Problem 12 - College Algebra Essentials

Question

Graph each inequality. $$y>-3$$

Step-by-Step Solution

Answer

The graph of the inequality \(y > -3\) is represented by shading the entire region above the dashed horizontal line y=-3

Step 1: Draw Horizontal Axis

Start by drawing a Cartesian coordinate system consisting of the x-axis (horizontal) and the y-axis (vertical).

Step 2: Identify the Line y=-3

Locate -3 on the y-axis. This line will be a horizontal line passing through y=-3. Mark it lightly on your graph; this is the line corresponding to y=-3.

Step 3: Draw the Line

Use a pencil to draw a dashed horizontal line through this point, extending across the graph to the right and left, this indicates that y=-3 is not included in the inequality.

Step 4: Shading the Area

The inequality given is \(y > -3\). Thus, for y to be greater than -3, we must consider the points above the line y=-3. Shade all the area above this horizontal line on the graph. That represents all possible solutions for \(y > -3\) over the entire x-axis.

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is the fundamental framework for graphing equations and inequalities in two dimensions. It consists of two perpendicular lines, known as axes, that intersect at a point called the origin. The horizontal axis is typically labeled as the x-axis, while the vertical axis is the y-axis. Each point in the coordinate system is determined by an ordered pair of numbers \( (x, y) \), which represents its position relative to the two axes. Understanding how to plot points on this system is crucial for drawing lines, shapes, and representing algebraic conditions like inequalities.

Horizontal Line Inequality

Inequalities that involve a simple 'greater than' \( (>) \) or 'less than' \( (<) \) without any 'x' variable represent horizontal lines. A horizontal line inequality like \( y > -3 \) means that the entire region above the line \( y = -3 \) is included in the solution set. Conversely, \( y < -3 \) would mean the area below that line is included. When graphing such inequalities, you draw the line as dashed to indicate that points on the line itself do not satisfy the inequality.

The concept can be visualized easily once you locate the y-value on the vertical axis. It's like looking at a horizontal 'barrier' on the graph: everything above or below the line (depending on the inequality) is part of the solution.

Inequality Shading

Inequality shading is the graphical technique used to depict which side of a boundary line contains the solutions to an inequality. For instance, if the inequality is \( y > -3 \), we shade above the dashed line \( y = -3 \) because \( y \) values that are greater than -3 are found in that direction. The shading represents all points that satisfy the condition set by the inequality.

This visual cue is vital for distinguishing the solution region on a graph. Always remember, the type of line used (dotted or solid) and the direction of shading (above or below, left or right) provide critical information about the inequality's solutions and whether the boundary line itself is included in the solution set.