Problem 58 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 3 x+y \leq 6 \\ x>-2 \\ y \leq 4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of each system of inequalities is the region that is below the line \(y = -3x + 6\), to the right of the line \(x = -2\), and below the line \(y = 4\).

Step 1: Graph Each Inequality

First we need to graph each inequality in the system.\n\nFor \(3x + y \leq 6\), rearrange the inequality to \(y \leq -3x + 6\). The region below the line \(y = -3x + 6\) is our solution set for this inequality. \n\nFor \(x > -2\), the area to the right of the vertical line \(x = -2\) represents our solution set.\n\nFor \(y \leq 4\), the area below the horizontal line \(y = 4\) is our solution set.

Step 2: Sketch the Solution Set

After that, the solution set of the system of inequalities would be the area of the graph that satisfies all of these inequalities at once. This would be the area that is below the line \(y = -3x + 6\), to the right of the line \(x = -2\), and below the line \(y = 4\). Thus, find the region of the graph that is true for all three conditions and shade it. This shaded region represents the solution set of system of inequalities.

Key Concepts

Solution Set

When working with inequalities, the term "solution set" refers to all the possible values of the variables that satisfy each inequality in the system. In other words, it is a collection of points on the Cartesian plane that fulfill the conditions given by all inequalities at once.

To determine the solution set, we need to consider overlaps of regions that satisfy each individual inequality. The solution set is the intersected region that remains common to all inequalities.

For instance, with the system given in the original exercise, we need to find the set of points that not only satisfy the first inequality, but also lie within the confines of the other two inequalities.

This common region will include:

• Points that are below the line derived from the inequality \(3x + y \leq 6\). This line acts as a boundary where points below it qualify.
• Points located to the right of the line \(x = -2\), as per the inequality \(x > -2\).
• Points situated below the line \(y = 4\), adhering to the inequality \(y \leq 4\).

The intersection of all these individual solution sets will form one cohesive solution set for the entire system of inequalities.

Systems of Inequalities

A system of inequalities consists of two or more inequalities that share the same variables. Solving a system means finding all values that can satisfy each inequality in the system simultaneously.

Unlike equations, where solutions are often singular points, inequalities describe regions on a graph. Therefore, the solution to a system of inequalities is a shaded region on the graph where all conditions are true.

Let's look closely at our current exercise. It combines multiple inequalities:

• \(3x + y \leq 6\)
• \(x > -2\)
• \(y \leq 4\)

To solve such a system, we adopt a graphical approach since it makes understanding the solution visually intuitive. Each inequality is represented on the graph as a line or boundary. The solution is the area where the shaded regions of each boundary overlap.

Graphical Representation

Graphical representation of inequalities is a powerful method for visualization and understanding. Each inequality corresponds to a portion of the Cartesian coordinate plane.

In the exercise, each inequality was translated into a line. For example, \(3x + y \leq 6\) turned into the line \(y = -3x + 6\). Points below this line satisfy the inequality, so this area is shaded. We do this for all inequalities:

• For \(3x+y\leq 6\) convert to \(y \leq -3x + 6\) and shade below the line.
• For \(x > -2\) represent as a vertical line and shade to the right.
• For \(y \leq 4\) draw a horizontal line and shade below.

The overlapping shaded region, where all are true, visually represents the solution set. This graphical method gives us an immediate picture of where all conditions are satisfied together.

Inequality Constraints

Inequality constraints define the limits or boundaries within which solutions must lie. Each inequality acts as a constraint, indicating which side of a line the solutions can exist.

For our exercise, the given inequalities serve as constraints:

• \(3x + y \leq 6\) means points must reside below or on this line.
• \(x > -2\) indicates that points should be to the right of this vertical boundary.
• \(y \leq 4\) confines points below or on this horizontal line.

Handling these constraints graphically helps visualize allowable regions quickly, but analytically, constraints guide us towards permissible solutions through algebraic manipulation. By understanding these constraints, we understand the boundaries of our solution set, allowing us to efficiently solve and verify our solutions.