Equality vs Approximation which one has the precedence? Which one of these statements is true?
4/3 ≈ 1.3 = 8/6

(i.e. 4/3 ≈ 1.3 and 4/3 = 8/6)
4/3 ≈ 1.3 ≈ 8/6

Edit:
Is this statement true or false?
4/3 ≈ 1.3 ≈ 8/6 = 12/9 = 16/12 ≈ 1 ≈ 1.3 ≈ 4/3

Edit:
This sign ≈ denotes approximation.
And this sign = obviously denotes equality.
 A: 4/3 ≈ 1.3 ≈ 8/6 = 12/9 = 16/12 ≈ 1 ≈ 1.3 ≈ 4/3 is shorthand for
(4/3 ≈ 1.3 and 1.3 ≈ 8/6 and 8/6 = 12/9 and 12/9 = 16/12 and 16/12 ≈ 1 and 1 ≈ 1.3 and 1.3 ≈ 4/3).  Your compound statement is true iff each part is true.
A: The second is true. In the second, you wrote that both $4/3$ and $8/6$ are 'close enough for our purposes' to $1.3$. In the first, you have that $1.3=8/6$, which it objectively does not.
I just reread your title. Each and every relative statement, involving $\cong, =, \equiv, ~, \leq$, etc. should be true along the way. Have I resolved your concern?
Edit: Responding to your added portion of your question, I would say that your statement at the end is true, provided you don't mind saying that $1$ is 'close enough' to $1.3$, for example. That very well may be true in some applications, but most people would do a double-take. Although, it is a matter of preference and opinion at that point. Also of note: your final conclusion seems to be that $4/3$ is 'close enough' to itself, which it darn well better be! 
