solve the following

Question

5        =       2      +      x-3

x²-x-6                          x-2

 

 

 

2x+1      +     2x-1

2x-1              2x+1  

2x+1      -     2x-1

2x-1             2x+1

Answer 1

5/(x^2 -x -6) = 2 + (x-3)/(x-2)

5/(x-3)(x+2) = 2 + (x-3)/(x-2)

multiply both sides by (x-3)(x+2)(x-2) to remove the fractions and get

5(x-2) = 2(x-3)(x-2)(x+2) + (x-3)(x-3)(x+2)

graph it and find the x intercepts = the real solutions = about 2.2, 3.4 and -2.3

cubics are hard to solve. there is a cubic formula but it's complicated and esssy

it's a cubic equation but with discontinuities, "holes" where the denominators = 0

2nd problem

let y = (2x+1)/(2x-1) then you have

(y + 1/y)/(y -1/y) which =

(y^2 +1)/y over (y^2-1)/y

which = (y^2+1)/(y2-1)

wich after some algebra becomes

x+1/4x a major simplification