Absolute value and sign of an elasticity In my microeconomics book, I read that when we have $1+\dfrac{1}{\eta}$ where $\eta$ is an elasticity coefficient, we can write $1-\dfrac{
1}{|\eta|}$ "to avoid ambiguities stemming from the negative sign of the elasticity".
What does this mean? Is it always legitimate to perform such a transformation?

If the elasticity coefficient $\eta$ is negative, then $|\eta|=-\eta$. The ambiguity arises because some people may suppress the negative sign and write it as a positive number instead. In this case, using $1+\frac{1}{\eta}$ becomes ambiguous.
For example, if $\eta=-2$ but people write it as $\eta=2$, then $1+\frac{1}{\eta}$ can mean $1-\frac{1}{2}$ or $1+\frac{1}{2}$.
