Problem 4 - Elementary Algebra Within Reach

Question

In Exercises \(1-6\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{aligned} x+y &<-3 \\ y &>3 x-4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution of the system of inequalities is the region below the line \(y = -x-3\) and above the line \(y = 3x-4\).

Step 1: Graph the inequality \(x + y < -3\)

This inequality can also be rewritten as \(y < -x-3\). To graph it, first plot a line \(y = -x-3\). Remember to draw this line as dashed because the original inequality doesn't contain the equal sign. The region satisfying this inequality is below the line.

Step 2: Graph the inequality \(y > 3x - 4\)

Graph this inequality by plotting the line \(y =3x-4\). As this inequality also doesn't contain the equal sign, you should draw a dashed line. The region satisfying this inequality is above the line.

Step 3: Determine the intersection of the regions

Having graphed both lines and determined the regions satisfying each inequality, observe the region where both conditions are satisfied, i.e., the region which is both below the line \(y = -x-3\) and above the line \(y = 3x-4\). This common region is the solution to the system of inequalities.

Key Concepts

System of Linear Inequalities

A system of linear inequalities consists of two or more inequalities that are considered together. It is essential to find a set of solutions that satisfies all inequalities in the system. For instance, in the given exercise, you need to deal with the inequalities: \(x + y < -3\) and \(y > 3x - 4\). These inequalities tell us not just about a line, but about regions on a coordinate plane relative to these lines.

To solve such systems, graph each inequality separately on the same axes. Each inequality divides the plane into two regions: one that satisfies the inequality and one that doesn't.

• Mark each corresponding line as either part of the region or not (use a solid line if it includes equal to, dashed if it doesn’t).
• The solution to the entire system is the overlapping region where all individual inequalities hold true.
• Each point in this overlapping region will satisfy all inequalities simultaneously.

Intersection of Regions

When solving a system of linear inequalities, an important concept is finding the intersection of regions. The intersection is the area on the graph where all the inequalities overlap when plotted. This region represents all solutions that satisfy each inequality at the same time.

In our example, to find the intersection, look at the graph of \(x + y < -3\) and the graph of \(y > 3x - 4\):

• The solution to the system is the set of points that are part of both regions.
• It means identifying where the region below the line \(y = -x-3\) overlaps with the region above the line \(y = 3x-4\).
• Any point in this common area satisfies both inequalities, making it a solution to the system.

Understanding this concept helps us graphically solve complex inequality systems by visualizing the solutions within the intersecting shaded areas.

Dashed Lines in Graphing

When graphing inequalities, the type of line drawn is crucial in indicating whether the solutions include the line itself. If an inequality is strictly less than or greater than (such as \(<\) or \(>\)), you represent it with a dashed line. This line shows that points exactly on the line do not make the inequality true.

In our example:

• For \(x + y < -3\), the line \(y = -x - 3\) should be dashed because \(x+y\) cannot be equal to \(-3\).
• Similarly, for \(y > 3x - 4\), the line \(y = 3x - 4\) is dashed since \(y\) cannot be exactly \(3x - 4\).
• This visual cue helps distinguish whether to include the line in the solution set or exclude it.

Dashed lines effectively communicate the subtle yet important restriction in the graph, aiding in accurate representation and interpretation of the solution region.