###### Example2.4.5

Is $$-5$$ a solution to $$\sqrt{169-y^2}=y^2-2y\text{?}$$

To find out, substitute in $$-5$$ for $$y$$ and see what happens.

\begin{align*} \sqrt{169-y^2}\amp=y^2-2y\\ \sqrt{169-\substitute{(-5)}^2}\amp\stackrel{?}{=}\substitute{(-5)}^2-2(\substitute{-5})\\ \sqrt{169-\highlight{25}}\amp\stackrel{?}{=}\highlight{25}-2(-5)\\ \sqrt{\highlight{144}}\amp\stackrel{?}{=}25-(\highlight{-10})\\ \highlight{12}\amp\stackrel{\text{no}}{=}\highlight{35} \end{align*}

So no, $$-5$$ is not a solution to $$\sqrt{169-y^2}=y^2-2y\text{.}$$

But is $$-5$$ a solution to the inequality $$\sqrt{169-y^2}\leq y^2-2y\text{?}$$ Yes, because substituting $$-5$$ in for $$y$$ would give you

\begin{equation*} 12\leq35\text{,} \end{equation*}

which is true.

in-context