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.