Problem 39 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x-y \leq 1 \\ x \geq 2 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set covers the region where both conditions, \(x \geq 2\) and \(x-y\leq 1\), are met. It's the region to the right of and including the line \(x = 2\), but below the line \(y = x-1\).

Step 1: Graph the first inequality

First, let's graph the inequality \(x-y \leq 1\). This is the line \(y = x - 1\) with everything on or below this line shaded. This is because the inequality allows all values of y that are less than or equal to \(x-1\). Plot a few points to find the line, such as (0,-1), (1,0), and (2,1), and then shade the region below the line.

Step 2: Graph the second inequality

Next, graph the inequality \(x \geq 2\). This is a vertical line at \(x = 2\). According to inequality, all values to the right and including line \(x = 2\) should be shaded.

Step 3: Find the overlapping region

The solution to the system of inequalities is the overlapping region from the two shaded regions. So, keep the overlapping shaded region and erase other parts. The overlapping region is in the first quadrant bounded by the line \(y = x - 1\) and the line \(x = 2\).

Key Concepts

Graphing Inequalities

Graphing inequalities is a visual method to display the solutions for inequality equations. It involves plotting a line on a graph to represent the equation part of the inequality. However, what's significant is the area that satisfies the inequality. Here’s how to graph inequalities:

• First, convert the inequality to an equation by replacing the inequality sign with an equal sign.
• If the inequality is not strictly greater or less, use a solid line to graph the equation. This indicates the points on the line are part of the solution.
• If it is a strict inequality (e.g., < or >), use a dashed line. This shows points on the line itself are not part of the solution.
• Identify which side of the line to shade by testing a point not on the line. If it satisfies the inequality, shade this region.

Graphing inequalities shows solutions visually and makes it easier to comprehend the regions fulfilling the conditions.

Solution Set

The solution set of a system of inequalities is the collection of all possible values fulfilling all inequalities simultaneously. In simpler terms, it’s the overlap where all conditions are true at once.
For our example, the solution includes the values of x and y that make both inequalities true:

• The first inequality, "\(x - y \leq 1\)," allows all points on or below the line \(y = x - 1\).
• The second inequality, "\(x \geq 2\)," includes points on or right of the vertical line \(x = 2\).

Therefore, the solution set is where these criteria intersect.

Overlapping Region

The overlapping region on a graph of a system of inequalities represents the solution set. It's where the shaded regions from each individual inequality meet. This concept is crucial because:

• It visually conveys which areas satisfy all parts of the system simultaneously.
• Finding this overlap eliminates any portion that does not fulfill every inequality in the system.

In the example provided, the overlapping area is bounded by the line \(y = x - 1\) and occurs from \(x = 2\) onwards. This area satisfies both given inequalities.

Linear Inequalities

Linear inequalities are expressions involving inequality signs (like \( \leq, \geq, <, > \)) with linear equations. They form straight lines on a graph when isolated. Understanding these inequalities involves:

• Recognizing the difference between the equality line and the shaded area that represents the inequality.
• Identifying whether to use dashed or solid lines based on the inequality type.
• Grasping the direction of the shading, which indicates the solution area.

Linear inequalities greatly assist in determining feasible regions in real-life problems where not all values are possible. This utility highlights why understanding their solutions graphically is so beneficial.