Problem 2 - Algebra 2

Question

Graph each inequality. $$ y<3 $$

Step-by-Step Solution

Answer

The graph of the inequality `y < 3` is represented by a shaded region below the dashed horizontal line at `y = 3`.

Step 1: Understanding the Inequality

The inequality given is of the form `y < c` where `c` is a constant. This inequality represents all points in the coordinate plane where the `y` value is less than 3. The boundary of this inequality will be a horizontal line at `y = 3`, which will not be included in the solution set since the inequality is strict (less than and not less than or equal to).

Step 2: Drawing the Boundary

Draw a horizontal dashed line at `y = 3`. This line is dashed to indicate that points on this line are not included in the solution set of the inequality.

Step 3: Shading the Solution Region

Since the inequality is `y < 3`, the area below the dashed line (where all points have a `y` value less than 3) represents the solution set. Shade this region to illustrate all the possible points that satisfy the inequality.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is the foundational setting for graphing inequalities. It's a two-dimensional surface consisting of two perpendicular lines, labeled the x-axis (horizontal) and y-axis (vertical), which intersect at a point known as the origin (0,0). Every point on this plane can be identified by a pair of numbers \( (x, y) \) which represent its coordinates; the first number corresponds to the x-axis, and the second to the y-axis.

To graph an inequality like \( y < 3 \) on the coordinate plane, you need to visualize the location of all points that have a y-value less than 3. These points don't lie on a single line but fill an entire area below the line \( y = 3 \). It's crucial to remember that every point on a coordinate plane represents a potential solution to an inequality, and your job is to determine which points make the inequality true.

Interpreting Linear Inequalities

or equal to). If the inequality were \( y \leq 3 \) (note the line under the < sign), the line would be solid, and points on the line would be included in the solution set. The key to understanding linear inequalities is to recognize the difference between 'less than' and 'less than or equal to' as this will determine how you represent the boundary on your graph.

Applying Inequality Shading

Inequality shading is a visual technique used on a graph to illustrate the solution set of an inequality. It involves shading the area where the inequality holds true. For the inequality \( y < 3 \) on a coordinate plane, you would shade below the dashed line at \( y = 3 \). This shaded region shows all the points where the y-value is less than 3.

When you're shading for an inequality, it's essential to remember that the shaded area continues indefinitely in the direction where the inequality is true. Hence, for \( y < 3 \) the shading would extend to the bottom of the graph, depicting that as y-value decreases, it continues to satisfy the inequality. The dashed boundary line is critical as it ensures someone reading the graph understands that points on the line \( y = 3 \) are not included in the solution set. Inequality shading is a powerful tool in visual learning, and when done correctly, it gives a quick and intuitive grasp of which points satisfy the inequality.