Square root of x^2

[infiniti007]

What is the value of x?

1.) √x^4 = 9
2.) √x^2 = -x

[Matt@VeritasPrep]

One useful identity: √x² = |x|, for any value of x.

(You can see this if you plug in a few numbers. For instance, √(-3)² = |-3|, and √(3²) = |3|.)

Using this, we have

S1:: |x²| = 9. Since x² is always nonnegative, this is really x² = 9, which has two solutions; NOT SUFFICIENT.

S2:: |x| = -x. This is a tricky identity that the GMAT likes. Though it may not look like it at first, this is just a funny way of saying x ≤ 0. (-x is really -1*x, so if -1*x is ≥ 0, then x itself ≤ 0.) So we know x ISN'T positive, but this isn't enough to find its value; NOT SUFFICIENT.

Together, however, only x = -3 satisfies both statements, so our answer is C.

[manik11]

S2:: |x| = -x. This is a tricky identity that the GMAT likes. Though it may not look like it at first, this is just a funny way of saying x ≤ 0. (-x is really -1*x, so if -1*x is ≥ 0, then x itself ≤ 0.) So we know x ISN'T positive, but this isn't enough to find its value; NOT SUFFICIENT.

Together, however, only x = -3 satisfies both statements, so our answer is C.

Matt,
How did you get to -1*X>=0 from |X| = -X? Can you please explain it with an example?

[mindful]

Let me take a jab at this. Interesting way of looking at it. I had never considered this before.

I in fact thought that while simplifying the second statement, when we reach the following stage:

|x| = -x

this basically means that the GMAT is giving us a condition, something like |x| = -p or |x| = -4x ; you know, something like a function? Meaning, each time we get the L.H.S., we should take it to mean that it is the R.H.S. Going by this reasoning, and combining it with the first statement, we of course get -3.

What I am happy about is that I just saw another way of looking at this! What Matt has done here essentially is that he has taken the R.H.S. first, and has taken it to (truly) equate it with L.H.S., rather than interpreting it as a "condition/function" (as I did.).

So he is saying: -x = |x|

In other words, -1 * x = Positive x (since |x| is always positive);

If -1 * x = positive, then x itself must be negative.

Yaay!

Pl excuse if my explanation doesn't make sense.

[Matt@VeritasPrep]

S2:: |x| = -x. This is a tricky identity that the GMAT likes. Though it may not look like it at first, this is just a funny way of saying x ≤ 0. (-x is really -1*x, so if -1*x is ≥ 0, then x itself ≤ 0.) So we know x ISN'T positive, but this isn't enough to find its value; NOT SUFFICIENT.

Together, however, only x = -3 satisfies both statements, so our answer is C.

Matt,
How did you get to -1*X>=0 from |X| = -X? Can you please explain it with an example?

Sure! Let's look for some numbers that satisfy |x| = -x.

POSITIVE:
Suppose x = 3. Then we'd have
|3| = -3
or
3 = -3

OK, that didn't work.

NEGATIVE:
Suppose x = -3. Then we'd have
|-3| = -(-3)
or
3 = 3

Success!

Last case is ZERO:
|0| = -0
or
0 = 0

Touchdown!

So x is zero or negative, and |x| = -x implies x ≤ 0.

[mindful]

Cheers. Truly clarifies stuff - the example you've just given.

[Matt@VeritasPrep]

Cheers. Truly clarifies stuff - the example you've just given.

The biggest takeaway here: -x doesn't mean "x is negative", it means "take x, then multiply it by -1". So as strange as it sounds, if x itself is negative, -x is actually positive!

[mindful]

"So as strange as it sounds, if x itself is negative, -x is actually positive!"
This in itself does not sound strange as independently we know of the multiplication property of negative numbers.

"The biggest takeaway here: -x doesn't mean "x is negative", it means "take x, then multiply it by -1". [/quote] Nice.

Wow. That's right...This is the hard part to process immediately...from |x| = -x. Realising that Math is quite a language, requiring interpretation.

I got stuck here (4th post from the top). Is it possible for me to "see" it differently? I couldn't see it the way theceo sees it right away...

https://www.beatthegmat.com/absolute-val ... tml#757220

[Max@Math Revolution]

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of x?

1) √x^4 = 9
2) √x^2 = -x

There is one variable (x) and 2 equations are given from the 2 conditions, making (D) our likely answer.
For condition 1, it is not sufficient as √x^4=√(x^2)^2=|x^2|=x^2=9 and x=-3,3
For condition 2, it is not sufficient as well because √x^2=|x|=-x ==>x<0
When looking at them together, we get x=-3, so this is sufficient. The answer becomes (C).

For cases where we need 1 more equation, such as original conditions with "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.