Equations with Two Moduli

$|u|=|v|$ means that $u$ and $v$ have equal size: they are equal or opposites. Both sides are already non-negative, so there is no extra sign restriction on $v$ .

The double arrow means the statements are equivalent: each implies the other.

At $1/2$ both insides are $11/2$. At $-9/4$ they are $-11/4$ and $11/4$. Both roots work.

Equal non-negative numbers have equal squares, and vice versa. Since $|u|^2=u^2$, we can also write $u^2=v^2$. Use the difference of two squares to avoid unnecessary expansion:

A product is zero only if at least one factor is zero. These factors give exactly the same two cases as before. For linear expressions, the two linear equations are usually shorter.