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Here's an example of an inconsistent equation:
3x/(x-2) = (4x2 - 8x)/(x2 - 4x + 4)
On its face, it looks perfectly fine. It is not immediately obvious that you can't solve for x and get a meaningful result. But if you take the time to factor the numerator and denominator of the righthand part of the equation, you'll start to see the problem. If you continue and try to solve for x using normal algebraic techniques, you will get the impossible result: 3 = 4.
That result shows that your starting equation is internally inconsistent; that is, it is not consistent with itself.
Solution:
3x/(x-2) = (4x2 - 8x)/(x2 - 4x + 4)
3x/(x-2) = 4x(x - 2)/(x - 2)2
3x = 4x
!!
Don't you like x=0?
As a solution to your full-consistent equation?
Beside that, your definition of self-consistent equation is right. On the contrary the specific example is not.
It is worth noting that often "self-consistent equation" is a misuse for "self-consistency equation", namely an equation whose role is to guarantee the self consistency of a theory (model, whatever). If the equation is satisfied then the theory is self-consistent.
Literally a "self-consistent equation" is a meaningful one.
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