Is x 0? (2) x^2 = 9x (2) |x| = -x

Question

Is  x > 0 ?

(1)  x^2 = 9x 
(2)  |x| = -x

chetan2u · Answer

Hi,
 lets see the statements 
1.  x^2 = 9x ..
 x^2 - 9x = 0 ..
 x(x-9)=0 ..
so x=0 or x=9... 
Insuff

2.  |x| = -x 
 this shows -x is +ive or 0, so x has to be non positive 
ans is NO 
Suff

B

Bunuel · Answer

Notice that for (2) x can be 0 too. The rest is correct.

chetan2u · Answer

hi,
I too thought of x as 0 or -ive ..
but since we are answering "is x>0?"
Ans will be NO in both cases and went with B

robu · Answer

But X=0 is true, then can we say stmt 2 is sufficient.?

Bunuel · Answer

Yes, B is correct. I just pointed out that  |x| = -x  means that  x \leq 0 , not x < 0.

chetan2u · Answer

Yes statement 2 will be suff in both cases..
1) if x=0..
Q is " is x>0?"----ans -NO

2) x is -ive
Q is " is x>0?"----ans -NO

so both cases ans is NO

Bunuel · Answer

From (2) x can be 0 or negative. In any case answer to the question is x > 0 is NO, thus sufficient.

andreroma · Answer

Why the first option is not enough?

we have: x^2=9x if we divide both sides for x we have x=9 .

Bunuel · Answer

Please read  this solution . You'll see that TWO solutions satisfy x^2=9x: x= 0 and x = 9. You cannot divide both parts by x because x can be 0 and you cannot divide by 0.

ashikaverma13 · Answer

Hi,

I have a difficulty understanding the second statement. Anything inside mod is positive right so how is that equal to a negative number? Can you please elaborate on this?

amanvermagmat · Answer

Hi

Anything inside mod is NOT necessarily positive. However, anything that comes out of mod (as a result) is either positive or zero. 
Please consider the following:

1) Let x = 5. Here |x| = 5. So the number inside mod is positive, and the result is also positive. We can see here that |x| = x only.

2) Let x = 0. Here |x| = 0. So the number inside mod is zero, and the result is also zero. We can see here that |x| = x, OR we can also say that |x| = -x, because putting a negative sign before 0 also is 0 only.

3) Let x = -4. Here |x| = 4. Here the number inside mod is negative, BUT the result is positive, which is actually negative of that negative number.
I mean -(-4) = 4. We can see here that |x| = 4 = -(-4) = -x.

So if we are given that |x| = -x, it could mean two things: Either x is 0, Or x is negative. That is what Bunuel wrote in his comment.

tamal99 · Answer

Hello  chetan2u ,
I couldnt understand the first condition, please explain why we cant consider 9X>0, as x^2 will always be +ve.

Regards,
Tamal

fskilnik · Answer

x\,\,\mathop > \limits^? \,\,0

\left( 1 \right)\,\,{x^2} = 9x\,\,\,\, \Leftrightarrow \,\,\,\,x\left( {x - 9} \right) = 0\,\,\,\, \Leftrightarrow \,\,\,\,\left\{ \matrix{
 \,x = 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr 
 \,\,\,{\rm{OR}} \hfill \cr 
 \,x = 9\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.

\left( 2 \right)\,\,\,\left| x \right| = - x\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \le 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

amanvermagmat · Answer

Hello

Its NOT necessary that x^2 will be positive only, x^2 can be 0 also (if x is 0). So we have to consider that possibility also for first statement.

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Is x > 0? (2) x^2 = 9x (2) |x| = -x

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