Question

Describe the graph of the system of inequalities. $$\begin{array}{l} 2 x+3 y>-6 \\ 2 x+3 y \geq 6 \end{array}$$

Short answer

The graph of this system of inequalities is a region on the coordinate plane between the two lines represented by the inequalities. The line \(2x + 3y = 6\) is a solid line while \(2x + 3y = -6\) is a dashed line. The solution set is the area enclosed between these lines.

Step-by-step solution

Graph the Inequalities

In order to visualize the system of inequalities, it is vital to graph them. The inequality signs give us a clue about whether the lines will be solid or dashed. A 'greater than or equal to' (\(\geq\)) or 'less than or equal to' (\(\leq\)) sign means that the line is included in the solution, therefore it will be a solid line. A 'greater than' (\(>\)) or 'less than' (\(<\)) sign means that the line is not included in the solution, therefore it will be a dashed line. The line for the inequality \(2x + 3y \geq 6\) will be solid, while the line for \(2x + 3y > -6\) will be dashed. Plot these lines on the same graph. Use the y-intercept and slope to plot these lines. For \(2x + 3y \geq 6\), the y-intercept is \(b=6/3=2\) and the slope is \(-2/3\). Therefore it will cross the y-axis at 2 and go down 2 for each 3 units to the right. The process for \(2x + 3y > -6\) is similar, the only difference will be the y-intercept \(b=-6/3=-2\), it will cross the y-axis at -2 and go up 2 for each 3 units to the right.

Find the Solution Region

The solution region for the system of inequalities is the area on the coordinate plane that fits all the given inequalities. In this case, the area that fits both \(2x + 3y \geq 6\) and \(2x + 3y > -6\). The inequality \(2x + 3y \geq 6\) means the region below the line (because y has to be less for given x to satisfy the inequality), and \(2x + 3y > -6\) represents the region above the line. The solution region is therefore the region between these two lines.

Shade the Solution Region

Now that the solution region has been determined, it can be shaded on the graph. The shaded area represents all the possible solutions that satisfy the system of inequalities.