Incorrect General Statement for Modulus Inequalities $$|x + 1| = x$$
This, quite evidently, has no solution.
Through solving many inequalities, I came to the conclusion that,

If,
$$|f(x)| = g(x)$$
Then,
$$f(x) = ±g(x)$$

And this was quite successful in solving many inequalities. However, applying the above to this particular inequality:
$$|x + 1| = x$$
$$x + 1 = ±x$$
When ± is +, there is no solution. However, when ± is -:
$$x + 1 = -x$$
$$2x = -1$$
$$x = -\frac{1}{2}$$
Which does not make sense. What is wrong with the supposedly general statement that seemed to work flawlessly?

solving ∣x+1∣=x amounts to finding A={x:∣x+1∣=x}. Now A⊂{x:x+1=x}∪{x:x+1=-x}=φ∪{1/2}={1/2}. Thus A⊂{1/2},A=φ. This is possible.
