Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Explain how to determine whether a parenthesis or a square bracket is used when graphing an inequality on a number line.

Step-by-Step Solution

Answer

Parentheses are used for strict inequalities, while square brackets are used for inclusive inequalities.

Step 1: Understanding Inequalities

Before determining whether to use a parenthesis or a square bracket on a number line, we need to understand the inequality expression. Inequalities can be either strict (e.g., ">" or "<") or inclusive (e.g., "≥" or "≤").

Step 2: Strict Inequalities

For strict inequalities, such as "<" or ">", we use a parenthesis. This indicates that the number at that point is not included in the solution set. For example, if the inequality is "x < 5", we use a parenthesis at 5.

Step 3: Inclusive Inequalities

For inclusive inequalities, such as "≤" or "≥", we use a square bracket. This indicates that the number at that point is included in the solution set. For instance, if the inequality is "x ≤ 5", we use a square bracket at 5.

Step 4: Graphing on a Number Line

When graphing inequalities on a number line, you start by identifying the critical value from the inequality. Depending on whether the inequality is strict or inclusive, you place either a parenthesis or a square bracket at this critical point.

Key Concepts

Understanding Inequalities

Inequalities are numerical expressions that show the relationship between two values. They help us identify when one value is bigger, smaller, or equal to another. When expressing inequalities, we use different symbols to represent different relations:

• > or < for which the expression is strict
• ≥ or ≤ for expressions that are inclusive

Understanding these symbols is the foundation for dealing with inequalities. For example, "x > 5" means that x can be any number greater than 5 but not 5 itself. In contrast, "x ≥ 5" tells us that x is greater than or equal to 5, so 5 is included in the set of possible solutions. By recognizing these symbols, we can determine key points when graphing these expressions.

Using a Number Line for Graphing Inequalities

A number line is a simple yet effective tool to visually represent inequalities. To graph an inequality on a number line, follow these easy steps:

• Draw a straight line and mark equal intervals to represent numbers.
• Identify the point that represents the value in the inequality.
• Decide between using a parenthesis or a bracket (more below on that).
• Shade the part of the number line that represents the solution set.

By depicting inequalities on a number line, you gain a visual understanding of which numbers make the inequality true. Similarly, graphing can clarify which numbers are not part of the solution set, enabling a clearer interpretation of the inequality at hand.

Parenthesis vs. Bracket Notation

When graphing inequalities on a number line, choosing between parenthesis and bracket is crucial. Here’s how you know which one to use:

• Parenthesis (< ): Use this for strict inequalities such as ">" or "<". It signifies that the boundary value is not included. For instance, if the inequality is "x < 5", the 5 is not part of the solution, imagined as a boundary that's open.
• Bracket [ ]: This is for inclusive inequalities like "≤" and "≥". It indicates that the boundary value is included as part of the solution set, like "x ≤ 5" meaning x can be 5 or any number less than 5.

Choosing the correct notation is essential. It ensures your graph accurately represents the solution set.

Distinguishing Strict and Inclusive Inequalities

Knowing if an inequality is strict or inclusive helps determine the correct graphical representation on a number line:

• Strict Inequalities (“<” and “>”): These expressions exclude the boundary number itself. The language of these inequalities is critical marker—"less than" and "greater than"—signifying that the boundary value itself is a sort of open door that the solution cannot touch.
• Inclusive Inequalities (“≤” and “≥”): Here, the relationship includes the boundary number. These expressions, "less than or equal to" and "greater than or equal to" indicate that the boundary point is part of the set of solutions. Such inequalities include the boundary as being part of the point, allowing the "]" or ")" to be used affirmatively on the boundary.

By differentiating between strict and inclusive inequalities, you ensure clarity in your mathematical expressions and graph representations, giving an accurate picture of the range of solutions.