To simplify the expression $b + \left(\frac{1}{b} - 4\right) - 3$, we will follow these steps:

1. Simplify the parentheses:

$b + \left(\frac{1}{b} - 4\right) - 3 = b + \frac{1}{b} - 4 - 3$

2. Combine like terms:

$b + \frac{1}{b} - 7$

This is the simplified expression.

Now, for the condition:

We must consider if there are any values of $b$ that make the expression undefined. The only potential issue arises from the term $\frac{1}{b}$, which is undefined when $b = 0$.

Therefore, the expression is undefined when $b = 0$.

Answer:

The simplified expression is $b + \frac{1}{b} - 7$, and the value of $b$ for which the equivalence does not hold is $b = 0$.

Would you like further details or have any questions?

Relative Questions:

1. What happens when $b = 0$ in rational expressions?
2. How can you identify terms that make an expression undefined?
3. How would this expression change if it were a fraction instead of a sum?
4. How do parentheses affect the order of operations?
5. Can the expression be factored further if written in a different form?

Tip: Always check for values that make a denominator zero when simplifying expressions with fractions.