tannumunu wrote:
hiranmay wrote:
Is \(x<0?\) ?
(1) \(x^2 - x > 0\) -->insuff: x(x-1)>0 => x <0 & x>1
(2) \(|x| < 1\) --> insuff: -1<x<+1
Combining (1) & (2) => -1<x<0, so sufficient
Answer: C
Hi
thanks for explanation.
will you please explain this equation
x(x-1)>0 => x <0
why x<0 why not x>0?
Hello tannumunu,
\(x^2\) – x>0 is a quadratic inequality. When you have a quadratic inequality, factorise the expression to obtain the roots of the expression. These represent the critical points on the number line. Since the expression is quadratic, it will clearly have two roots.
When you mark two points on a number line, you will see that these points will divide the ENTIRE number line into three segments. Consider
the right most segment as positive,
the segment in the middle as negative and
the left most segment as positive. Reason enough to call this the wavy curve method since the graph sweeps down from right to left and then rises up.
If your inequality has a ‘>’ sign, the segments with the positive sign represent the solutions to your inequality. On the other hand,
if your inequality has a ‘<’ sign, the segment with the negative sign represents the solutions to your inequality.
Let’s simplify \(x^2\)-x>0 to obtain x(x-1)>0. What are the roots here? They are 0 and 1, right? Let’s mark them on the number line which will look like this:
Attachment:
03rd Apr 2020 - Reply 3.jpg [ 31.01 KiB | Viewed 2650 times ]
Since we have a ‘>’ sign in our inequality, the range of x which will satisfy our inequality is x>1 OR x<0. Important keyword here – OR. x can be greater than 1 OR lesser than 0.
As we do not know which range it is, we cannot conclusively say if x>0 or not.
Talking about your other argument – why not x<0?
x(x-1)>0. What does this mean? This means that the product of x and (x-1) has to be positive. This can happen only when both are positive or both are negative.
If x<0, (x-1)<0. If (x-1)<0, x<1. But, if x<0, then x<1 automatically, isn’t it? But, is x<0 the only range? No.
If x>0, (x-1)>0. If (x-1)>0, x>1. But, if x>1, then x>0 automatically, isn’t it?
Hope that clarifies your doubt. Here’s a link to our post on Quadratic inequalities. This will help you understand this concept better.
https://bit.ly/2X35jno