Problem 75 - Precalculus: Mathematics for Calculus

Question

DISCUSS: Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$\left\\{\begin{array}{r} x+2 y>4 \\ -x+y<1 \\ x+3 y<9 \\ x<3 \end{array}\right.$$

Step-by-Step Solution

Answer

Both methods yield the same solution region, but the unshaded method may provide a clearer visual representation. Preference depends on ease of graph interpretation.

Step 1: Graph the First Inequality

Start with the first inequality: \(x + 2y > 4\). First, rewrite it in equation form as \(x + 2y = 4\) to find the boundary line. Choose a test point, say \((0,0)\), and substitute it in the inequality. \(0 + 0 > 4\) is false, so shade the side opposite the point \((0,0)\) along the line. Use a dashed line for \(x + 2y = 4\) since it's a strict inequality.

Step 2: Graph the Second Inequality

Next, graph \(-x + y < 1\). Rewrite it as \(-x + y = 1\) for the boundary. Choose a test point, \((0,0)\), substitute: \(-0 + 0 < 1\), true. Shade the side including \((0,0)\). Use a dashed line since it is also strict.

Step 3: Graph the Third Inequality

For \(x + 3y < 9\), rewrite as \(x + 3y = 9\). Use \((0,0)\) as a test point: \(0 + 0 < 9\), true. Shade the side containing the point \((0,0)\). Use a dashed line again since the inequality is strict.

Step 4: Graph the Fourth Inequality

Consider \(x < 3\), which can be written as \(x = 3\) for the boundary. Choose \((0,0)\) and test: \(0 < 3\), true. Shade the region to the left of the line. Use a dashed line.

Step 5: Identify the Solution Region (Standard Method)

The solution region for the system of inequalities is where all shaded areas overlap. If visualized correctly, this overlaps in a region that satisfies all inequalities.

Step 6: Graph the Solutions using Unwanted Regions Method

Begin by shading the opposite side of each inequality's boundary line (the region not satisfying the inequality). For each inequality, the procedure is similar—identify the side that fails the inequality and shade it.

Step 7: Identify Solution Region (Alternate Method)

With unwanted regions shaded, the region left unshaded on the graph is the intersection satisfying all inequalities, presenting the system's solution.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines of each inequality and determining which side of the line is part of the solution. This process is essential in visualizing the solutions to a system of inequalities. Here's how you can do it successfully:

• Start by converting each inequality into an equation. For example, the inequality \(x + 2y > 4\) becomes the equation \(x + 2y = 4\).
• Graph these equations. They form what is called 'boundary lines' on a coordinate plane.
• Use a test point, like \((0,0)\), to determine which side of the boundary line should be shaded. Substitute the test point into the inequality. If it makes the inequality true, shade that side of the line.
• In cases of strict inequalities (\(>\) or \(<\)), use dashed lines to denote that the points on the line are not included in the solution set.

Graphing is a visual method that helps see where solutions overlap and provides an intuitive understanding of which regions satisfy all inequalities involved.

Solution Region

The solution region in a system of inequalities is where all the conditions of the inequalities are met simultaneously. Finding this region is crucial for solving systems:
- The basic idea is to first graph all inequalities on the same coordinate plane. - Shade the areas of the plane that satisfy each individual inequality. - The intersection of all these shaded areas is your solution region.
This overlap of regions shows the range of values (or coordinates) that fulfill every inequality in the system. If using the alternative method of shading unwanted regions, the part that remains unshaded is the solution region. It is often simpler to spot, minimizing the shading confusion. This unshaded area contains values where each inequality holds true.

Boundary Line

The boundary line of an inequality is a crucial visual guide to the solution. Here's what you need to know:

• The boundary line is created by turning the inequality into an equation by replacing inequality symbols with an equals sign.
• For example, \(x + 2y > 4\) becomes \(x + 2y = 4\), marking where the value transitions from true to false which segments the graph.
• If the inequality is non-strict, such as \(\geq\) or \(\leq\), use a solid line, indicating that points on this line are included in the solution set.
• For strict inequalities (\(>\) or \(<\)), a dashed line is used, clarifying that the boundary itself is not included in the solution space.

Understanding the nature of the boundary line helps in accurately determining which parts of the graph you should shade or consider when identifying solution regions.

Strict Inequalities

Strict inequalities, noted by symbols \(>\) and \(<\), play a significant role in determining solution boundaries:

• These inequalities indicate that the boundary, where the relationship is exactly equal, is not part of the solution.
• When graphing these, you create a dashed boundary line, reminding you that the line itself does not satisfy the inequality.
• This differentiation is crucial because it affects which points (or coordinates) we include when describing the solution set.
• It bears repeating, always check with a test point to confirm shading since we are focusing only on parts strictly greater or lesser than the boundary outcome.

Remember, strict inequalities help define limits in a sharp, clear manner and are essential when showing exclusions from the solution set.