Question

is (3, -1) a solution of the graphed system of inequalities? choose 1 answer:

Explanation:

Step1: Identify line equations

First, find the equations of the two lines.

• Solid line: passes through $(0,-2)$ and $(4,0)$. Slope $m=\frac{0-(-2)}{4-0}=\frac{1}{2}$. Equation: $y=\frac{1}{2}x-2$. The shaded region is above this solid line, so inequality: $y\geq\frac{1}{2}x-2$.
• Dashed line: passes through $(0,-4)$ and $(4,0)$. Slope $m=\frac{0-(-4)}{4-0}=1$. Equation: $y=x-4$. The shaded region is above this dashed line, so inequality: $y>x-4$.

Step2: Test point $(3,-1)$ in first inequality

Substitute $x=3, y=-1$ into $y\geq\frac{1}{2}x-2$:

$$\begin{align*} -1 &\geq \frac{1}{2}(3)-2 \\ -1 &\geq 1.5-2 \\ -1 &\geq -0.5 \end{align*}$$

This is false.

Step3: Verify with second inequality (optional)

Substitute $x=3, y=-1$ into $y>x-4$:

$$\begin{align*} -1 &> 3-4 \\ -1 &> -1 \end{align*}$$

This is also false.

Answer:

No, $(3,-1)$ is not a solution of the graphed system of inequalities.