battlefield wrote:Hi Pete,
I don't understand how you have solved this problem(Bit difficult to grasp it)
Can you please explain why don't we take a scenario where any of the conditions are satisfied with any positive number...including 0<x<1 and 1 < x < infinity.
Please explain if there is some conceptual problem in my doubt!
Hi. I'm not 100% sure I understand your question, but let me know if this helps. The problem with just checking if numbers from 0<x<1 and 1<x<infinity work is that not all numbers from those regions will work out the same way. x=1/2 gives us one way of ordering the expressions, whereas x=7/8 gives us another way. So if you just plug in one number from 0<x<1, you might think that whatever answer you get will be true for EVERY value of 0<x<1, but that is not the case.
Look at the following graph where I have plotted y=x^2 in blue, y=1/x in red, and y=2x in green and we are looking only at the part of the graph where x>0:
Blue(B)=x^2, Red(R)=1/x, and Green(G)=2x. Notice that when x is just slightly bigger than 0, the blue graph is the lowest, green is in the middle, and red is highest. B<G<R or x^2<2x<1/x. Then, the green and red graphs cross a little short of x=1 and it becomes B<R<G or x^2<1/x<2x. Then, at x=1, the blue and red graphs cross and it becomes R<B<G or 1/x<x^2<2x. Finally, the green and blue graphs cross at x=2 and it becomes R<G<B or 1/x<2x<x^2. To summarize, the possibilities, if x is positive, are:
x^2<2x<1/x
x^2<1/x<2x
1/x<x^2<2x
1/x<2x<x^2.
The key thing to realize is that you shouldn't assume that 0<x<1 and 1<x<infinity are the ONLY regions you need to consider. Notice that the inequalities changed every time the graphs intersect. They intersect when the expressions are equal to each other. If you want to solve this by plugging in numbers from different zones on the number line, you could do the following:
Set each pair of equations equal to each other and determine intersection points(keeping in mind that we are only interested in solutions where x>0:
x^2=1/x ---> x^3=1 ---->
x=1
x^2=2x
x^2-2x=0
x(x-2)=0
x=2
2x=1/x
2x^2=1
x^2=1/2
x=sqrt(1/2)
x=sqrt(2)/2
Let these three values determine the regions from which you are choosing numbers to plug in:
0<x<sqrt(2)/2, sqrt(2)/2<x<1, 1<x<2, 2<x<infinity.
Plug in numbers from each of these zones to the expressions listed, and see which of the choices are valid.
