Question

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

Explanation:

Step1: Find solid line equation

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

Step2: Find dashed line equation

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

Step3: Match with options

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

Answer:

$y \geq x - 2$ and $x + 2y < 4$