Problem 2 - Algebra and Trigonometry

Question

1–14 Graph the inequality. $$y \geq-2$$

Step-by-Step Solution

Answer

Shade above the line \(y = -2\) and use a solid line.

Step 1: Identify the Inequality

The problem asks us to graph the inequality \( y \geq -2 \). This inequality states that \( y \) can be any number greater than or equal to \(-2\).

Step 2: Draw the Boundary Line

First, draw the boundary line where \( y = -2 \). This line should be parallel to the x-axis and pass through the point where \( y = -2 \) on the y-axis.

Step 3: Determine Line Style

Since the inequality is \( y \geq -2 \), the boundary line should be a solid line. This indicates that points on the line itself are solutions to the inequality, because \( y \) can be equal to \(-2\).

Step 4: Shade the Solution Area

Shade the region above the line because the inequality \( y \geq -2 \) tells us that \( y \) is greater than or equal to \(-2\). Every point in this shaded region will satisfy the inequality.

Key Concepts

Inequalities in Two Variables

When dealing with inequalities in two variables, you are essentially comparing one variable to another, typically using symbols such as >, <, ≥, or ≤. In the context of a graph, these inequalities define a region of the coordinate plane that is a solution set. Each point within this region will satisfy the inequality.

Understanding inequalities in two variables is crucial because they often describe real-world situations, such as financial constraints or resource limits.

• An inequality like \( y \geq -2 \) represents all possible values for \( y \) that are either equal to or greater than \(-2\).
• This relationship is visually represented on the graph by a combination of a boundary line and a shaded region.
• Both the boundary line and the shaded region together show all the solutions that the inequality encompasses.

Shading the Solution Region

Shading the solution region on a graph is a vital step in visualizing inequalities. Once the boundary line is established, the next step is to shade all the areas that satisfy the inequality. This shaded area represents the solution set.

The key to correct shading lies in understanding the inequality symbol:

• If the inequality symbol is > or ≥, you will shade the region above the boundary line.
• If the symbol is < or ≤, you will shade the region below the boundary line.

For the inequality \( y \geq -2 \), you should shade all the space above and including the horizontal line at \( y = -2 \). Each point in this shaded area is a solution to the inequality.

Boundary Lines

A boundary line in the context of graphing inequalities is the line that separates the solutions from the non-solutions. This line is based on the equation obtained by changing the inequality sign to an equality.

There are two styles you might encounter:

• Solid line: Used when the inequality includes equality, such as ≥ or ≤. This means the points on the line are part of the solution set.
• Dashed line: Applied when the inequality doesn't include equality, using symbols like > or <. With a dashed line, the points on the line are not solutions.

For the inequality \( y \geq -2 \), the boundary line is solid because it includes equality. This solid line runs horizontally through \( y = -2 \) and any point right on this line, as well as above it, satisfies our inequality.