How to denote that an equation is true?

Question

If I have a simple equation such as this:

$$x+5-1=x+4$$

how can I denote that this equation is true? More specifically, if I refer to that equation as P(x), then is there a mathematical notation for saying "P(2) is true"?

Answer 1

Beyond calculus's answer, there is not a super short way to write it. If the problem gave you a domain, you can write that the input is a member of the domain. Or, if it is not a too rigorous mathematics course, you can show that 1=1 when you have a particular x value.

Answer 2

One name for an equation that always holds, no matter what values the variables involved take, is an "identity": "$x+5-1=x+4$ is an identity". Really important ones get names: $\sin^2\theta+\cos^2\theta=1$ is the Pythagorean Identity.

If the equation holds only for certain values of the variables -- say $3x + 6 = 0$ -- then you instead say the equation has a solution at $x=2$ (or whatever).