Question

1. what is the system of inequalities associated with the following graph?

options:
\\(\

$$\begin{cases} x + y > 2 \\\\ 4x + y > -1 \\end{cases}$$

\\)
\\(\

$$\begin{cases} x + y < 2 \\\\ 4x + y < -1 \\end{cases}$$

\\)
\\(\

$$\begin{cases} x + y \\leq 2 \\\\ 4x + y \\leq -1 \\end{cases}$$

\\)
\\(\

$$\begin{cases} x + y \\geq 2 \\\\ 4x + y \\geq -1 \\end{cases}$$

\\)

Explanation:

Step1: Identify first line equation

The solid line passes through $(0,2)$ and $(2,0)$. Using intercept form: $\frac{x}{2}+\frac{y}{2}=1$, simplified to $x+y=2$. The shaded region is above this line, and the line is solid, so the inequality is $x+y\geq2$.

Step2: Identify second line equation

The dashed line passes through $(0,-1)$ and $(-0.25,0)$. Using slope-intercept form: slope $m=\frac{0-(-1)}{-0.25-0}=-4$, so equation is $y=4x-1$, or $4x+y=-1$. The shaded region is above this line, and the line is dashed, so the inequality is $4x+y>-1$.

Step3: Match to options

Combine the two inequalities:

$$\begin{cases} x+y\geq2 \\ 4x+y>-1 \end{cases}$$

Answer:

$\boldsymbol{

$$\begin{cases} x+y\geq2 \\ 4x+y>-1 \end{cases}$$

}$ (corresponding to the last option:

$$\begin{cases} x + y \geq 2 \\ 4x + y \geq -1 \end{cases}$$

)