Problem 40 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$\left\\{\begin{array}{l}x \leq 3 \\\x \geq 3\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is a single line at \(x = 3\).

Step 1: Interpret the Inequalities

We are given the system of inequalities: \(x \leq 3\) and \(x \geq 3\). This implies that \(x\) must be both less than or equal to 3, and greater than or equal to 3.

Step 2: Find the Overlapping Region

Since \(x\) must satisfy both \(x \leq 3\) and \(x \geq 3\), the only value \(x\) can take that satisfies both conditions is \(x = 3\).

Step 3: Graph the Solution

The solution is a single vertical line at \(x = 3\) on the rectangular coordinate system. Plot a solid vertical line at \(x = 3\) to indicate that it satisfies both inequalities.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is essential for graphing equations and inequalities. Sometimes known as the Cartesian plane, it consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, denoted by the coordinates (0,0). This system allows us to represent graphical solutions of equations and inequalities by plotting points or lines.
Here are a few key points:

• The x-axis runs left to right (horizontally) and measures the horizontal distance of points.
• The y-axis runs up and down (vertically) and measures the vertical distance of points.
• Points on the plane are represented as ordered pairs (x, y).
• The plane is divided into four quadrants, each representing a unique combination of positive and negative values for x and y.

Understanding this system is crucial for locating and visualizing the solution sets of inequalities.

System of Inequalities

A system of inequalities involves multiple inequalities that we interpret and graph together. Unlike equations, where we're finding precise solutions, inequalities define a range of possible solutions, often shown as a shaded region on the graph. In this case, however, we have a special example with the inequalities:
\[ x \leq 3 \]\[ x \geq 3 \]These inequalities together specify all x-values that are both less than or equal to 3 and greater than or equal to 3. This means x can only be equal to 3.
Key concepts include:

• Intersecting solutions: Where solutions of each inequality overlap.
• Graphical representation: Often involves shading regions or drawing lines to represent possible solutions.
• Unique solutions: Sometimes, as in this example, the solution is not a shaded area but a specific line or point.

This system's solution is unique and graphically represented by a vertical line, due to the nature of both inequalities demanding x to be exactly 3.

Vertical Line

A vertical line in the coordinate plane has a constant x-value for all points on the line. In this exercise, we focus on the line defined by the equation \(x = 3\). This vertical line graphically represents the solution to the provided system of inequalities.
Characteristics of a vertical line:

• All points on the line have the same x-coordinate.
• There is no variation in the x value, so all solutions of the system lie on this line.
• In our graph, the line extends indefinitely in the vertical (up and down) direction.

The graphical representation of this problem results in a solid line at x = 3, since it meets the condition for equality, which is indicated by the inequalities involving "less than or equal to" and "greater than or equal to." It visually shows the single point of intersection where both inequalities are satisfied.