Suruchim12 wrote:
Bunuel wrote:
Three points T, U and V on the number line have coordinates t, u, and v respectively. Is T between points U and V?
(1) \(t^2<4<u^2<v^2\). Since all parts of the inequality are non-negative, we can safely take the square root from it: \(|t| < 2 < |u| < |v|\). It's possible that T is between U and V, for example, if t=0, u=-3 and v=4, as well as it's possible that it's not, for example, t=0, u=3 and v=4. Not sufficient.
(2) \(u<0<v\). Clearly insufficient.
(1) Since from (2) u<0<v, then from (1) we get \(|t| < 2 < -u < v\). If break it: we'll get \(u<-2\) (from 2 < -u), \(-2<t<2\) (from |t| < 2), and \(2<v\). T is between U and V. Sufficient.
Answer: C.
Hope it's clear.
Bunuel Hi, can you please help clarify my doubt below:
For st. 1 when we take the sq. root. why do we take them as the absolute values for u and v? why don't we simply say take the +ve values of u and v since u and v can't be -ve value of the sq. root if they are bigger than 2?
TIA!
Why not?
From 4 < u^2, u can be more than 2 as well as less than -2, for example, -3:
4 < (-3)^2.
Generally, \(\sqrt{x^2}=|x|\). Consider this, say we have \(\sqrt{x^2}=5\). What is the value of x? Well, x can obviously be 5: \(\sqrt{5^2}=\sqrt{25}=5\) but it can also be -5: \(\sqrt{(-5)^2}=\sqrt{25}=5\). So, as you can see \(\sqrt{x^2}=5\) means that \(\sqrt{x^2}=|x|=5\), which gives x = 5 or x = -5.
So, after taking the square root from 4 < u^2 we get 2 < |u| and 2 < |u| in turn means u < -2 or u > 2.
Hope it's clear.