Problem 7 - Algebra for College Students

Question

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{r} 2 x-y \geq 4 \\ x+3 y<3 \end{array}\right) $$

Step-by-Step Solution

Answer

Graph two inequalities and highlight where shaded regions overlap.

Step 1: Understand the Inequalities

We have two inequalities: 1. \(2x - y \geq 4\) 2. \(x + 3y < 3\) Both need separate graphical interpretations to find their solution set.

Step 2: Graph the First Inequality

Rearrange \(2x - y \geq 4\) to get the equation of the line: \(y = 2x - 4\) Plot this line on a graph. The inequality \(\geq\) means we need to include the line as part of the solution and shade the area above it.

Step 3: Graph the Second Inequality

Rearrange \(x + 3y < 3\) to get the equation of the line: \(y = -\frac{1}{3}x + 1\) Plot this line on the graph. The inequality \(<\) means we should not include the line itself in the solution set and shade the area below it.

Step 4: Identify the Solution Set

The solution set is the region where the shaded areas of both inequalities overlap. The region must satisfy both inequalities simultaneously.

Step 5: Draw the Final Shaded Region

Ensure a clear distinction by recoloring or re-emphasizing the overlapping region that represents the solution set of the system. This is the common area which satisfies both inequalities.

Key Concepts

Graphical Solutions

Graphical solutions help us visually identify the solution set for a system of inequalities. We convert each inequality into a linear equation, which helps us draw the boundary lines on a graph. Each of these boundary lines divides the plane into two halves. One of these halves will hold the possible solutions for each inequality.

To solve graphically, follow these steps:

• Convert each inequality into an equation by treating the inequality sign as an equal sign.
• Draw the line represented by each equation on a coordinate plane.
• Determine which side of the line satisfies the inequality.
• Shade the appropriate side for each inequality.

The final graphical solution is the overlapping shaded region where the individual solutions of each inequality meet. This region will satisfy all the conditions given by the system of inequalities. Graphical solutions provide a powerful visual method for solving such systems.

Shading Regions

Shading regions is a key step in graphically solving systems of inequalities. Once you have plotted the boundary lines from the inequality equations, the next task is to decide which portion of the graph to shade. Shading represents the set of points that satisfy the inequality.

Here's what you need to know about shading:

• For inequalities like \( \geq \) or \( \< \), shading will involve finding one side of the line to shade.
• For inequalities like \( \geq \) or \( \leq \), the line itself is included in the solution set, typically represented by a solid line.
• For inequalities like \( \> \) or \( \< \), the line is not part of the solution set and is usually drawn as a dashed line.

Always test a point not on the line (often (0,0) for simplicity) to determine which side of the line to shade. If the test point satisfies the inequality, then you shade that side of the line. Properly shading allows you to visually find the solution set that satisfies all the conditions in the system of inequalities.

Linear Inequalities

Linear inequalities are similar to linear equations but instead of equalities, they involve inequality signs like \(\geq\), \(\leq\), \( \< \), and \( \> \). Solving linear inequalities requires understanding how to plot and analyze these inequalities on a graph.

Here are essential points about linear inequalities:

• They describe a region of the coordinate plane bounded by a straight line.
• The inequality relationship determines which side of the line is included in the solution set.
• Each inequality can be rearranged into the form \(y = mx + b\) to identify the slope and intercept for graphing purposes.
• Once graphed, the inequality represents all points that lie in the region specified, either including or excluding the boundary line based on the inequality symbol used.

Linear inequalities are foundational for solving systems of inequalities. Their primary tool is the graph, allowing students to visualize the problem and find solutions that may not be immediately obvious from the algebraic expressions alone. Understanding the properties of linear inequalities guides you in shading the correct region and combining these to find a solution set.