Chapter 8.9, Problem 35E

a.

To determine

To tell:

Whether the ordered pair $(0,-4)$ is a solution for the given system of inequalities.

Answer to Problem 35E

The ordered pair $(-2,5)$ is not a solution for the given system of inequalities.

Explanation of Solution

Given:

A system of linear inequalities: $\begin{array}{l}y<x+3\\ y\ge -2x-3\end{array}$

Calculation:

To check whether the ordered pair $(0,-4)$ is a solution for the given system of inequalities, we will substitute coordinates of given point in both inequalities as shown below:

  $\begin{array}{l}y<x+3\left(\text{Given inequality}\right)\\ -4<0+3\left(\text{Substituting }x=0\text{and }y=-4\right)\\ -4<3\end{array}$

  $\begin{array}{l}y\ge -2x-3\left(\text{Given inequality}\right)\\ -4\ge -2(0)-3\left(\text{Substituting }x=0\text{and }y=-4\right)\\ -4\ge 0-3\\ -4\ge -3\end{array}$

Since the given point only satisfiesone inequality, therefore, the ordered pair $(0,-4)$ is not a solution for the given system of inequalities.

b.

To determine

To tell:

Whether the ordered pair $(1,3)$ is a solution for the given system of inequalities.

Answer to Problem 35E

The ordered pair $(1,3)$ is a solution for the given system of inequalities.

Explanation of Solution

Given:

A system of linear inequalities: $\begin{array}{l}y<x+3\\ y\ge -2x-3\end{array}$

Calculation:

To check whether the ordered pair $(1,3)$ is a solution for the given system of inequalities, we will substitute coordinates of given point in both inequalities as shown below:

  $\begin{array}{l}y<x+3\left(\text{Given inequality}\right)\\ 3<1+3\left(\text{Substituting }x=1\text{and }y=3\right)\\ 3<4\end{array}$

  $\begin{array}{l}y\ge -2x-3\left(\text{Given inequality}\right)\\ 3\ge -2(1)-3\left(\text{Substituting }x=1\text{and }y=3\right)\\ 3\ge -2-3\\ 3\ge -5\end{array}$

Since the given point satisfies both inequalities, therefore, the ordered pair $(1,3)$ is a solution for the given system of inequalities.

c.

To determine

To tell:

Whether the ordered pair $(-2,1)$ is a solution for the given system of inequalities.

Answer to Problem 35E

The ordered pair $(-2,1)$ is not a solution for the given system of inequalities.

Explanation of Solution

Given:

A system of linear inequalities: $\begin{array}{l}y<x+3\\ y\ge -2x-3\end{array}$

Calculation:

To check whether the ordered pair $(-2,1)$ is a solution for the given system of inequalities, we will substitute coordinates of given point in both inequalities as shown below:

  $\begin{array}{l}y<x+3\left(\text{Given inequality}\right)\\ 1<-2+3\left(\text{Substituting }x=-2\text{and }y=1\right)\\ 1<1\end{array}$

  $\begin{array}{l}y\ge -2x-3\left(\text{Given inequality}\right)\\ 1\ge -2(-2)-3\left(\text{Substituting }x=-2\text{and }y=1\right)\\ 1\ge 4-3\\ 1\ge 1\end{array}$

Since the given point only satisfies one inequality, therefore, the ordered pair $(-2,1)$ is not a solution for the given system of inequalities.