Question

which system of linear inequalities is represented by the graph?\\(\bigcirc\\ y\geq x - 2\\) and \\(x - 2y < 4\\)\\(\bigcirc\\ y\geq x + 2\\) and \\(x + 2y < 4\\)\\(\bigcirc\\ y\geq x - 2\\) and \\(x + 2y < 4\\)\\(\bigcirc\\ y > x - 2\\) and \\(x + 2y < -4\\)

Explanation:

Step1: Identify solid line equation

The solid line passes through $(0,-2)$ and $(2,0)$. Slope $m=\frac{0-(-2)}{2-0}=1$. Using $y=mx+b$, $b=-2$, so line is $y=x-2$. The shaded region is above the solid line, so inequality is $y\geq x-2$.

Step2: Identify dashed line equation

The dashed line passes through $(4,0)$ and $(0,2)$. Slope $m=\frac{2-0}{0-4}=-\frac{1}{2}$. Using $y=mx+b$, $b=2$, so line is $y=-\frac{1}{2}x+2$. Multiply by 2: $2y=-x+4$, rearranged to $x+2y=4$. The shaded region is below the dashed line, so inequality is $x+2y<4$.

Step3: Match to options

Combine the two inequalities: $y\geq x-2$ and $x+2y<4$.

Answer:

$y\geq x-2$ and $x+2y<4$ (the third option)