Question 1188994
<pre>

Here is the way I think your teacher expected you to solve the problem:

{{{(x-sqrt(x+1))/(x+sqrt(x+1)) = 11/5}}}

{{{5(x-sqrt(x+1))=11(x+sqrt(x+1))}}}

{{{5x-5sqrt(x+1)=11x+11sqrt(x+1)}}}

{{{-6x = 16sqrt(x+1)}}}

Here you should observe that since the right side is non-negative, the
left side is also non-negative. That can be true if but only if x ≤ 0.

{{{-3x = 8sqrt(x+1)}}}, x ≤ 0

{{{(-3x)^2 = (8sqrt(x+1))^2}}}, x ≤ 0

{{{9x^2=64(x+1)}}}, x ≤ 0

{{{9x^2=64x+64}}}, x ≤ 0

{{{9x^2-64x-64=0}}}, x ≤ 0

{{{(9x+8)(x-8)=0}}}, x ≤ 0

9x+8=0;    x-8=0
  9x=-8;     x=8
   x=-8/9;  

Since x ≤ 0, we discard x=8, and x=-8/9 is the only solution.

x=-8/9 is the only solution for x, so the sum of all values of x that 
satisfy the equation is -8/9 itself.

The negative reciprocal of -8/9 is +9/8 which when changed to a mixed
number is {{{1&1/8}}}, choice C).

Edwin</pre>