Problem 44 - Algebra 1

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential step in solving systems of inequalities. When graphing, you treat each inequality like an equation to plot a line on a coordinate plane. The type of line used depends on the inequality sign:

• A solid line is used if the inequality includes ">=" or "<=". This means points on the line are included in the solution set.
• A dashed line is used if the inequality uses ">" or "<". This indicates points on the line are not part of the solution set.

For example, in the system:
\[2x + 3y \geq 6\]
the line is solid because of the "\(\geq\)" sign. Meanwhile, for
\[2x + 3y > -6\]
the line is dashed to reflect the exclusion of the boundary points.

Solution Region

The solution region is the overlapping area on the graph that satisfies all the inequalities in the system. Once you've plotted the lines, determining which areas they encompass allows you to find this region. Each inequality divides the plane into two halves:

• For \(2x + 3y \geq 6\), the solution region lies below and on the solid line because it includes all points that make the inequality true.
• For \(2x + 3y > -6\), it is the area above the dashed line, as only points strictly greater than the line satisfy it.

You locate the combined solution region by identifying where these two areas overlap. This shared space is what satisfies both inequalities.

Inequality Signs

Inequality signs determine how you construct and interpret lines on a graph. Each type of inequality provides specific instructions on how to include or exclude points in your solutions:

• "\(\geq\)" or "\(\leq\)" signs mean that solutions include the boundary line, which is represented with a solid line.
• Conversely, "\(>\)" or "\(<\)" mean solutions do not include the line itself, hence using a dashed line instead.

Understanding these signs helps you accurately visualize and shade the correct regions of your graph. They guide you in determining where solutions begin and end in relation to the plotted line.