Problem 58 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 3 x+y \leq 6 \\ x>-2 \\ y \leq 4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set is the triangular region formed by the intersection of \(y \leq -3x + 6\), \(x > -2\), and \(y \leq 4\).

Step 1: Graph the First Inequality

The first inequality is \(3x+y \leq 6\). This can be rewritten as \(y \leq -3x + 6\) by rearranging the terms. When graphing, this looks like a line with a negative slope. The line intercepts the y-axis at 6 and because the inequality is less than or equal, every point on the line and the area beneath it is considered part of the solution.

Step 2: Graph the Second Inequality

The second inequality, \(x > -2\), is a vertical line along x = -2. However, because the inequality is greater than, the area to the right of this line, excluding the line itself, is the solution.

Step 3: Graph the Third Inequality

The third inequality is \(y \leq 4\). This corresponds to a horizontal line at y = 4. Since the inequality is less than or equal to 4, the solution includes every point on or beneath this line.

Step 4: Identifying the Solution Set

Now, the region of the graph that satisfies all three inequalities is the solution to the system of inequalities. From the graph, this would be the triangular area defined by the intersection of the graphs of the three inequalities. This region represents every possible solution to the system of inequalities.

Key Concepts

Graphing Inequalities

The process of graphing inequalities is crucial when dealing with systems of inequalities. Each inequality can be represented as a half-plane on a coordinate system.
This involves visualizing where the solutions to an inequality lie in relation to a line that represents equality. For instance, when dealing with an inequality like \(3x + y \leq 6\), we first graph the line \(y = -3x + 6\). This serves as a boundary. Since the inequality is "less than or equal to," we shade the area below the line, indicating where the solutions to the inequality exist.
It is important to consider if the line itself is part of the solution, which is true in cases of \(\leq\) or \(\geq\). In contrast, with \( < \) or \( > \), only one side of the line is included, and not the line itself.

Solution Sets

In systems of inequalities, the solution set is the region that simultaneously satisfies all the inequalities given. It's essentially the overlap of all the individual regions defined by each inequality.
For example, after graphing each inequality in the given system, the solution set is the area where all the shaded regions meet. This region represents all possible solutions to the system.
Understanding the concept of solution sets ensures students can identify valid solutions and evaluate specific scenarios, helping in assessing whether a point falls within the solution set.

Inequality Intersection

The intersection of inequalities is where solutions to different inequalities meet on a graph. It's the area that lies within all the half-planes defined by each inequality.
This concept is vital because it reflects the practical range of solutions that satisfy a complex condition set by multiple inequalities.
Graphically, this is seen as the common shaded area where the regions defined by each inequality overlap. For the system given, it is a triangular region where all inequalities are satisfied.

Algebraic Graphing Methods

Algebraic graphing methods help in determining how individual inequalities form boundaries on a graph. Each inequality is translated into an equation for graphing, often requiring rearrangement to a more graphical-friendly format, such as solving for \(y\).
The line represents a boundary that helps in identifying the solution region. For instance, the inequality \(3x + y \leq 6\) is graphically represented by the line \(y = -3x + 6\) and its corresponding shaded area. The same concept applies to both vertical (e.g., \(x > -2\)) and horizontal (e.g., \(y \leq 4\)) inequalities.
Using algebraic methods in graphing ensures precision and clarity, making it easier to identify intersection points and the solution set effectively.