Chapter 1.7, Problem 6E

To determine

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

Answer to Problem 6E

  $S=\{1,\sqrt{2},2,4\}$

Explanation of Solution

Given:

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

Inequality,

  $2x-1\ge x$

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

For $x=-2$ ,

  $\begin{array}{l}2x-1\ge x\\ 2(-2)-1\ge -2\\ -4-1\ge -2\\ -5\ge -2\end{array}$

-5 is not greater than or equal to -2 so $x=-2$ does not satisfy the inequality

For $x=-1$ ,

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

-4 is not greater than or equal to -1 So, $x=-1$ does not satisfy the inequality

For $x=0$

  $\begin{array}{l}2x-1\ge 0\\ 2(0)-1\ge 0\\ 0-1\ge 0\\ -1\ge 0\end{array}$

-1 is not greater than or equal to 0 So , $x=0$ does not satisfy the inequality

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

  $\begin{array}{l}2x-1\ge x\\ 2(\frac{1}{2})-1\ge \frac{1}{2}\\ 1-1\ge \frac{1}{2}\\ 0\ge \frac{1}{2}\end{array}$

0 is not greater than or equal to $x=\frac{1}{2}$ So, $x=\frac{1}{2}$ doesnot satisfy the inequality

For $x=1$

  $\begin{array}{l}2x-1\ge x\\ 2(1)-1\ge 1\\ 2-1\ge 1\\ 1\ge 1\end{array}$

1 is equals to 1 so, $x=1$ satisfies the inequality

For $x=\sqrt{2}$

  $\begin{array}{l}2x-1\ge x\\ 2(\sqrt{2})-1\ge \sqrt{2}\\ 2(1.41)-1\ge 1.41\\ 2.82-1\ge 1.41\\ 1.82\ge 1.41\end{array}$

1.82 is greater than 1.41 So, $x=\sqrt{2}$ satisfies the inequality

For $x=2$

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

3 is greater than 2 So, $x=2$ satisfies the inequality

For $x=4$

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

7 is smaller than 2 So $x=4$ satisfies the inequality

Conclusion:

Hence the values of S which satisfies the inequality are $1,\sqrt{2},2,4$