Problem 53 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} -2 < x < 2 \\ y > 1 \\ x-y > 0 \end{array}$$

Step-by-Step Solution

Answer

The solution set is a triangular region where the conditions \(-2 < x < 2\), \(y > 1\), and \(y < x\) overlap.

Step 1: Understanding the Inequalities

First, we analyze each inequality separately: \(-2 < x < 2\), which tells us that \(x\) is between \(-2\) and \(2\). Next, \(y > 1\), meaning any \(y\)-value must be greater than 1. Finally, \(x - y > 0\), which rearranges to give \(y < x\), meaning \(y\) must always be less than \(x\).

Step 2: Graphing x-Range

We graph the inequality \(-2 < x < 2\) by drawing two vertical dotted lines, one at \(x = -2\) and another at \(x = 2\). The solution is in the interval between these two lines, excluding the lines themselves. Highlight this area on the graph.

Step 3: Graphing y Condition

Next, graph \(y > 1\) by drawing a horizontal dotted line at \(y = 1\). All \(y\) values above this line are part of the solution. Shade the region above this line, indicating \(y > 1\).

Step 4: Graphing the Line x = y

Graph the inequality \(y < x\) by plotting the line \(y = x\). This line needs to be dashed, as it represents \(y = x\) (where \(y = x\) values are not included in the solution). The solution set is located below this dashed line, so shade the area below it.

Step 5: Identify the Solution Set

The solution set is the region where all three graphed areas overlap. It is a triangular region bordered by the vertical lines at \(x = -2\) and \(x = 2\), above \(y = 1\), and below the line \(y = x\).

Key Concepts

Graphical Solution of Inequalities

When dealing with systems of inequalities, graphical solutions can be an incredibly useful method. Graphing each inequality separately allows us to visualize their individual constraints on the coordinate plane. This visualization helps us find the possible values (or solution set) that satisfy all inequalities at once. Consider the system given:

• \(-2 < x < 2\)
• \(y > 1\)
• \(x - y > 0\) or equivalently, \(y < x\)

Each inequality in this system is graphed as a distinct series of lines on a coordinate plane, with either solid or dashed boundaries indicating whether the boundary is or is not part of the solution. In graphical terms, a solid line shows the points included in the solution, while a dashed line indicates they are not included but help shape the boundary of the solution set. Together, these graphs allow us to identify areas of overlap, which is the final solution set.

Inequality Graphing Techniques

Different techniques are applied when graphing inequalities, and understanding these can greatly aid in solving such systems. Graphing linear inequalities typically involves:

• Identifying whether the inequality is strict (\(<\) or \(>\)) or inclusive (\(\leq\) or \(\geq\)). This dictates whether you use a dashed line (strict) or a solid line (inclusive).
• Plotting boundaries first, in this problem represented by vertical lines for the \(x\)-range, a horizontal line for \(y > 1\), and a diagonal line for \(y = x\).
• Shading appropriate regions that satisfy each inequality: inside or outside boundaries, above or below a line.

For the given system, graph the first inequality by using dashed vertical lines at \(x = -2\) and \(x = 2\). Next, graph \(y > 1\) with a dashed horizontal line at \(y = 1\) and shade above it. Finally, for \(y < x\), use a dashed diagonal line for \(y = x\) and shade below the line. These steps help translate algebraic inequalities into visual representations.

Solution Set Identification

Once each inequality is graphed, the solution set is identified as the area where all shaded regions intersect. It's essentially the graphical representation of all potential solutions that satisfy the entire system of inequalities.For our system, after graphing:

• The overlap occurs between the vertical dashed lines \(-2 < x < 2\).
• Above the horizontal line \(y = 1\), which satisfies \(y > 1\).
• And below the diagonal line \(y = x\), meeting \(y < x\).

The intersection of these constraints forms a bounded triangle. To confirm solution set accuracy, check a point within this triangular region against all inequalities. If the point satisfies the entire system, the shaded area accurately represents the solution set. Visualizing these constraints helps students see both the overarching limitations and permitted solutions in one coherent picture.