Problem 22 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} x+y>1 \\ x-y<1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is the region above \( x + y = 1 \) and below \( x - y = 1 \), excluding the boundary lines.

Step 1: Understand and Rewrite the Inequalities

The given system of inequalities is \( x + y > 1 \) and \( x - y < 1 \). These need to be solved concurrently to find the solution set.

Step 2: Plot the Line for First Inequality

Rewrite the first inequality \( x + y > 1 \) as the line \( x + y = 1 \). This is the boundary line. Plot this line on a graph. Use a dashed line to indicate that the points on the line are not included in the solution (since it's a \(>\) inequality).

Step 3: Determine the Shaded Region for First Inequality

To find the region of \( x + y > 1 \), choose a test point such as \((0,0)\). Plugging into the inequality gives \(0 + 0 = 0 > 1\), which is false. Thus, the region above the line \(x + y = 1\) is the solution.

Step 4: Plot the Line for Second Inequality

Rewrite the second inequality \( x - y < 1 \) as \( x - y = 1 \). Similarly, this is the boundary line. Plot this line on the same graph. Use a dashed line here as well, indicating that points on the line aren't part of the solution.

Step 5: Determine the Shaded Region for Second Inequality

For \( x - y < 1 \), use the test point \((0,0)\). Substitute into the inequality to get \(0 - 0 = 0 < 1\), which is true. The region below this line \( x - y = 1 \) is the solution.

Step 6: Identify Intersection of Solutions

The solution to the system is the region where the two shaded regions from the inequalities overlap. This is the area above \( x + y = 1 \) and below \( x - y = 1 \). Be sure to exclude the lines themselves from the solution region.

Key Concepts

Graphing Inequalities

When we graph inequalities, we're essentially drawing a picture of all possible solutions to these inequalities. For the inequality system we are handling, the two inequalities to graph are:

• For the first inequality, \( x + y > 1 \), we treat it like an equation \( x + y = 1 \) to find the associated line.
• For the second inequality, \( x - y < 1 \), rewrite it as \( x - y = 1 \) to find the line here too.

Once the lines are plotted on a graph, remember that these lines act as boundaries, not solutions. Use dashed lines to indicate this. This difference is crucial because the inequalities use greater than (\(>\)) and less than (\(<\)), meaning the lines themselves are not part of the solution set.

By choosing test points, like \((0, 0)\), you can determine which side of the boundary line holds the solutions. This insight helps you shade the correct region that represents all solutions to the inequality.

Solution Sets

The solution set to a system of inequalities is the set of all points that satisfy both inequalities simultaneously. In this problem:

• For the inequality \( x + y > 1 \), the solution includes all points above the line \( x + y = 1 \).
• For \( x - y < 1 \), the solution comprises all points below the line \( x - y = 1 \).

The true magic happens in the overlap of these two sets on the graph. Only the region where the shaded areas intersect represents the solution set for the entire system. This intersection area is where both inequalities are satisfied at once. By properly shading the intersection, you visually capture all the feasible solutions.

Boundary Lines

Boundary lines are the "edges" that define where the inequality transitions. They come from the equalities when you replace the inequality symbols with equal signs:

• The boundary line for \( x + y > 1 \) is \( x + y = 1 \).
• And for \( x - y < 1 \), it's \( x - y = 1 \).

Using dashed lines to denote these boundaries signals that the actual line is excluded from the solution. Compare with solid lines which would represent \( \leq \) or \( \geq \). Dash marks help us understand that, although they're not part of the solution, these lines guide us on where the solutions start.

Understanding the role of boundary lines is key in solving and graphing systems of inequalities. They serve as visual guides to separating regions of valid solutions from those that don't satisfy the inequality.