please help

Question

Solve and show work

 

x-6  +     16       = 0

x-4      x²-16

Answer 1

we first need to bring both fractions to the same denominator, so we will multiply the first one by x+4.   (because (x-4)(x+4)=x^2-16) 

 

(x-6)(x+4)/(x^2-16) + 16/(x^2-16)=0

 

(x-6)(x+4)+16=0 (if a fraction is equal to 0, the numerator must be 0, so we can theoretically ignore the denominator)

 

x^2-2x+8=0

 

(x-4)(x+2)=0 therefore x could be 4 or x could be -2.

 

Now, the only solution is actually -2 because in the beginning of the exercise we had some restrictions on x. The denominator of the fractions couldn't be 0, therefore x couldn't be 4 or -4.

 

Final solution: x=-2