Bunuel wrote:
If \(z = \frac{(x+y)}{2xy}\) and \(0<y<x<1,\) which of the following must be true about range of z?
A. \(z<0.5\)
B. \(z>0.5\)
C. \(z\geq{1}\)
D. \(z<1\)
E. \(z>1\)
Here's a different approach....
To narrow down the answer choices let's first
test a pair of values If \(0<y<x<1\), then it COULD be the case that y = 0.2 and x = 0.5
In this case, \(z = \frac{0.5+0.2}{2(0.5)(0.2)}=\frac{0.7}{0.2}=3.5\)
If it is possible for \(z = 3.5\), then we can ELIMINATE A and D, because they suggest that z CANNOT equal 3.5
At this point, I notice that some of the remaining answer choices suggest that z COULD equal 1.
So let's see if this is possible.
In other words, given the possible values of \(x\) and \(y\), is it possible that \(1 = \frac{(x+y)}{2xy}\)?
Let's first multiply both sides of the equation by \(2xy\) to get: \(2xy = x+y\)
Now subtract x and y from both sides of the equation: \(2xy - x - y= 0\)
Rewrite \(2xy\) as \(xy + xy\) to get: \(xy + xy - x - y= 0\)
Rewrite as: \(xy - x + xy - y= 0\)
Factor in parts to get: \(x(y - 1) + y(x - 1)= 0\)
At this point we might be able to see that there can be no solution to this equation. Here's why:
Since y < 1, we know that (y - 1) is NEGATIVE
Likewise, since x < 1, we know that (x - 1) is NEGATIVE
We also know that x and y are both POSITIVE.
So our equation, \(x(y - 1) + y(x - 1)= 0\), becomes (POSITIVE)(NEGATIVE) + (POSITIVE)(NEGATIVE) = 0
Simplify to get: NEGATIVE + NEGATIVE = 0
This is impossible.
Since we are unable to find a solution to be equation \(1 = \frac{(x+y)}{2xy}\), we can conclude that
z cannot equal 1This means we can ELIMINATE B and C, because they suggest that z CAN equal 1
By the process of elimination the correct answer is E.
Cheers,
Brent