I selected option B because -|x| = -x.

How is it the D? I still do not understand. Could anyone further explain both question and answer?

Thank you.

It is not always the case that -|x| = -x

If x is a POSITIVE number, then -|x| = -x
For example, if x = 2, we get: -|2| = -2
Simplify to get: -2 = -2

However, if x is a NEGATIVE number, then it is NOT the case that -|x| = -x
For example, if x = -3, we get: -|-3| = -(-3)
Simplify to get: -3 = 3, which is NOT TRUE

Does that help?

Cheers,
Brent

I have learnt that any negative number under mod becomes positive.

If so, -|-3| = -(3) i.e. = -3

Did I learn the wrong concept of absolute value?

You learned the correct concept.
In my solution, you'l see that I also state that -|-3| = -(3) = -3

Cheers,
Brent

Yes.

Regards

is it possible to have negative sign in absolut value

I attempted this sum in a different way:

|x ^ 3| < 64 , hence x needs to be less than 4 . As x can be positive or negative, it can have the values : +1,-1 ; +2 , -2 ;
+3 , -3 .

Lets consider x = 3

QA : -3
QB : -3

a= b

Lets consider x = -3

QA : 3
QB : -3

a>b

hence Ans is D

\(|x^3| < 64 \) \(\text { }\) implies the value of \(x\) could be positive or negative. So, we could write two equations

\(x^3 < 64 \) ......(1)

\((-x)^3 < 64\)......(2)

Now, Quantity B, \(-|x|\) will always be negative.

So, one of the two inequalities above, (1) or (2), has negative value for \(x\) and is equivalent to \(-|x|\), So both quantities are equal.

The other inequality will give \(x\) a positive value and hence it will be greater than Quantity B.

Therefore, the answer is D.