if equations, y=1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) and y=k*x has exactly four solution.

Question

please find total number of possible values of k  with explanation.please show graph if possible for you.thank you.

Answer 1

See graphs: http://hwmath.net/solutions/eq01.jpg

plotted solutions: http://hwmath.net/solutions/eq01A.jpg

Answer 2

I'm really not sure what your asking for, so here's what I've got; hope it helps.

[Assuming k to be a constant, is that true?]

 

k*x = 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4)

k*x(x-1)(x-2)(x-3)(x-4) = ..., where ... is a cubic polynomial on x ≠ 1,2,3,4.

k*x(x-1)(x-2)(x-3)(x-4) - ... = 0, we now have (at most) a quintic polynomial on x ≠ 1,2,3,4.

 

The polynomial, being quintic, could have at most five solutions.