<5 (however, this doesn't tell us anything about the values of x)

Hence, from statement 1 we can say that the range of x which will satisfy the given inequality is x > -1 i.e. x can be a positive number, a negative number between 0 and -1, OR Zero.

If x = 0 then xy will be 0, which would give us a definite no for "is xy > 0?". However, if x is a negative number between 0 and -1, for instance, -0.25 or a positive number then to get a definite no for "is xy>0?" we would need information about the sign of y. Therefore, statement 1 is INSUFFICIENT.

Statement 2: |2 - 2y| < y - 1

Again, this is an absolute value inequality with variables on both sides. On solving we get two inequalities:
(i) 2-2y < y-1 and (ii) -2+2y < y-1

On further simplification, we get (i) 1 < y and (ii) y < 1
Hence, statement 2 isn't sufficient.

On further combining Statement 1 and 2, we still don't know anything about the sign of y.

Hence, answer is (E) Both statements are insufficient.

Papist · Answer

Good question! S1: Because of how absolute values works, we have to solve for two equations: 3-xx+5. We get x>-1 OR x<-1 as solutions. So x could be any number except -1. We are told nothing about y, so this is clearly insufficient. S2: Again, we have two equations: 2-2y>y-1 and 2-2y1 OR y<1 as solutions, so y could be any number except 1. We are told nothing about x, so this is clearly insufficient. S1+S2: Based on the possible values of x and y, the product could be any number, positive, negative, or 0. We cannot determine whether xy>0, so the answer is E.

Mo2men · Answer

Dear  GMATGuruNY

Can you please help with statement 2?

The left side is zero or positive and hence the right hand side must be positive .....then y-1> 0 ...y>1.......There is no number positive number that make inequality true

I tried to find solution but there is no solution at all. I got y >1 and y <1

is that allowable in GMAT test?

GMATGuruNY · Answer

As you have discovered, there is no solution for Statement 2. Statement 2: |2 - 2y| < y - 1 Since an absolute value cannot be less than 0, the right side of this inequality must be positive: y-1 > 0 y > 1 For all values greater than 1, |2-2y| = 2y-2 Substituting |2-2y|= 2y-2 into |2-2y| < y-1, we get: 2y-2 < y-1 y < 1 Since Statement 2 is valid only for values greater than 1, the resulting inequality in red is not possible. Implication: Statement 2 has no valid solutions. The problem is flawed. To my knowledge, no official problem has included a statement for which there is no solution. I would disregard this problem.

Mo2men · Answer

Thanks a lo for your support

cledgard · Answer

WRONG, Bad question. Statement (2) is not true for any value of Y; therefore, this question is flawed.

cledgard · Answer

Check your answer to statement one.
For -5<x<3, statement (1) will be true. WRONG.
Try x = -4 NOT TRUE
Try x= 4 TRUE
The answer to statement 1 is x > -1

As for statement (2), you are right, it can never be true. So this is an incorrect GMAT question. All statements must be true.

bumpbot · Answer

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Is xy > 0? (1) |3 - x| < x + 5 (2) |2 - 2y| < y - 1

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