SAT Question c9355dec: Linear Inequalities answer and explanation with Desmos solution

Choice A is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form $y=mx+b$, where $m$ is the slope and $\left(0,b\right)$ is the y-intercept of the line. For the graph shown, the boundary line passes through the points $\left(0,-6\right)$ and $\left(9,0\right)$. Given two points on a line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, the slope of the line can be calculated using the equation $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting the points $\left(0,-6\right)$ and $\left(9,0\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in this equation yields $m=\frac{0-\left(-6\right)}{9-0}$, which is equivalent to $m=\frac{6}{9}$, or $m=\frac{2}{3}$. Since the point $\left(0,-6\right)$ represents the y-intercept, it follows that $b=-6$. Substituting $\frac{2}{3}$ for $m$ and $-6$ for $b$ in the equation $y=mx+b$ yields $y=\frac{2}{3}x-6$ as the equation of the boundary line. Since the shaded region represents all the points on and above this boundary line, it follows that the shaded region shown represents the solutions to the inequality $y\ge \frac{2}{3}x-6$.