Question

Solutions? Let $S=\left\{-5,-1,0,\frac{2}{3},\frac{5}{6},1,\sqrt{5},3,5\right\}.$ Determine which elements of satisfying the inequality.

$-2\le 3-x<2$

Short answer

The set$\left\{\sqrt{2},2,4\right\}$ satisfies the inequality.

Step-by-step solution

Step-1. Consider the given inequality.

The given inequality is ,

$-2\le 3-x<2$

$-2+2\le 3-x+2<2+2$ (Add ).

$0\le 5-x<4\cdots \left(1\right).$

Step-2. Substitute 4 for x into the equation (1).

$\begin{array}{c}0\le 5-4<4\\ 0\le 1<4\end{array}$

Therefore,$x=4$ satisfies the equation (1)

Step-3. Again substitute 2 for x into equation (1).

$\begin{array}{c}0\le 5-2<4\\ 0\le 3<4\end{array}$

Therefore,$x=4$ satisfies inequality (1)

Substitute$\sqrt{2}$ for into equation (1)

$0\le 5-\sqrt{2}<4$

Therefore, $x=\sqrt{2}$also satisfies inequality (1)

Step-4. Substitute 1 for x into inequality (1).

$\begin{array}{c}0\le 5-1<4\\ 0\le 4<4\end{array}$

Hence, $x=1$does not satisfies inequality (1)

$\therefore$The set$\left\{\sqrt{2},2,4\right\}$ satisfies the inequality.