Chapter 3, Problem 5PT

To determine

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

  $\begin{array}{c}x+y\le 4\\ y\ge x\end{array}$

Answer to Problem 5PT

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

  

Explanation of Solution

Given information:

The system of inequalities,

  $\begin{array}{c}x+y\le 4\\ y\ge x\end{array}$

Calculation:

Consider the system of inequalities,

  $\begin{array}{c}x+y\le 4\\ y\ge x\end{array}$

Plot the inequalities $x+y\le 4$ and $y\ge x$ on the coordinate plane.

  

Region 1 and 3 depicts the inequality $x+y\le 4$ and region 2 and 3 depicts the inequality $y\ge x$.

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 $\left(-10,6\right)$ from region 3 in the inequalities and check.

For the inequality $x+y\le 4$ , substitute x as $-10$ and y as $6$.

  $\begin{array}{c}-10+6\le 4\\ -4\le 4\end{array}$

For the inequality $y\ge x$ , substitute x as $-10$ and y as $6$.

  $6\ge -10$

Both the inequalities satisfy the point.

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