Problem 34 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The graph of \(y > -3\) is a horizontal line drawn at \(y = -3\), represented by a dashed line, with the area above the line shaded in to indicate that all the y-values greater than -3 are part of the solution.

Step 1: Draw a Horizontal Line

Draw a horizontal line corresponding to \(y = -3\). As this is a simple 'greater than' inequality, this line won't be included in the solution. This can be represented by using a dashed line.

Step 2: Chart the Inequality Region

Since the inequality is \(y > -3\), the solution set will include all the y-values above this line. Thus, shade the area above the dashed line to represent the solution.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves to represent algebraic equations visually.
It consists of two perpendicular axes:

• The horizontal line, known as the x-axis.
• The vertical line, called the y-axis.

These two axes intersect at the origin (0,0), dividing the plane into four quadrants. Each point on the plane is identified by an ordered pair \(x, y\), showing its position relative to the origin.
When graphing inequalities, the coordinate plane becomes a useful tool to visualize which regions satisfy the inequality conditions. Knowing how to navigate and plot points on the coordinate plane is essential for understanding and solving problems involving inequalities. By plotting boundaries and determining regions that meet the inequality criteria, students can see a graphical representation of the solution set.

Linear Inequality

A linear inequality resembles a linear equation but employs inequality signs—such as \(>\), \(<\), \(\geq\), or \(\leq\)—instead of the equal sign.
Unlike equations, inequalities define a range of values rather than a specific point, describing one region on the coordinate plane.
For example, in the inequality \(y > -3\),

• The line \(y = -3\) serves as a boundary.
• As it is a strict inequality (greater than \(-3\), not greater than or equal to), the line is not part of the solution set and is represented using a dashed line.

Linear inequalities split the plane into distinct regions, with one side containing solutions that satisfy the inequality. An easy way to remember this is:

• A dashed line conveys that points on the line itself are not included in the solution set.
• A solid line is used when the inequality includes the boundary, such as \(\leq\) or \(\geq\).

Inequality Solution Set

The solution set of an inequality encompasses all the points on the coordinate plane that satisfy the inequality condition.
In the case of \(y > -3\), the solution involves all points where the y-coordinate is greater than \(-3\).
To visualize this solution set, follow these steps:

• Draw the boundary line (in this case, \(y = -3\) as a dashed line to indicate the line itself is not included.
• Shade the region above the dashed line.

Shading confirms that all the points in this region are part of the solution set.
By observing the shaded area, you can see where all the applicable solutions lie, making it evident which values satisfy the inequality mentioned in the problem. Understanding and correctly interpreting the shaded area is crucial for accurately determining the solution to a graphing inequality problem.