What is the solution set for x in the equation below square root of x + 1 - 1 = x?

The notation in your question is not clear, but I presume that the equation is:

√ (x + 1) - 1 = x

In order to solve irrational equation, you need to separate the radicand from the rest:

√ (x + 1) = x + 1

Now, you need to set the condition in order to have a positive number as a radicand (you cannot have a square root of a negative number):

(x + 1) ≥ 0

x ≥ - 1

Once the condition is set, you can elevate everything to the second power:

(√ (x + 1)) ² = (x + 1) ²

x + 1 = x² + 2x + 1

x ² + x = 0

x (x + 1) = 0

x = 0 and x = - 1

Now that you have the solutions, check if they are relevant according to the condition you have previously found: in this case, yes, they are both greater than or equal to - 1.

Therefore, the set of solutions is {-1, 0}