College Algebra, Chapter 1, 1.6, Section 1.6, Problem 10

Suppose that $\displaystyle s = \left\{ -2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4 \right\}$. Determine which element of $s$ satisfy the inequality $\displaystyle x^2 + 2 < 4$.


$
\begin{equation}
\begin{aligned}

& x^2 + 2 < 4
&& \text{Subtract } 4
\\
\\
& x^2 - 2 < 0
&& \text{Difference of squares}
\\
\\
& (x + \sqrt{2})(x - \sqrt{2}) \text{ and } x + \sqrt{2} < 0
&&
\\
\\
& x - \sqrt{2} < 0 \text{ and } x + \sqrt{2} < 0
&&
\\
\\
& x< \sqrt{2} \qquad x < - \sqrt{2}

\end{aligned}
\end{equation}
$


Thus, the solution set is the union of the two intervals $\displaystyle \left\{ -2, -2, 0, \frac{1}{2}, 1 \right\}$ satistfy the inequality.