Is x^2 greater than x ?

(1) x is less than -1.

If x is less than -1, then x^2 will always be greater than x, because x^2 is positive, and x is negative. SUFFICIENT

(2) x^2 is greater than 1.

The inequality x^2 > 1 can be broken down into the following:

x < -1 or x > 1

We've already determined in Statement (1) that x^2 is greater than x if x < -1. If x > 1, then no matter what value of x you choose, x^2 will always be greater than x. SUFFICIENT

You could also approach this problem by deducing some information from the prompt. The questions asks if x^2 > x. Ask yourself: In what situations is x^2 greater than x?

When faced with questions dealing with inequalities and exponents, consider the four important ranges:

1. x < -1

2. -1 < x < 0

3. 0 < x < 1

4. x > 1

Let's look at each of these ranges and apply them to a comparison between x^2 and x:

1. x < -1 (in this situation, x^2 is greater than x)

2. -1 < x < 0 (in this situation, x^2 is greater than x)

3. 0 < x < 1 (in this situation, x^2 is smaller than x)

4. x > 1 (in this situation, x^2 is greater than x)

Also, in this case, we have to consider that x^2 is equal to x if x is either 0 or 1.

So, the only case in which x^2 is not greater than x is if 0 <= x <= 1 (i.e. if x is 0, 1, or a proper fraction).

So really, the question in the prompt can be rephrased as "Is x anything other than 0, 1, or a proper fraction?". And on a DS question, that's the same thing as asking "Is 0 <= x <= 1?", because answering NO to that question is the exact same thing as answering YES to the actual question. Look how much easier the question becomes when we substitute that rephrasing:


Is 0 <= x <= 1?

(1) x is less than -1.

(2) x^2 is greater than 1.


Be on the lookout for ways to rephrase the prompt if possible! Hope that helps!

Hey raz1024,

I am really glad with your explanantion. this is something I was looking for. I know all basic rules but I was missing some options to consider for this tye of questions(may be I need some more practice).
ut after reading your explanation I feel that I am going to solve all these type of questions.

Thanks a lot.

Hi Rich,
if x^2 is greater than 1.

The inequality x^2 > 1 can be broken down into the following:

x > -1 or x > 1 , did not understand how you got x < -1 or x > 1 ?

Though the above doesn't make a difference to the answer, was thinking how you got these... Thank You...

Regards,
Narendra.

narenkr wrote:Hi Rich,
if x^2 is greater than 1.

The inequality x^2 > 1 can be broken down into the following:

x > -1 or x > 1 , did not understand how you got x < -1 or x > 1 ?

Though the above doesn't make a difference to the answer, was thinking how you got these... Thank You...

Regards,
Narendra.

Suppose, x^2 = 1 and solve this for x. You will get either (i) x=1 or (ii) x=-1 (because 1^2=1 and (-1)^2=1)

Then think about it, for x^2 to be greater than 1. For (i) x must be greater than 1 (x>1). I think this is obvious, then for (ii) Take -2 for example, what is the square of -2? It is 4. 4 is greater than 1, right?.. Then x must be smaller than -1 for (ii). So x<-1.

Hence, either x>1 or x<-1.

Hey, Thank You.. I think it is fine now... I was solely looking at the statement rather than understanding the same...

Regards,
Narendra.