(X^2-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1.
If you then subtract your original number, x, you are left with 1. The reason this doesn't work with 1 is that you would have 1-1=0 in the denominator and you cannot divide by zero.
This uses the skill of simplifying rational expressions by using the difference of squares formula (x squared minus y squared = (x+y) (x-y), in this case x squared minus 1 squared = (x+1) (x-1) and then noticing that you have x-1 in both the numerator and the denominator and cancelling them out.
To simplify the expression (x^2-1)/(x-1), you can indeed use the difference of squares formula. The difference of squares formula states that if you have an expression in the form of a^2 - b^2, it can be factored as (a + b)(a - b).
In this case, you have x^2 - 1, which can be rewritten as (x + 1)(x - 1) using the difference of squares formula. Now, you have (x + 1)(x - 1)/(x - 1).
To simplify this expression further, you can cancel out the common factor of (x - 1) in both the numerator and denominator. This leaves you with (x + 1). Therefore, (x^2-1)/(x-1) is equivalent to x + 1.
Now, if you subtract x from the simplified expression x + 1, you indeed get 1. This is because the x term cancels out: (x + 1) - x = 1.
However, it's important to note that this simplification is only valid when x is not equal to 1. This is because if x equals 1, you would have 1 - 1 = 0 in the denominator, which is undefined. Division by zero is not allowed in mathematics.
So, in summary, the simplified expression (x^2-1)/(x-1) is equal to x + 1, except when x equals 1.