SOLUTION: find k such that for every real x we have (1 + kx)/(1 + x^2) < k The answer is k &#8712; (4/3,&#8734;). I want to know how to get this answer without plugging in random values.

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Question 741468: find k such that for every real x we have (1 + kx)/(1 + x^2) < k
The answer is k ∈ (4/3,∞). I want to know how to get this answer without plugging in random values.
Answer by tommyt3rd(5050) (Show Source):
You can put this solution on YOUR website!
Assuming k is any real number...

leads to

then

and combining them we get

after simplifying the numerator...

Notice that is always positive, so to determine when the sign of the expression will be less than zero we only need to look at the numerator:

using the quadratic formula to find x...

then...

then...

which only has a positive real value when
and

or
and
So we notice x is only real on the interval

This interval gives us boundary conditions to test. After testing (using a graphing utility) we see only the values k>4/3 satisfies the inequality for ALL values of x

SOLUTION: find k such that for every real x we have (1 + kx)/(1 + x^2) < k The answer is k &#8712; (4/3,&#8734;). I want to know how to get this answer without plugging in random values.

SOLUTION: find k such that for every real x we have (1 + kx)/(1 + x^2) < k The answer is k ∈ (4/3,∞). I want to know how to get this answer without plugging in random values.