which of the following systems of inequalities matches the graph shown? y ≤ -4x - 2 and y ≥ -12x + 6 y ≤ -9x…

Explanation:

Step1: Find the slope of the two lines

For a linear equation (y = mx + b), (m) is the slope. Let's assume the two lines in the system of inequalities. Take two points on one line: say for the steeper - looking line (the left - hand side of the solution region). If we assume two points ((x_1,y_1)=(- 2,6)) and ((x_2,y_2)=(0,-2)) The slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2 - 6}{0+2}=\frac{-8}{2}=-4) For the less - steep line (the right - hand side of the solution region). Assume two points ((x_1,y_1)=(0,6)) and ((x_2,y_2)=(2,-18)) The slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-18 - 6}{2-0}=\frac{-24}{2}=-12)

Step2: Find the (y) - intercept of the two lines

The (y) - intercept (b) is the value of (y) when (x = 0). For the line with slope (m=-4), when (x = 0), (y=-2) (from the point ((0,-2))), so the equation of the line is (y=-4x - 2) For the line with slope (m=-12), when (x = 0), (y = 6) (from the point ((0,6))), so the equation of the line is (y=-12x+6)

Step3: Determine the inequalities

The shaded region is below the line (y=-4x - 2) (since the line (y=-4x - 2) is solid and the region is below it), so (y\leq-4x - 2) The shaded region is above the line (y=-12x + 6) (since the line (y=-12x + 6) is solid and the region is above it), so (y\geq-12x + 6)

Answer:

(y\leq-4x - 2) AND (y\geq-12x + 6)