Problem 16 - 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}{l} y

Step-by-Step Solution

Answer

The solution region is where \( y < x \) and \( y \leq 2 \) overlap.

Step 1: Understand the Inequalities

We are given two inequalities: \( y < x \) and \( y \leq 2 \). Our goal is to find the solution set by graphing each inequality and shading the region that satisfies both.

Step 2: Graph \( y < x \)

First, consider the inequality \( y < x \). The line \( y = x \) is a diagonal line that passes through the origin and divides the plane into two regions. Since it is a 'less than' inequality (\(<\)), we will use a dashed line to represent it and shade below the line, where \( y \) is less than \( x \).

Step 3: Graph \( y \leq 2 \)

Now, consider the inequality \( y \leq 2 \). The line \( y = 2 \) is a horizontal line. Since it is a 'less than or equal to' inequality (\(\leq\)), we will use a solid line for \( y = 2 \) and shade below this line where \( y \leq 2 \).

Step 4: Determine the Overlapping Region

The solution set for the system of inequalities is the region that satisfies both conditions \( y < x \) and \( y \leq 2 \). Look for the region where the shading for both inequalities overlaps. This is the area below both the line \( y = x \) and the line \( y = 2 \).

Step 5: Sketch the Final Solution Region

On the graph, highlight or shade the region where the two shaded areas from steps 2 and 3 overlap. This region is below the line \( y = x \) and below or on the line \( y = 2 \). This is the solution set for the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting linear equations on a coordinate plane and identifying regions that satisfy these inequalities. When graphed, inequalities create either solid or dashed lines depending on the inequality symbol. This helps determine which side of the line represents the solution set.

• For 'less than' (<) or 'greater than' (>) inequalities, use a dashed line. This indicates that points on the line are not part of the solution.
• For 'less than or equal to' (≤) or 'greater than or equal to' (≥) inequalities, use a solid line. Here, points on the line are included in the solution set.

For the given inequalities, plot each inequality separately, using the correct line type to ensure clarity in identifying the solution set.

Solution Sets

A solution set is a collection of all possible points that satisfy the given system of inequalities. Once you have graphed each inequality, the solution set is the area where the shadings from each inequality overlap.

• This overlap is visually identified on the graph.
• The shading acts as a guide to find intersections of valid points.

In this exercise, we are interested in identifying the region where both inequalities are true at the same time. In real-world problems, solution sets help us determine feasible regions, such as in optimization tasks.

Linear Inequalities

Linear inequalities are expressions involving a linear function where one side is compared to another using inequality symbols (<, >, ≤, ≥). These inequalities define different regions on the coordinate plane.
For example, a linear inequality like "y < x" involves an infinite set of points making the y-coordinate always less than the x-coordinate. When written in the form "y ≤ 2," it confines the y-values to being at most 2.
Linear inequalities are crucial in modeling situations where relationships between quantities need to be expressed with limits, such as budgets or resource constraints.

Coordinate Plane Shading

Shading on the coordinate plane is a way to visually represent solutions to inequalities. The key idea is that one side of the line represents values that satisfy the inequality, while the other does not.

• For each inequality, decide which region to shade by testing a point not on the line (e.g., (0,0)).
• If the point satisfies the inequality, shade the side containing the point. If it doesn't, shade the opposite side.

For the inequality "y < x," shading will be below the dashed line. For "y ≤ 2," shade below or on the solid horizontal line. The intersection of these shaded areas forms the solution set for the system of inequalities.