two - variable inequalities from their graphs\nfind the inequality represented by the graph.

Explanation:

Step1: Find the equation of the line

The line passes through points $(0,2)$ and $(2,0)$. The slope $m$ of the line using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ is $m=\frac{0 - 2}{2-0}=- 1$. Using the slope - intercept form $y = mx + b$ with $b = 2$ (the y - intercept), the equation of the line is $y=-x + 2$.

Step2: Determine the inequality

The line is solid, so the inequality is either $y\leq -x + 2$ or $y\geq -x + 2$. We test a point in the shaded region, say $(0,3)$. Substituting $x = 0$ and $y = 3$ into $y\geq -x + 2$, we get $3\geq-0 + 2$ which is true. So the inequality is $y\geq -x + 2$.

Answer:

$y\geq -x + 2$