Chapter 3, Problem 23SGR

To determine

To calculate: The solution set for the system of inequalities with the help of a graph.

  $\begin{array}{c}y\ge x+3\\ 2y\le x-5\end{array}$

Answer to Problem 23SGR

The solution set for the system of inequalities is Region 3.

  

Explanation of Solution

Given information:

The system of inequalities,

  $\begin{array}{c}y\ge x+3\\ 2y\le x-5\end{array}$

Calculation:

Consider the system of inequalities,

  $\begin{array}{c}y\ge x+3\\ 2y\le x-5\end{array}$

Plot the inequalities $y\ge x+3$ and $2y\le x-5$ on the coordinate plane.

  

Region 1 and 3 depicts the inequality $y\ge x+3$ and region 2 and 3 depicts the inequality $2y\le x-5$ .

The region common to both the inequalities is region 3.

Therefore, the solution set of the system of inequalities is region 3.

To verify substitute any arbitrary point say $\left(-33,-21\right)$ from region 3 in the inequalities and check.

For the inequality $y\ge x+3$ , substitute x as $-33$ and y as $-21$ .

  $\begin{array}{c}-21\ge -33+3\\ -21\ge -30\end{array}$

For the inequality $2y\le x-5$ , substitute x as $-33$ and y as $-21$ .

  $\begin{array}{c}2\left(-21\right)\le -33-5\\ -42\le -38\end{array}$

Both the inequalities satisfy the point.

Thus, the solution set for the system of inequalities is region 3.