If x = -|w|, which of the following must be true

Given x = –|w|.

Arrange the equation and plug the values.

Absolute value properties:

When x≤0 then |x|=−x, or more generally when some expression≤0 then |some expression|=−(some expression). For example: |−5|=5=−(−5)

When x≥0 then |x|=x, or more generally when some expression≥0 then |some expression|=some expression. For example: |5|=5|5|=5

Then x + |w| = 0 is the question. Here we are not sure whether w has to be postive or negative. Follow the above property.

x or w has to be negative ( for ex x = 2 and w = -2 then we get 0 )

1. x= - w.

=> x + w = 0. if w ≤ 0 ; then w = -ve ; then we get 0 total.
if w > 0 ; then w = +ve ; then we get some total.

Not true. Two different answers.

2. x = w.

Same as the above explanation, depending upon w as +ve or -ve value we get 0 or some total.

3. \(x^2\) = w. Same as above explanation.

4. \(x^2\) = \(w^2\) .

Then \(x^2\) - \(w^2\) if the result has to be 0 then consider w = +ve or -ve , in both the cases we get 0 as the result. ( ex: x = 2 or w = +2/-2) .

5. \(x^3\) - \(w^3\) ; when w = +ve we get 0 as the result or when w = -ve we some total. For ex: ( \(2^3\) - \((-2)^3\) = 16 ).

For the 5th option I think it has to be x power cube and w power cube.

|w| = w when w>0 OR |w| = -(w) when w<0

According to question, there are two possible cases

x = (w) or x = (-w). These two options can be eliminated as the question asks "Which of the following MUST be true?". IN this case, one is not, when the other is.

The only other option that is for sure in all cases imaginable is when both x and w are same in magnitude as well as quantity. Possible only in Option D

x and w must be the same distance from zero. Therefore raising each to the same power will result in values that are also the same distance from zero.

And if the power they are raised to is EVEN, the result will be the same distance from zero and will be NON-NEGATIVE (if they were raised to the same odd value, the original values would keep their signs, so there would be a possibility that one would end up pos and one would end up neg)

Therefore, D MUST be true, because x any y (which have the same magnitude) are raised to the same even value.

ScottTargetTestPrep, JeffTargetTestPrep. Please explain how have you eliminated other options. Thanks.

We cannot be sure of answer choice A, because -w is negative if w is positive; but is positive if w is negative.

We cannot be sure of answer choices B and C, because, as explained in the previous response, we have no information about the sign of w.

Finally, we cannot be sure of answer choice E, because w^3 is negative if w is negative and is positive if w is positive.

For answer choice D, on the other hand, no matter the sign of w (and x), w^2 (and x^2) will always be positive. Needless to say that it also holds for the case where x = w = 0. That's why D is the correct option.

i just cannot seem to understand the rule of absolutes and always get these questions wrong. Just for absolutes, can someone recommend good studying source?

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

We can square both sides of \(x = –|w|\); Because \(|w|=\sqrt{w^2}\)

\(x^2 = (–\sqrt{w^2})^2\)

So, \(x^2 = w^2\)

The Answer is D

Let's test some values that satisfy the given equation \(x = –|w|\)

For example, it could be the case that \(x = -2\) and \(w = -2\)
So we'll plug these values into the answer choices to see which ones yield a valid equation.
A. \(-2 = –(-2)\) not true. ELIMINATE.
B. \(-2 = -2\) TRUE
C. \((-2)^2 = -2\) not true. ELIMINATE.
D. \((-2)^2 = (-2)^2\) TRUE
E. \((-2)^3 = (-2)^3\) TRUE

We're down to 3 answer choices.

It could also be the case that \(x = -2\) and \(w = 2\), since those values also satisfy the given equation \(x = –|w|\)
Plug these values into the remaining answer choices....
B. \(-2 = 2\) not true. ELIMINATE.
D. \((-2)^2 = 2^2\) TRUE
E. \((-2)^3 = 2^3\) not true. ELIMINATE.

Answer: D

BrentGMATPrepNow

I was curious about setting up the different cases for the absolute value equation as one normally would to solve...
If w is positive --> x=-w
If w is negative -->x=w

So, then I saw that because of the positive and negative w, only squaring both sides made sense to eliminate the sign difference.

Does this approach make sense/did I set up the possible cases correctly? Many thanks

Yes, it's true that:
If w is positive --> x=-w
If w is negative -->x=w

The symbol |w| represents the absolute value of w, which is always non-negative. When you take the negative of that, i.e., -|w|, you get a non-positive value. So, x must be less than or equal to 0.

Now, let's analyze the choices:

A. x = -w: This is not necessarily true. If w is positive, -w is negative and could be equal to x, but if w is negative, -w is positive and cannot be equal to x.

B. x = w: This is not necessarily true. If w is positive, its absolute value is the same as w itself, but the negative sign in front of the absolute value means that x would be negative, not equal to the positive w.

C. x² = w: This is not necessarily true. Square of x would always be non-negative, while w can be either positive, negative, or zero.

D. x² = w²: This is true. Because whether x or w is negative or positive, the square of both will always be non-positive, and since x is defined as -|w|, x² will always be equal to w².

E. x³ = w³: This is not necessarily true. If w is positive, its cube will also be positive, but x³ (since x is non-positive) will always be non-positive.

So, the correct choice is D: x² = w².

ScottTargetTestPrep, JeffTargetTestPrep, can you please re-explain why the answer won't be A?

I was able to understand why not B, C and E. But why not A?

 


­The absolute value of a number is always non-negative, so it's either 0 or positive. This implies that -|w| is either 0 or negative. Hence, \(x = –|w|\) implies that x, being equal to -|w|, is also either 0 or negative. However, from \(x = –|w|\), we cannot determine anything about w itself - it can be positive, negative, or 0. For example, we can have:



You can observe that either the second case or the third case is not true with all the options except option D, which is true for all the cases. For example, option A is not true if x = -2 and w = -2, which satisfies \(x = –|w|\) but does not satisfy \(x = –w\).

Essentially, any option that preserves the sign of w, such as w, -w, or w raised to an odd power, may not always hold true. This is because these options yield either positive or negative results depending on the value of w in different scenarios.

I hope it's clear.­