Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$y>1$$

Step-by-Step Solution

Answer

Graph a dashed horizontal line at \( y = 1 \) and shade above.

Step 1: Understand the Inequality

The inequality given is \( y > 1 \). This means we are considering all the points in the coordinate plane where the \( y \)-value is greater than 1.

Step 2: Graph the Boundary Line

Start by graphing the line \( y = 1 \), which serves as the boundary for the inequality. This line is a horizontal line passing through all points with a \( y \)-coordinate of 1.

Step 3: Determine the Type of Line

Since the inequality is \( y > 1 \) (and not \( y \geq 1 \)), the boundary line should be dashed. This indicates that points on the line \( y = 1 \) are not solutions to the inequality.

Step 4: Shade the Solution Area

Shade the area above the dashed horizontal line. This region represents all points where the \( y \)-coordinate is greater than 1, satisfying the inequality \( y > 1 \).

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and graphs. It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at the origin, which is the point (0, 0).
The coordinate plane is divided into four quadrants:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive and y is negative.

When graphing inequalities, like the provided exercise of graphing the inequality \(y > 1\), it is important to understand how these quadrants help us determine the areas of the plane that must be shaded or bounded. The coordinate plane provides a framework for visualizing solutions and exploring relationships between variables.

Boundary Line

The boundary line is a crucial part of graphing inequalities on the coordinate plane. It represents the set of points that satisfy the equation derived from the inequality. In the exercise, we derived the equation \(y = 1\) from the inequality \(y > 1\).
This line acts as a dividing path between the solution area and the non-solution area. It defines the edge of the region we are interested in. Depending on the inequality, the boundary line can be:

• Solid, if the inequality is \(\geq\) or \(\leq\), which means the points on the line are included in the solution set.
• Dashed, like in our exercise with \(y > 1\), indicating that points on this line are not part of the solution.

In our specific graphing practice, the boundary line \(y = 1\) will be dashed because the inequality is strictly greater than (>), not including the line itself. This visual cue helps quickly identify that the boundary line separates areas that satisfy the inequality from those that do not.

Shading Solution Area

Shading the solution area is the final step in graphing an inequality, such as \(y > 1\). The shading represents all points on the coordinate plane where the inequality holds true, showing the region that satisfies it.
For our exercise:

• The boundary line is dashed, indicating it's not part of the solution.
• We shade the region above the line \(y = 1\), reflecting all points where the \(y\)-value is greater than 1.

This visual method effectively communicates which portion of the plane is included in the solution set. Always ensure your shading is clear and extends to the edges of the graph for greater clarity. Using shading also helps in problems involving multiple inequalities, as it allows for quick visualization of overlapping solution regions.

Horizontal Line

Horizontal lines are unique because they run parallel to the x-axis. In graphing, they have the equation form \(y = c\), where \(c\) is a constant. In our inequality graph \(y > 1\), the horizontal line is \(y = 1\).
Important characteristics of horizontal lines include:

• All points along the line have the same y-value.
• They are perpendicular to vertical lines, which have the form \(x = c\).

Understanding horizontal lines is essential in the context of inequalities. Since they define a constant value for y, changes in x do not affect the line. For the inequality \(y > 1\), this horizontal boundary clearly marks the starting point for shading the solution area. It shows the region where \(y\) is consistently greater than a constant value, helping visualize the entire solution space effortlessly.