Given the following equation, how do I solve for both values of α ? $$\Delta=\frac{\lvert\alpha-\beta\rvert}{\alpha}$$ For context, in Numerical Analysis $\beta$ represents an estimate, to an actual value, $\alpha$, and $\Delta$ represents their difference.
I have tried to experiment with inequality properties of the absolute value function, but that might give me a range of values, instead of two values of $\alpha$. Additionally, I couldn't find any helpful absolute value properties online.
Bonus points for recommending good reads/cheat-sheets on absolute value properties with example problems.
Multiply both sides by $\alpha$, such that $$|\alpha-\beta|=\alpha \Delta$$
Now, since $|x|=y \implies x=\pm y$, we get $$\begin{align} \alpha-\beta =\pm \alpha \Delta &\implies \alpha\mp \alpha\Delta = \beta \\ &\implies \alpha(1\mp \Delta)=\beta \\ &\implies \alpha_{1,2}=\frac{\beta}{1\pm \Delta}. \end{align}$$