help simplify expressions ans solve equations 1/x-2

Question

 Please help simplify and solve

 

 

 

1. ___1          +    2x       =     __x___

      x - 2             (x-2)(x-8)       2(x-8)

 

 

 

 

 

2.  (m^2 + 2m + 1/m^3 + 3m^2 + 3m + 1)   (m^2/m  -  3m/3)

Answer 1

1.

 

1/(x-2) + 2x/((x-2)(x-8)) = x/(2(x-8))

 

Can't divide by zero so x ≠ 2 or 8.

 

Multiply both sides by 2(x-2)(x-8):

 

2(x-2)(x-8)/(x-2) + 4x(x-2)(x-8)/((x-2)(x-8)) = 2x(x-2)(x-8)/(2(x-8))

 

2(x-8) + 4x = x(x-2)

 

2x - 16 + 4x = x^2-2x

 

0 = x^2 - 8x + 16

 

0 = (x-4)^2

 

x = 4 is the answer.

 

check:

 

1/(4-2) + 2(4)/((4-2)(4-8)) =? 4/(2(4-8))

 

1/2 + 2(4)/((2)(-4)) =? 4/(2(-4))

 

1/2 - 1 =? -1/2

 

-1/2 = -1/2 √

 

2.

 

(m^2 + 2m + 1/m^3 + 3m^2 + 3m + 1) (m^2/m - 3m/3)

 

Can't divide by zero so m ≠ 0.

 

= (m^2 + 2m + m^-3 + 3m^2 + 3m + 1) (m - m)

= (m^2 + 2m + m^-3 + 3m^2 + 3m + 1) (0)

 

= 0