SOLUTION: Let x and y be nonzero, real numbers such that -1 < x < 1 and -2 < y < 2. Suppose x/y + y/x = z, where z is some real number. Which of the following is true for all possible z? cho

Question 474645: Let x and y be nonzero, real numbers such that -1 < x < 1 and -2 < y < 2. Suppose x/y + y/x = z, where z is some real number. Which of the following is true for all possible z? choose the answer and explain the reason to choose your response
z is less than 10^k for some k>1
z is positive
xyz can be negative
The other three answers are false
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
We have

.

The first choice "z is less than 10^k for some k>1" is not necessarily true (even if we let k get really large) because z can also go to infinity.

z does not have to be positive (e.g. let x and y have different signs).

xyz can be negative, but xyz can also be positive (let x,y be positive).

Hence, the last choice "the other three answers are false" is correct because the first three choices have counterexamples; i.e. the statements are not true for all possible z.
RELATED QUESTIONS
Let x, y, and z be nonzero real numbers, such that no two are equal, and x + {1}/{y} = y (answered by CubeyThePenguin,ikleyn)

Let x, y, and z be nonzero real numbers. Find all possible values of (x + y + z)/(|x| + (answered by yurtman)

Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of (answered by CPhill,ikleyn)

Find the smallest possible value of $$\frac{(y-x)^2}{(y-z)(z-x)} +... (answered by Alan3354)

Let x, y, and z be nonzero real numbers. Find all possible values of (x)/(|x|) +... (answered by ikleyn,Edwin McCravy)

For positive real numbers x, y, and z such that 3x = y(sqrt2)/3 = z(sqrt2)/3.1, which of... (answered by CubeyThePenguin)

Let x, y, z be nonzero real numbers such that {{{x + y + z = 0}}}, and {{{xy + xz + yz != (answered by Edwin McCravy)

Please help me solve this: If z is a complex number such that the real part of... (answered by solver91311)

Let x, y, and z be real numbers such that x + y + z, xy + xz + yz, and xyz are all... (answered by ikleyn)