Chapter 1.7, Problem 8E

To determine

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

Answer to Problem 8E

  $S=\{\sqrt{2},2,4\}$

Explanation of Solution

Given:

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

Inequality,

  $-2\le 3-x<2$

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

For $x=-2$ ,

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-(-2)<2\\ -2\le 3+2<2\\ -2\le 5<2\end{array}$

5 is greater than -2 but not smaller than 2 so $x=-2$ does not satisfy the inequality

For $x=-1$ ,

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-(-1)<2\\ -2\le 3+1<2\\ -2\le 4<2\end{array}$

4 is greater than -2 but not smaller than 2 So, $x=-1$ does not satisfy the inequality

For $x=0$

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-(0)<2\\ -2\le 3<2\end{array}$

3 is greater than -2 but not smaller than 2 So , $x=0$ does not satisfy the inequality

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

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-\frac{1}{2}<2\\ -2\le \frac{5}{2}<2\\ -2\le 2.5<2\end{array}$

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

For $x=1$

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-1<2\\ -2\le 2<2\end{array}$

2 is greater than -2 but not smaller than 2 so, $x=1$ does not satisfies the inequality

For $x=\sqrt{2}$

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-\sqrt{2}<2\\ -2\le 3-1.41<2\\ -2\le 1.59<2\end{array}$

1.59 is greater than -2 and smaller than 2 So, $x=\sqrt{2}$ satisfies the inequality

For $x=2$

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-2<2\\ -2\le 1<2\end{array}$

1 is greater than -2 and smaller than 2 So, $x=2$ satisfies the inequality

For $x=4$

  $\begin{array}{l}-2\le 3-x<2\\ -2\le 3-4<2\\ -2\le -1<2\end{array}$

-1 is greater than -2 and smaller than 2 So $x=4$ satisfies the inequality

Conclusion:

Hence the values of S which satisfies the inequality are $\sqrt{2},2,4$