Problem Solving

Hello,

I am a little confused about inequalities. Could someone please help me with the following question?

x^2>x

I have two ways of solving this one and they both give me different answers, please let me know which one is the correct method and why.

Method 1
x^2>x
x^2-x>0
x(x-1)>0
This means either x and x-1 are both +ive or both -ive
Case 1 (both +ive)
x>0 and x-1>0
=> x>1
Case 2(both -ive)
x<0 and x-1<0
=> x<0 and x<1 hence x<0

Hence Method 1 gives me x>1 or x<0

Method 2
x^2>x
Consider x to be +ive, we can divide by x
x>1
conside x to be -ive, we divide by a -ive number and change the inequality
x<-1

Hence Method 2 gives me x>1 and x<-1

Please let me know which method is correct.

Thanks.

The first method is correct.

First off, just think about the second method. x < -1 is not true. As long as x is NEGATIVE x^2 will be greater than x because x^2 will always be positive and x will always be negative. (so it should be x < 0)

Now look at it mathematically, x^2 > x (assuming x negative).

means x < 1 (not -1 because x/x equals 1 whether pos or neg).

Furthermore since x is negative it is actually saying...

x < 0 (because x is negative it can't be between 0 and 1).


That is a looking at it a little more complicated than necessary. The basic rule is not to divide by inequalities if you are not sure of the sign. Basically, use the first method.

Oops, made mistake in math on second part.

But just remember, rule is do not divide variables in inequalities unless you know the sign.

When in doubt, use logic to find out.

Second Method can not be right because it is saying that x CAN NOT be between -1 and 0, which is incorrect. It can be any negative number.

Hint: The GMAT loves to make you factor. Look at Erics Flashcards, whenever it can be factored, DO IT! Most likely that will get you to the answer.

Thanks a lot Mike, your explanation really helped.