Question

graph the inequality on the axes below. -2x + y ≥ -6

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-2x + y\geq - 6\). To get it in the form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept), we add \(2x\) to both sides of the inequality.
\(y\geq2x - 6\)

Step2: Graph the boundary line

The boundary line for the inequality \(y\geq2x - 6\) is the line \(y = 2x-6\).

• Find the y - intercept: When \(x = 0\), \(y=2(0)-6=-6\). So the line passes through the point \((0,-6)\).
• Find the x - intercept: When \(y = 0\), we set up the equation \(0=2x - 6\). Adding 6 to both sides gives \(2x=6\), and dividing by 2 gives \(x = 3\). So the line passes through the point \((3,0)\).
• Since the inequality is \(\geq\), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step3: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\) (as long as it is not on the boundary line).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\geq2x - 6\):
\(0\geq2(0)-6\)
\(0\geq - 6\), which is a true statement.
So we shade the region that includes the point \((0,0)\), which is the region above the line \(y = 2x-6\).

(Note: Since the problem asks to graph the inequality, the final answer is the graph with a solid line \(y = 2x - 6\) and the region above the line shaded. If we were to describe the key steps for graphing, the above steps are the main ones. If we consider the equation of the boundary line and the direction of shading, the process is as described.)

Answer:

The graph of the inequality \(-2x + y\geq - 6\) (or \(y\geq2x - 6\)) has a solid boundary line \(y = 2x-6\) (passing through \((0,-6)\) and \((3,0)\)) and the region above the line is shaded.