Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ (3,7) $$

Step-by-Step Solution

Answer

\(3 < x < 7\)

Step 1: Understanding the Interval Notation

The interval notation \((3, 7)\) indicates an open interval. This means that the interval includes all numbers between 3 and 7, but not the endpoints 3 and 7 themselves.

Step 2: Writing the Inequality

Since the interval is open and does not include the endpoints, it can be expressed as an inequality: \(3 < x < 7\). This inequality indicates that \(x\) is greater than 3 and less than 7.

Key Concepts

Understanding Open Intervals

An open interval is a way to describe a set of numbers between two endpoints, without including the endpoints themselves. It's like saying, "Think about numbers between two points, but don't count the end numbers." For example, the open interval

• \((3, 7)\) includes numbers like 4, 5.5, and 6.9, but not 3 or 7.

We use parentheses in the interval notation to show we aren't including the endpoints. This is a key feature that separates open intervals from closed intervals (which do include the endpoints and use square brackets). So, if you see parentheses, remember - the border numbers are out of the game!

Exploring Interval Notation

Interval notation is a mathematical shorthand used to represent a set of numbers along the number line. It is a concise, visually clear way to display a range of values. Here's what you need to know about interval notation:

• Parentheses \((\) and \()\) are used for open intervals. This indicates the endpoints are not included in the set.
• Square brackets \([\) and \()]\) are used for closed intervals, where endpoints are included.
• An interval can also be half-open (or half-closed): one end is included, the other is not. For instance, \([3, 7)\).

When you see an interval written like \((3, 7)\), it effectively communicates a range without using words - extremely handy in mathematics.

Expressing Intervals as Inequalities

One fundamental skill in math is translating between different ways of expressing the same idea, such as going from interval notation to inequalities. When you perform this translation, you help clarify what numbers belong to a set. Let's look at how we do this:

• For the open interval \((3, 7)\), we don't include the 3 and 7. This translates to the inequality \(3 < x < 7\).
• If the interval had been closed, say \([3, 7]\), the corresponding inequality would be \(3 \leq x \leq 7\).

By using inequalities, you put into words the rules of the set, showing clearly which numbers make sense in the context. This makes inequalities a powerful way to ensure everyone understands the boundaries of a problem space.