91Ó°ÊÓ

Change each inequality into an equation by replacing the inequality symbol with an equal sign to find their boundary lines. 1. \(2x - 3y = -9\) 2. \(x + 6y = 12\) Now, we can find the x and y-intercepts for the boundary lines. 1. For \(2x - 3y = -9\): x-intercept is found by setting \(y = 0\), and y-intercept is found by setting \(x = 0\). x-intercept: \((2x = -9) \implies x = -\frac{9}{2}\) y-intercept: \((3y = -9) \implies y = -3\) 2. For \(x + 6y = 12\): x-intercept is found by setting \(y = 0\), and y-intercept is found by setting \(x = 0\). x-intercept: \((x = 12)\) y-intercept: \((6y = 12) \implies y = 2\) Plot these two lines on a coordinate plane.

To find the solution region for each inequality, pick any test point that is not on the boundary line and insert its coordinates into the inequality. If the inequality is true, that means the entire region that contains the test point is the solution region. If not, the region on the opposite side of the line is the solution region. 1. For \(2x - 3y < -9\): We choose the test point \((0,0)\). \((2 \cdot 0) - (3 \cdot 0) < -9\) \(0 < -9\), which is false. Thus, the solution region is on the opposite side of the line (the side that doesn't contain the origin). 2. For \(x + 6y < 12\): We choose the test point \((0,0)\) again. \((0) + (6 \cdot 0) < 12\) \(0 < 12\), which is true. Thus, the solution region is the region that contains the origin.

Graph the solution regions for both inequalities by shading the regions using different patterns such as diagonal lines or cross-hatching. Find the region where both shaded areas overlap to determine the compound inequality solution region. This overlapping region is the graph of the given compound inequality: \(2x - 3y < -9\) and \(x + 6y < 12\).