When does $\,2x=14\iff x\neq7\,$ hold? I am a non-mathematiciam who is taking some classes in computer engineering. My question is:

For which real numbers $x$ does the following hold? $$\,2x=14\iff x\neq7\,$$

I am not interested in just the answer, I would prefer to know how to solve this type of problem.
 A: $$2x = 14 \iff x = 7$$
Logically, you're statement can be simplified to $$x = 7 \iff x\neq 7$$ which is a contradiction: it is false no matter what $x$ is.

Recall that $a \leftrightarrow b$ is true if and only if both $a, b$ are true, or both $a, b$ are false. 
In the example above, if we put $a: x = 7$, and $b: x\neq 7$, then if $a$ is true, $b$ must be false, and if $b$ is false, $a$ must be true. That is, $a, b$ cannot both be true, nor can they both be false. So the statement $a \iff b$ is necessarily false, irrespective of the value of $x$.
A: For $x=7$, $2x=14$ is true and $x\ne 7$ is false, so that $2x=14\iff x\neq7$ does not hold.
For $x\ne7$, $2x=14$ is false and $x\ne 7$ is true, so that $2x=14\iff x\neq7$ does not hold.
This covers all real numbers.
A: When you're in characteristic $\;2\;$ , your equation is
$$0=2x=14=0\rlap{\;\;\;\;/}\implies x=7=1\pmod 2$$
In any other case one can either divide by two and then $\;x=7\;$ , or else $\;2\;$ isn't invertible in a particular algebraic structure.
