Question

graph the inequality. -2x - 3y ≤ -9

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-2x - 3y\leq - 9\). First, we want to solve for \(y\) to get it in the form \(y = mx + b\) (or in this case, an inequality).
Add \(2x\) to both sides of the inequality: \(-3y\leq2x - 9\).
Then, divide every term by \(- 3\). When we divide an inequality by a negative number, the direction of the inequality sign changes. So we have \(y\geq-\frac{2}{3}x + 3\).

Step2: Graph the boundary line

The boundary line for the inequality \(y\geq-\frac{2}{3}x + 3\) is the line \(y = -\frac{2}{3}x+3\). This is a linear equation in slope - intercept form \(y=mx + b\), where the slope \(m=-\frac{2}{3}\) and the \(y\) - intercept \(b = 3\).

• To find the \(y\) - intercept, we set \(x = 0\). Then \(y=-\frac{2}{3}(0)+3=3\). So the line passes through the point \((0,3)\).
• To find another point on the line, we can use the slope. The slope \(m = \frac{\text{rise}}{\text{run}}=-\frac{2}{3}\). From the point \((0,3)\), we can go down 2 units (because the rise is - 2) and to the right 3 units (because the run is 3). So we get the point \((3,3 - 2)=(3,1)\).

Since the inequality is \(y\geq-\frac{2}{3}x + 3\) (the "greater than or equal to" sign), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step3: Shade the solution region

We test a point that is not on the line to determine which side of the line to shade. A common test point is the origin \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\geq-\frac{2}{3}x + 3\):
\(0\geq-\frac{2}{3}(0)+3\)
\(0\geq3\), which is false. So we shade the side of the line that does not contain the origin. In other words, we shade the region above the line \(y = -\frac{2}{3}x + 3\) (since the inequality is \(y\geq\) the line, we shade above the line).

(Note: Since this is a graphing problem, the final answer is the graph with the solid line \(y = -\frac{2}{3}x + 3\) and the region above the line shaded. If we were to describe the key features: the boundary line has a slope of \(-\frac{2}{3}\), \(y\) - intercept at \((0,3)\), is solid, and the solution region is above the line.)

Answer:

The graph has a solid line \(y = -\frac{2}{3}x + 3\) (with slope \(-\frac{2}{3}\) and \(y\) - intercept \((0,3)\)) and the region above the line is shaded.