Problem 24 - Intermediate Algebra

Question

Graph each inequality. $$ y \geq-2 $$

Step-by-Step Solution

Answer

Draw a solid line at \( y = -2 \) and shade above it.

Step 1: Identify the inequality

The inequality given is \( y \geq -2 \), which means that the value of \( y \) is greater than or equal to \(-2\).

Step 2: Understand the inequality context

This inequality represents a horizontal region on the coordinate plane where the \( y \)-coordinate is greater than or equal to \(-2\).

Step 3: Draw the boundary line

Draw a horizontal line at \( y = -2 \). Since the inequality is \( y \geq -2 \) (not just \( y > -2 \)), the line at \( y = -2 \) should be solid, indicating that points on this line are included in the solution set.

Step 4: Shade the solution area

Since \( y \) is greater than or equal to \(-2\), shade the area above the line \( y = -2 \). This indicates all points where \( y \) is \(-2\) or more.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to represent mathematical equations and inequalities. It is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. These lines intersect at the origin, denoted as (0,0). Points on the coordinate plane are identified by their x and y coordinates, written as (x,y). For example, the point (3, -2) is 3 units to the right on the x-axis and 2 units down on the y-axis.

The axes divide the plane into four quadrants. Quadrant I is the top-right section, Quadrant II is the top-left, Quadrant III is the bottom-left, and Quadrant IV is the bottom-right. The coordinate plane is essential for graphing equations and inequalities, allowing us to visualize solutions and behaviors of equations.

Graphing Inequalities

Graphing inequalities helps us visualize the solutions that satisfy the inequality. Unlike equations, which have exact equalities, inequalities represent ranges of values. To graph an inequality like \(y \geq -2\), follow these steps:

• First, transform the inequality into an equation: \(y = -2\).
• Graph this equation as a line on the coordinate plane. This line is the boundary of the inequality.
• Determine the type of line. A solid line indicates that the boundary values are included (\(\geq\) or \(\leq\)), while a dashed line indicates they are not (\(>\) or \(<\)).
• Finally, shade the region of the coordinate plane that represents the solution set for the inequality. For \(y \geq -2\), you would shade above the line \(y = -2\).

Graphing inequalities allows us to clearly see which areas fulfill the condition, making problem-solving more intuitive.

Boundary Line

The boundary line in graphing inequalities acts as the dividing line on the coordinate plane. It distinguishes the solution set that satisfies the inequality from the one that does not. When dealing with \(y \geq -2\), the boundary line is the graph of the equation \(y = -2\).

This line is horizontal, parallel to the x-axis, and runs through all points where the y-coordinate equals -2. The nature of this line is crucial:

• If the inequality includes "greater than or equal to" (\(\geq\)) or "less than or equal to" (\(\leq\)), the boundary line is solid. This means the points on the line are part of the solution.
• If it is just "greater than" (\(>\)) or "less than" (\(<\)), the boundary line is dashed, excluding the points on the line from the solution set.

Understanding how to correctly draw and interpret the boundary line ensures the graph accurately reflects the inequality.

Solution Set

The solution set is the group of all points on the coordinate plane that make the inequality true. For the inequality \(y \geq -2\), the solution set includes all points with a y-coordinate of -2 or more.

To identify and visualize the solution set:

• Recognize the boundary line, which is \(y = -2\) for this inequality.
• Decide how to incorporate the boundary itself. Since the inequality is \(\geq\), the line is included in the solution set.
• Shade the region that satisfies the inequality. For \(y \geq -2\), this area is above the line. Every point in this region represents a valid solution to the inequality.

The solution set provides a visual and concrete way to understand the range of possibilities that satisfy the inequality. This approach makes analyzing mathematical relationships and trends more accessible and intuitive.