Example2.2.5

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