Q. 282 - Intermediate Algebra

Question

In the following exercise, determine whether each ordered pair is a solution to the system.

$$\begin{gathered} y>\frac{2}{3}x-5 \\ x+\frac{1}{2}y\le 4 \end{gathered}$$

$$\begin{gathered} ⓐ (6, -4) \\ ⓑ (3, 0) \end{gathered}$$

Step-by-Step Solution

Answer

Part (a). The ordered pair $\left(6,-4\right)$ is not the solution of the given system of inequalities.

Part (b). The ordered pair $\left(3,0\right)$ is the the solution of the given system of inequalities.

Part (a) step 1. Substitute ( 6 , - 4 ) in the given system of inequalities.

Substituting $$\begin{gathered} x=6 \\ y=-4 \end{gathered}$$ in the inequality

$$\begin{gathered} y>\frac{2}{3}x-5 \\ -4>\frac{2}{3}\times 6-5 \\ -4>-1 \end{gathered}$$:

This is not true.

Substituting $$\begin{gathered} x=6 \\ y=-4 \end{gathered}$$ in the inequality

$x+\frac{1}{2}y\le 4$:

$$\begin{gathered} x+\frac{1}{2}y\le 4 \\ 6+\frac{1}{2}(-4)\le 4 \\ 6-2\le 4 \\ 4\le 4 \end{gathered}$$

This is true.

Thus, the ordered pair $\left(6,-4\right)$ is not the solution.

Part (b) Step 1. Substitute 3 , 0 in the given system of inequalities.

Substituting $$\begin{gathered} x=3 \\ y=0 \end{gathered}$$in the inequality

$x+\frac{1}{2}y\le 4$:

$$\begin{gathered} y>\frac{2}{3}x-5 \\ 0>\frac{2}{3}\left(3\right)-5 \\ 0>2-5 \\ 0>-3 \end{gathered}$$

This is true.

Substituting $$\begin{gathered} x=3 \\ y=0 \end{gathered}$$ in the inequality

$x+\frac{1}{2}y\le 4$:

$$\begin{gathered} x+\frac{1}{2}y\le 4 \\ 3+\frac{1}{2}\left(0\right)\le 4 \\ 3+0\le 4 \\ 3\le 4 \end{gathered}$$

This is true.

Thus, the ordered pair $\left(3,0\right)$ is the solution.