Chapter A.9, Problem 101AYU

To determine

To express: The fact that $x$ differs from 2 by less than $\frac{1}{2}$ as an inequality involving an absolute value. Solve for x.

Answer to Problem 101AYU

$\left\{x|\frac{3}{2} < x < \frac{5}{2}\right\}\text{ or }\left(\frac{\frac{3}{2, 5}}{2}\right)$

Explanation of Solution

$\begin{array}{l}|x-2| < \frac{1}{2}\hfill \\ -\frac{1}{2} < -2 < \frac{1}{2}\hfill \\ -\frac{1}{2} + 2 < x-2 + 2 < \frac{1}{2} + 2\hfill \\ \frac{3}{2} < x < \frac{5}{2}\hfill \end{array}$

The solution set is $\left\{x|\frac{3}{2} < x < \frac{5}{2}\right\}$ or using interval notation, all numbers in the interval $\left(\frac{\frac{3}{2, 5}}{2}\right)$.