Problem 36 - Introductory Algebra for College Students

Question

Graph each inequality. $$y \leq 0$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \leq 0\) is a solid line along the x-axis \(y = 0\) and the entire region shaded below it.

Step 1 - Understanding the inequality

Notice that the inequality is \(y \leq 0\). This means all points on the y-axis where y is 0 or lesser will be included as the solution.

Step 2 - Graphing the line

Plot the line \(y = 0\) on the graph. Since \(y \leq 0\) here, the line will be a solid one, to indicate that the points on the line are included in the solution.

Step 3 - Shading the solution region

For the inequality \(y \leq 0\), you need to shade every point where \(y\) is less than or equal to 0. This will be below the line \(y = 0\), or, effectively, the entire region below the x-axis. Shade this region on the graph.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that compare two values, showing the relationship between them in terms of size. When we talk about inequalities in graphing, they let us know which parts of the plane hold true for the relationship expressed by the inequality. For example, the inequality \( y \leq 0 \) suggests that we're looking at all the points where the "y" value is less than or equal to zero.

Here's a quick checklist to remember when working with inequalities:

• The symbols \( < \) and \( > \) represent "less than" and "greater than," respectively.
• \( \leq \) and \( \geq \) include the point where the values are equal, meaning it's "less than or equal to" or "greater than or equal to."
• For graphical solutions, these inequalities help determine which regions of the graph contain solutions to the inequality.

Understanding these basics will aid in visualizing and solving a variety of problems that involve inequalities.

Interpreting the Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of two perpendicular lines called axes:

• The horizontal axis, known as the x-axis, runs left to right.
• The vertical axis, known as the y-axis, runs up and down.
• The point where these axes intersect is called the origin, represented as (0,0).

The coordinate plane is divided into four quadrants. This division aids in understanding the sign and location of points. For example, a point like (2, -3) indicates positive movement along the x-axis and a negative position along the y-axis.

When graphing inequalities, knowing the structure and layout of the coordinate plane is crucial since it helps us to correctly shade and identify regions.

How to Shade Regions Effectively

Once we graph the line for a particular inequality, the next step is shading the correct region. This shading visually represents all possible solutions to the inequality.

For the inequality \( y \leq 0 \):

• You start by plotting the line \( y = 0 \). Here, it's a solid line because the inequality includes "equal to".
• Next, you determine which side of the line to shade. Since we need all points where y is less than or equal to zero, shade below the line \( y = 0 \).
• Solid lines represent \( \leq \) or \( \geq \), while dashed lines show \( < \) or \( > \) without including the line itself.

Effective shading is key as it provides a clear visual answer to which areas of the plane satisfy the inequality.