Gurshaans wrote:
Hi guys, thank you for your explanations
Bunuel chetan2u . I have a question, I wanted to understand why we cant use an algebraic approach on the second statement? Since we have squares on either side, can't we take take the square root on either side, which would give us x - 3 < y - 3, then by adding 3 on either side we get x < y .. ? This is exactly what I did and marked D as the answer when I saw this question for the first time. From both the posts, my understanding is that we cant do what i mentioned earlier because we wouldn't know after taking the root whether x-3 and y-3 are positive or negative... Is this right? or am I mistaken? I'm asking this because I do not want to be confused in the exam under time pressure.. for example what would have happened had the question not specified that x and y are positive... and the first statement was x^2<y^2... ? would the answer in that case be E... Sorry about the long post, I just wanted clarity on this.. I do NOT want to get a 600-700 level question wrong in the exam.
MUST KNOW: \(\sqrt{x^2}=|x|\):The point here is that since
square root function cannot give negative result then \(\sqrt{some \ expression}\geq{0}\).
So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?
Let's consider following examples:
If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);
If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).
So we got that:
\(\sqrt{x^2}=x\), if \(x\geq{0}\);
\(\sqrt{x^2}=-x\), if \(x<0\).
What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).
BACK TO THE QUESTION:
According to the above if you take the square root from \((x-3)^2 < (y-3)^2\) you'll get |x - 3| < |y - 3|, which means that the distance between x and 3 is less than the distance between y and 3, which is not sufficient to say whether x < y.
Hope it helps.
|x - 3| < |y - 3|, which means that the distance between x and 3 is less than the distance between y and 3.