Question

graph the inequality on a plane. (click to shade a region below or above the line). -4x - 2y ≥ 8

Explanation:

Step1: Rewrite the inequality

Start with the given inequality \(-4x - 2y \geq 8\). We want to solve for \(y\) to get it in slope - intercept form (\(y=mx + b\)). First, add \(4x\) to both sides: \(-2y\geq4x + 8\). Then divide every term by \(-2\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So we have \(y\leq - 2x-4\).

Step2: Analyze the boundary line

The boundary line for the inequality \(y\leq - 2x - 4\) has the equation \(y=-2x - 4\). The slope \(m=-2\) and the \(y\) - intercept \(b = - 4\). Since the inequality is \(\leq\), the boundary line should be a solid line (because the inequality includes equality).

Step3: Determine the shading region

To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\leq - 2x-4\). We get \(0\leq-2(0)-4\), which simplifies to \(0\leq - 4\). This is false. So we shade the region that does not contain the point \((0,0)\). In other words, we shade below the line \(y=-2x - 4\) (since the inequality is \(y\leq\) the expression, we shade the region where \(y\) values are less than or equal to the values on the line).

Answer:

First, rewrite the inequality \(-4x - 2y\geq8\) as \(y\leq - 2x - 4\). The boundary line \(y = - 2x-4\) is solid (because of the \(\geq\) in the original inequality, after manipulation we have \(\leq\), but the equality is included). Then, using a test point (e.g., \((0,0)\)) which does not satisfy \(y\leq - 2x - 4\) (since \(0\leq - 4\) is false), we shade the region below the line \(y=-2x - 4\). The graph of the inequality \(-4x - 2y\geq8\) has a solid line with slope \(-2\) and \(y\) - intercept \(-4\), and the region below the line is shaded.