Is x = -4 ?

Question: Is x = -4?

Statement 1: \(\sqrt{x^2} = 4\)

\(\sqrt{x^2} = |x| = 4\)

We know that |x| is 4 but we don't know the value of x. It could be 4 or -4. Not sufficient.

Statement 2: \(|x| = - x\)
No value given for x. 4 is no where in the picture so not sufficient.

Taking both statements together, |x| = 4 = -x.
So x = -4. Sufficient.

Answer (C)

Hi,
On statement two, how can absolute value of X be equal to -X. My understanding is that the absolute value of a number can never be negative.

I see how the absolute value of positive X can never -X so X must be negative. But the equal sign really confused me. Can someone explain how to make sense then out of statement two?

Thanks in advance.

Good question. You are correct that absolute values are always non-negative (\(\geq\) 0). So let us consider 2 cases of numbers (+ and -) and see how do you obtain this non-negative value when you take an absolute value.

Case 1: when x>0, say x=4, |x|= |4|=4

Case 2: when x<0, say x=-3, |-3|=3 = -1*-3 = -x , when x is <0. So in effect you need to multiply negative numbers by -1 when you want to take their absolute values.

Statement 2 merely mentions that as |x|=-x ---> x <0. You still do not know whether x = -1 or -4 or -1000000. All 3 of these numbers will satisfy |x|=-x.

Hope this helps.

Engr2012 has already explained you the concept mathematically and perfectly.

I would just like to add the layman's method of explaining this:

When x is positive, -x becomes negative, right?
So if x is 4, -x is -4.

But what if x itself is negative? x or any other variable can certainly stand for a negative number, right? When we say, "what is the value of x?" x could take a negative value too. So x could be -10 or -2 etc
If x is already negative, what happens to -x? It becomes positive!
So if x is -10, -x is -(-10) = 10

So if we say that |x| is always positive, it means no matter what x is (positive or negative), |x| will always be positive.
So if x is already positive, |x| = x only
But if x is negative, |x| = - x (which gives us a positive number assuming x is not 0)

So |-2| = - (-2) = 2

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.

Is x = - 4 ?

(1) √x^2=4

(2) |x|=−x


In the original condition, there is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), |x|=4, x=-4,4, which is not unique and not sufficient.
For 2), |x|=-x -> x<=0, which is not sufficient either.
When 1) & 2), x=-4, which is unique and not sufficient.
Therefore, the answer is 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.