Chapter 1.7, Problem 7E

To determine

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

Answer to Problem 7E

  $S=\left\{4\right\}$

Explanation of Solution

Given:

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

Inequality,

  $1<2x-4\le 7$

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

For $x=-2$ ,

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(-2)-4\le 7\\ 1<-4-4\le 7\\ 1<-8\le 7\end{array}$

-8 is not greater than 1 but smaller than 7 so $x=-2$ does not satisfy the complete inequality

For $x=-1$ ,

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(-1)-4\le 7\\ 1<-2-4\le 7\\ 1<-6\le 7\end{array}$

-6 is not greater than 1 but smaller than 7 So, $x=-1$ does not satisfy the complete inequality

For $x=0$

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(0)-4\le 7\\ 1<0-4\le 7\\ 1<-4\le 7\end{array}$

-4 is not greater than 1 but smaller than 7 So , $x=0$ does not satisfy the complete inequality

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

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(\frac{1}{2})-4\le 7\\ 1<1-4\le 7\\ 1<-3\le 7\end{array}$

-3 is not greater than 1 but smaller than 7 So, $x=\frac{1}{2}$ does not satisfy the complete inequality

For $x=1$

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(1)-4\le 7\\ 1<2-4\le 7\\ 1<-2\le 7\end{array}$

-2is not greater than 1 but smaller than 7 so $x=1$ does not satisfies the inequality

For $x=\sqrt{2}$

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(\sqrt{2})-4\le 7\\ 1<2(1.41)-4\le 7\\ 1<2.82-4\le 7\\ 1<-1.18\le 7\end{array}$

-1.18 is not greater than 1 but smaller than 7 So, $x=\sqrt{2}$ does not satisfies the inequality

For $x=2$

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(2)-4\le 7\\ 1<4-4\le 7\\ 1<0\le 7\end{array}$

0 is not greater than 1 but smaller than 7 So, $x=2$ does not satisfies the inequality

For $x=4$

  $\begin{array}{l}1<2x-4\le 7\\ 1<2(4)-4\le 7\\ 1<8-4\le 7\\ 1<4\le 7\end{array}$

4 is greater than 1 and smaller than 7 So $x=4$ satisfies the inequality

Conclusion:

Hence the values of S which satisfies the inequality is4