Problem 111 - College Algebra

Question

When graphing the solutions of an inequality, what does a parenthesis signify? What does a bracket signify?

Step-by-Step Solution

Answer

In the context of graphing solutions of an inequality, a parenthesis indicates that the boundary point is not included in the solution set, representing an open interval. A bracket, on the other hand, denotes that the boundary point is included in the set, representing a closed interval.

Step 1: Understanding Parentheses

In the context of graphing an inequality, a parenthesis is used to denote that the boundary point is not included in the solution set. It indicates an open interval. For instance, if the inequality is \(x > 3\), on the number line, a parenthesis would be placed at 3 to indicate that although numbers greater than 3 are included in the solution set, the number 3 itself is not a part of the solution.

Step 2: Understanding Brackets

On the other hand, a bracket is used to signify that the boundary point is included in the solution set. It indicates a closed interval. For example, if the inequality is \(x \geq 3\), a bracket would be placed at 3 on the number line to show that the number 3 is included in the solution set, along with numbers that are greater than 3.

Step 3: Summary

In conclusion, parentheses and brackets are used in the graphing of inequalities to signify whether a boundary point is included in the solution set or not. Parentheses denote that a number is not included in the set (open interval), whereas brackets signify the inclusion of the number in the set (closed interval).

Key Concepts

Open Intervals

In mathematics, open intervals are used to describe a range of numbers where the boundary points are not included. Think of it like an open-ended question; it is not quite complete. With open intervals, parentheses are used to graphically emphasize that those particular numbers are not part of the solution. For instance, if we have an inequality like \(x > 3\), the numbers greater than 3, such as 3.1, 3.5, etc., are part of the solution, but the number 3 is not.
Open intervals are written as \((a, b)\) and include all real numbers \(x\) such that \(a < x < b\). Both of these endpoints \(a\) and \(b\) will not be part of the set itself. The use of parentheses signals this exclusion.

• Example: The interval \((2, 5)\) includes numbers like 2.1, 3, 4.99, but excludes 2 and 5.

Open intervals are particularly useful in scenarios where you are interested in trends or behaviors occurring between two points rather than at those points themselves.

Closed Intervals

Closed intervals, on the other hand, include their boundary points. Think of them as inclusive or a closed-door scenario where everything inside and at the door gets considered. You can spot a closed interval by its use of brackets.
For example, when you graph \(x \geq 3\), the bracket at 3 on a number line indicates that 3 is included as part of the solution set. This means that 3, along with numbers greater than 3, are all valid solutions.
A closed interval is written as \([a, b]\), which includes all numbers \(x\) such that \(a \leq x \leq b\). Unlike open intervals, here, the endpoints are included in the set. The brackets make this inclusion visually obvious.

• Example: In the interval \([1, 4]\), numbers like 1, 2.5, and 4 are part of the interval.

Closed intervals are ideal for situations where inclusion of boundary values impacts the context you're dealing with, such as physical measurements where starting or end points are critical.

Boundary Points

Boundary points are numbers at the edges of intervals and play a crucial role in determining whether those points are part of a solution set. The treatment of these points depends on whether you are working with open or closed intervals.

• If an interval is open, boundary points are excluded. Parentheses around the boundary point in an inequality graph show this clearly. For example, \((3, 7)\) means 3 and 7 are not part of the solution.
• When an interval is closed, boundary points are included. Brackets encompass these points, thereby making them part of the solution set. So, \([3, 7]\) includes 3 and 7.

Along with the interval notation, boundary points help in defining precisely where a solution set begins and ends. They offer clarity and precision in problems involving inequalities by providing visual cues on number lines or graphs regarding inclusivity or exclusivity.