Chapter 12.3, Problem 4MRE

To determine

To calculate: The solution set for open sentence $0<2z+1\le 9$ and graph it.

Answer to Problem 4MRE

The solution set is $\left(-\frac{1}{2},4\right]$ . The graph is provided below,

Explanation of Solution

Given information:

The open sentence $0<2z+1\le 9$ .

Calculation:

Consider the open sentence $0<2z+1\le 9$ .

Subtract 1 in each part of the above inequality,

  $\begin{array}{c}0-1<2z+1-1\le 9-1\\ -1<2z\le 8\end{array}$

Divide each part of above inequality by 2,

  $\begin{array}{c}0-1<2z+1-1\le 9-1\\ -1<2z\le 8\\ -\frac{1}{2}<\frac{2z}{2}\le \frac{8}{2}\\ -\frac{1}{2}<z\le 4\end{array}$

Write it in the interval form $\left(-\frac{1}{2},4\right]$ .

The solution set is the open interval that is all values of z that lie between $-\frac{1}{2}\text{ and }4$ and including 4.

On the real number mark the points $z=-\frac{1}{2}\text{ and }z=4$ and darken the line between the points and the solid end point at 4 means that $z=4$ is included and hollow dot at $z=-\frac{1}{2}$ means that this point is not included.

Thus, the solution set is $\left(-\frac{1}{2},4\right]$ .