If у + |y| = 0, which of the following must be true?

Question

If y + |y| = 0 which of the following must be true?

A. y > 0
B. y ≥ 0
C. y < 0
D. y ≤ 0
E. y = 0

Bunuel · Accepted Answer

Absolute value properties: 
When  x\leq{0}  then  |x|=-x , or more generally when  some \ expression\leq{0}  then  |some \ expression|={-(some \ expression)} . For example:  |-5|=5=-(-5) ;

When  x\geq{0}  then  |x|=x , or more generally when  some \ expression\geq{0}  then  |some \ expression|={some \ expression} . For example:  |5|=5 ;

So,  y+|y|=0  -->  |y|=-y , which means that  y\leq{0} .

Answer: D.

As for your doubt: question asks which of the following MUST be true, not COULD be true. Since all negative values of y satisfy  |y|=-y  then it's not necessarily true that  y=0 .

Hope it's clear.

Bunuel · Answer

y+|y| = 0  -->  |y|=-y . Now, LHS is an absolute value which is never negative hence RHS must also be non-negative:  -y\geq{0}  -->  y\leq{0} .

Or you can see right away that in order  y+|y|=0  to hold true y must be either negative number (for example -3+|-3|=0) or zero (0+|0|=0).

Answer: D.

Bunuel · Answer

y+|y|=0  means either  y=0  or  y<0  (in this case  |y|=-y  and  y + |y| = 0  becomes  y-y= 0 ). So must be true that  y\leq{0} .

Answer: D.

pavanpuneet · Answer

I have a question here.. if the the question was y + |y| = 2y , then can we say y>=0? given then 0+0 = 2(0). Please let me know in case I am doing something wrong. Thanks in advance.

Bunuel · Answer

y+|y|=2y  -->  |y|=y  -->  y\geq{0} .

Ousmane · Answer

For me the answer is C. (since the question asks MUST be true?)
Can you anyone tell me why C is not correct

Bunuel · Answer

Check the solution here: if-y-y-0-which-of-the-following-must-be-true-131099.html#p1076758 y<0 (C) is not correct because y+|y|=0 hods true when y=0 too, so y<0 is not necessarily true (not a must true statement). Hope it's clear.

EMPOWERgmatRichC · Answer

Hi All,

This question is perfect for TESTing VALUES.

We're told that Y + |Y| = 0. We're asked which of the following MUST be true (which means "which of the following is ALWAYS TRUE no matter how many different examples we can come up with?")

The obvious value for Y is 0 (since 0 + |0| = 0)
Eliminate A and C.

Since there are still 3 values remaining, there MUST be other options to consider....

IF....
Y = -1
-1 + |-1| = 0
Eliminate B and E.

There's only one answer left....

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

generis · Answer

y + |y| = 0 
 |y| = 0 - y 
 |y| = -y

The last expression means that  y\leq{0} . That rule can seem odd or counterintuitive.

The variable has a "hidden" negative sign. With the variable, it's hard to remember that there ARE two negative signs on RHS. We just do not (cannot) write the minus sign twice with the variable. These equations are equivalent, where y = -2:

|-2| = -(-2) = 2
|y| = -(y) = -y

So if  y + |y| = 0 , then  |y| = -y  and

y\leq{0}

Answer D

If none of the above occurs to you or if it makes no sense, pick and list three numbers: negative, 0, and positive.

Use them to try to DISPROVE the answers. Even one example that defies the rule being tested makes "must be true" false.

-2, 0, and 2

A. y > 0
 y + |y| = 0 . Try y = 0
 0 + |0| = 0 . That works.  y  does not have to be positive. REJECT

B. y≥0. Use -2
 y + |y| = 0 
 -2 + |-2| = 0 . That works.  y  can be negative. REJECT

C. y < 0. We know from (A) that  y  CAN equal 0. REJECT

D. y≤0. Try 2 
 y + |y| = 0 
 2 + |2| 
eq{0}

We know from (A) that  y  can equal 0. 
We know from (B) that  y  can be negative.

And having tested +2, we know that  y  CANNOT be positive.

This expression MUST be true. KEEP

E. y = 0
We know from (B) that  y  can be negative. Yes,  y   can  also be 0. But it does not  have  to be 0 -- it can be negative, e.g. -2. REJECT

Answer D

BrentGMATPrepNow · Answer

STRATEGY : Upon reading any GMAT Problem Solving question, we should always ask,   Can I use the answer choices to my advantage?   
In this case, we can quickly find solutions to the given equation, and then use those solutions to test the answer choices. 
Now let's give ourselves up to 20 seconds to identify a faster approach. 
I don't see a faster approach. So let's start testing . . .

If  y + |y| = 0 , we can see that it could be the case that   y = 0  . 
  More strategy: When looking for solutions to a given equation or inequality, it's useful to first check whether 0 is a solution, since zero is such an easy value to work with.

Now we’ll plug   y = 0   into the five answer choices to see which one(s) is/are true: 
A.   0   > 0. Not true. Eliminate. 
B.   0   ≥ 0. True. Keep.
C.   0   < 0. Not true. Eliminate.
D.   0   ≤ 0. True. Keep.
E.   0   = 0. True. Keep.

We’ve already reduced the options to B, D and E. 
Now we’ll test another y-value that satisfies the equation  y + |y| = 0 
We can see that   y = -1   is another possible solution. So, we’ll plug   y = -1   into the remaining three answer choices: 
B.   -1   ≥ 0. Not true. Eliminate.
D.   -1   ≤ 0. True. Keep.
E.   -1   = 0. Not true. Eliminate.

By the process of elimination, the correct answer is D.

Sourav8888 · Answer

stem is y + |y| = 0, we know IYI = -Y if Y <0 & IYI = Y if Y >0

1. if IYI = -Y, then Y + (-Y) = 0 (condition satisfied). so, Y<0
2. if IYI = -Y, then Y + Y = 0 which leads to Y = 0.

So the answer is Y ≤0

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If у + |y| = 0, which of the following must be true?

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If у + |y| = 0, which of the following must be true?

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