Chapter 1.7, Problem 9E

To determine

To Check: The given values of S are satisfy the inequality or not.

Answer to Problem 9E

  $S=\{-2,-1,0,2,4\}$

Explanation of Solution

Given:

  $S=\{-2,-1,0,\frac{1}{2},1,\sqrt{2},2,4\}$

Inequality,

  $\frac{1}{x}\le \frac{1}{2}$

Calculation, Let’s check for each value of S whether it satisfies the inequality or not,

For $x=-2$ ,

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{-2}\le \frac{1}{2}\\ -\frac{1}{2}\le \frac{1}{2}\end{array}$

  $-\frac{1}{2}$ is smaller than $\frac{1}{2}$ so $x=-2$ satisfies the inequality

For $x=-1$ ,

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{-1}\le \frac{1}{2}\\ -1\le \frac{1}{2}\end{array}$

-1 is smaller than $\frac{1}{2}$ So, $x=-1$ satisfies the inequality

For $x=0$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{0}\le \frac{1}{2}\\ \infty \le \frac{1}{2}\end{array}$

  $\infty$ is always greater than $\frac{1}{2}$ So , $x=0$ satisfies the inequality

For $x=\frac{1}{2}$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{\frac{1}{2}}\le \frac{1}{2}\\ 2\le \frac{1}{2}\end{array}$

2 is not smaller than $\frac{1}{2}$ So, $x=\frac{1}{2}$ does not satisfy the inequality

For $x=1$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{1}\le \frac{1}{2}\\ 1\le \frac{1}{2}\end{array}$

1 is not smaller than $\frac{1}{2}$ so, $x=1$ does not satisfies the inequality

For $x=\sqrt{2}$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{\sqrt{2}}\le \frac{1}{2}\\ \frac{1}{1.41}\le \frac{1}{2}\\ 0.7\le \frac{1}{2}\end{array}$

0.7 is not smaller than $\frac{1}{2}$ So, $x=\sqrt{2}$ does not satisfies the inequality

For $x=2$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{2}\le \frac{1}{2}\end{array}$

  $\frac{1}{2}$ is equals to $\frac{1}{2}$ So, $x=2$ satisfies the inequality

For $x=4$

  $\begin{array}{l}\frac{1}{x}\le \frac{1}{2}\\ \frac{1}{4}\le \frac{1}{2}\end{array}$

  $\frac{1}{4}$ is smaller than $\frac{1}{2}$ So $x=4$ satisfies the inequality

Conclusion:

Hence the values of S which satisfies the inequality are $-2,-1,0,2,4$