Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$y \leq 3$$

Step-by-Step Solution

Answer

A horizontal line is drawn at \(y = 3\), with the area below the line, including the line itself, shaded.

Step 1: Set Up the Axes

First, set up the x and y axes with appropriate scales. Identify the line \(y = 3\) on the y-axis.

Step 2: Draw the Line

Draw a horizontal line across at \(y = 3\). Make sure it is a solid line, this signifies that points on the line are included in the solution set because the inequality includes equal to ( \(\leq\) ).

Step 3: Shade the Region

Shade the region below the line, including the line itself. This is because the inequality is \(y \leq 3\), which means the solutions can be less than or equal to 3.

Key Concepts

Inequality Notation

Inequality notation is a way to express a range of values that a variable can take. It is essential to understand this notation to work effectively with mathematical inequalities. In our example, the inequality is given as \( y \leq 3 \). This tells us that the value of \( y \) can be any number less than or equal to 3.

Here are some typical symbols used in inequality notation:

• \( \leq \): Less than or equal to
• \( \geq \): Greater than or equal to
• \( < \): Less than
• \( > \): Greater than

When solving an inequality or graphing it, knowing the correct notation is crucial. It determines how you will depict the solution on a graph, specifically whether you will use a solid line or a dashed one. A solid line indicates that the boundary is part of the solution (\( \leq \) or \( \geq \)), while a dashed line is used when it is not (\( < \) or \( > \)).

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can graph equations and inequalities. It consists of two perpendicular lines called axes:

• x-axis: This is the horizontal axis.
• y-axis: This is the vertical axis.

These axes intersect at a point called the origin, denoted by \((0,0)\).

Each point on the coordinate plane is identified by an ordered pair \((x, y)\). This represents the distance from the origin along the x-axis and the y-axis. When working with inequalities, it's crucial to first set up a coordinate plane to accurately plot the solution sets. In our example, we need to mark the line \( y = 3 \), which involves locating and drawing a horizontal line at \( y = 3 \) on the plane.

Understanding how to navigate the coordinate plane is a basic skill for graphing any function or inequality. It helps visualize the relationships between different variables and understand where solutions exist.

Graphical Representation

Graphical representation is a visual way to show mathematical solutions, such as inequalities. For our exercise, which involves the inequality \( y \leq 3 \), we will use the graph to demonstrate all the possible values of \( y \).

Here are the steps to create a graphical representation of this inequality:

• Draw the Line: Start by drawing a horizontal line for \( y = 3 \). This line should be solid since the inequality includes \( \leq \), meaning the boundary itself is part of the solution set.
• Shade the Region: Since our inequality is \( y \leq 3 \), shade the region below this line, as it represents all the points where \( y \) can be less than or equal to 3.

Graphically representing an inequality helps in understanding not just the boundary line (or curve), but also the direction in which solutions lie. This technique is particularly useful in visualizing complex systems where multiple inequalities intersect.