Problem 57 - Intermediate Algebra

Question

Explain how to decide whether the boundary of the graph of a linear inequality should be drawn as a solid or a dashed line.

Step-by-Step Solution

Answer

Use a dashed line for \(<\) or \(>\); a solid line for \(\leq\) or \(\geq\).

Step 1: Understand the Inequality Symbols

The first step is to recognize the type of inequality symbol used in the linear inequality. The inequality symbols can be \(<\), \(>\), \(\leq\), or \(\geq\).

Step 2: Determine the Line Type

The type of line used to draw the boundary of the graph depends on the inequality symbol. If the inequality is \(<\) or \(>\), a dashed line is used to indicate that the boundary is not included in the solution set. If the inequality is \(\leq\) or \(\geq\), a solid line is used to indicate that the boundary is included in the solution set.

Key Concepts

Graphing Inequalities

Working with linear inequalities involves graphing solutions on a coordinate plane. Unlike equations, where lines or curves represent exact solutions, inequalities show a range of solutions. To graph an inequality, start by thinking of it as an equation, to find the boundary line. You can then determine which side of the line satisfies the inequality.
This involves creating a graph where:

• The x-axis (horizontal) and y-axis (vertical) help establish positions.
• The boundary line (linear equation) acts as a reference point.

After plotting the boundary line, choose a test point that isn't on the line, like \(0,0\) if possible. Substitute it into the inequality. If true, shade the side of the line containing that point, representing the solution set. If false, shade the opposite side. Remember, the shaded region represents all points that satisfy the inequality.

Boundary Lines

Boundary lines are crucial in representing linear inequalities on a graph. They separate the plane into two half-planes: one where the inequality is true, and one where it's false. These lines can be either solid or dashed.

• Solid Lines: Used when the inequality includes the boundary (\(\leq\) or \(\geq\)). This means points on the line satisfy the inequality.
• Dashed Lines: Used when the boundary is not part of the solution (\(<\) or \(>\)). This indicates points on the line do not satisfy the inequality.

For example, for the inequality \(y \geq 2x + 3\), the line \(y = 2x + 3\) is drawn solid because the solution includes the boundary line itself. In contrast, \(y > 2x + 3\) would be drawn with a dashed line, signifying exclusion of the boundary.

Inequality Symbols

Understanding inequality symbols is key to graphing linear inequalities correctly. These symbols depict the relationship between two expressions:

• \(<\) and \(>\): These indicate strict inequalities, meaning the expressions on either side are not equal. Use these when the graph should not include the boundary line. Hence, you use a dashed line to show exclusion.
• \(\leq\) and \(\geq\): These show inclusive inequalities. They mean the expression can be equal to the boundary. This necessitates a solid line to indicate inclusion.

Recognizing what each symbol represents helps in deciding whether the boundary should be part of the solution set or not. For example, with \(x \leq 5\), the solutions include the line \(x = 5\). But with \(x < 5\), the line itself doesn't count as a solution, affecting how we represent it graphically.