Problem 2 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Graph a solid line at \(y = -2\) and shade below it.

Step 1: Identify the Inequality Type

The inequality given is \(y \leq -2\). This indicates that all points where the \(y\) coordinate is less than or equal to \(-2\) are part of the solution. A horizontal line on the graph will represent this inequality.

Step 2: Graph the Equation Part

Start by graphing the line \(y = -2\). Since this line represents all points where \(y\) is equal to \(-2\), it will be a horizontal line crossing the \(y\)-axis at \(-2\).

Step 3: Determine Line Type

The inequality uses \(\leq\), which signifies 'less than or equal to'. To represent 'or equal to', use a solid line when graphing \(y = -2\).

Step 4: Shade the Appropriate Region

Since the inequality is \(y \leq -2\), shade the region below the line \(y = -2\), as this region represents all \(y\) values less than or equal to \(-2\).

Key Concepts

Horizontal Line

When you have an inequality like \(y \leq -2\), the first step in graphing this is to consider the equation part, \(y = -2\). This is a classic example of a horizontal line in a graph.

A horizontal line means that all the points along the line have the same \(y\)-value. For \(y = -2\), the line stretches across the entire graph horizontally, maintaining the \(y\)-value of -2 no matter what the \(x\)-value is.

Horizontal lines are unique because they do not lean or curve; they remain constant as they pass through the graph.

To plot the line \(y = -2\), simply draw a line parallel to the \(x\)-axis that crosses the \(y\)-axis at \(-2\). Regardless of the \(x\) coordinate, \(y\) stays \(-2\):

• Has the same \(y\)-value at all points
• Crosses the \(y\)-axis at a particular point (here at -2)

Solid Line

After drawing the basic line for \(y = -2\), it’s time to decide the type of line that represents the inequality \(y \leq -2\). This is where we introduce the concept of the solid line.

A solid line is used when an inequality includes "or equal to," symbolized by \(\leq\) or \(\geq\). This means that the points on the line itself are part of the solution to the inequality.

For the inequality \(y \leq -2\), you'll graph the line \(y = -2\) as a solid line, which indicates that points on this line satisfy the inequality:

• Solid line shows inclusion of the boundary
• Symbols \(\leq\) or \(\geq\) require solid lines
• Points on the line are part of the solution

Using solid lines is important for accurately representing the inequality and its complete range of solutions.

Shading Regions

Once you've established the line type, the next step is shading the appropriate region of the graph to represent the inequality \(y \leq -2\). This part of graphing inequalities visually shows all the potential solutions.

For the graph of \(y \leq -2\), we shade the area that falls below the line because we want to include all points where the \(y\)-values are less than or equal to \(-2\). This shaded region becomes a visual cue of the inequality's solution.

There are a few key points to remember when shading:

• Shading always covers the set of solutions allowed by the inequality.
• If the inequality were \(y < -2\), you would still shade below the line but use a dashed line to indicate the \(-2\) isn't included.
• The shaded area helps you visualize solutions that meet the inequality's condition.

Thus, shading the correct region ensures that we understand all potential solutions from a graphical standpoint.