Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Using the variable \(x,\) write each interval using set-builder notation. $$[2,7)$$

Step-by-Step Solution

Answer

\( \{ x \mid 2 \leq x < 7 \} \)

Step 1: Understand Interval Notation

The interval \([2, 7)\) indicates that 2 is included in the set, but 7 is not. This means the set contains all real numbers \(x\) such that \(2 \leq x < 7\).

Step 2: Write Using Set-Builder Notation

Set-builder notation is a way of describing a set by specifying a property that its members must satisfy. For the interval \([2, 7)\), we write the set as \( \{ x \mid 2 \leq x < 7 \} \). This reads as 'the set of all \(x\) such that \(x\) is greater than or equal to 2 and less than 7.'

Key Concepts

Interval Notation

Interval notation is a simple yet powerful way to depict a range of numbers on the real number line. It uses parentheses "(" or ")" and brackets "[" or "]" to indicate whether the endpoints of the interval are included or excluded.

• A bracket "[" or "]" means the endpoint is included in the interval.
• A parenthesis "(" or ")" means the endpoint is not included.

For example, in the interval \([2, 7)\), the number 2 is included (indicated by the bracket "["), but the number 7 is not (indicated by the parenthesis ")"). Hence, this interval captures all real numbers between 2 and 7, including 2 but excluding 7. This makes interval notation an efficient way to specify parts of the number line without verbosity. It is particularly useful in describing solutions to inequalities, ranges in functions, and datasets in statistics.

Real Numbers

Real numbers are the broad spectrum of numbers that include both rational and irrational numbers. They form a fundamental part of the number system and are depicted on the number line which extends infinitely in both directions.

• Rational numbers include fractions and integers, like 3, -4, and \( \frac{1}{2} \).
• Irrational numbers cannot be expressed as a simple fraction, such as \( \pi \) or \( \sqrt{2} \).

Real numbers are essential in interval notation, where intervals specify a range of real numbers satisfying certain conditions, like being less than, greater than, or between specific endpoints. They are utilized in various mathematical contexts, from elementary algebra to calculus, due to their comprehensive representation of quantities.

Inequalities

Inequalities are expressions that compare two values, showing if one is larger, smaller, or equal to the other with a certain restriction. They are essential when dealing with ranges of values, often represented by intervals.There are several types of inequalities:

• "\(<\)" means less than.
• "\(>\)" means greater than.
• "\(\leq\)" means less than or equal to.
• "\(\geq\)" means greater than or equal to.

In the context of the interval \([2, 7)\), the corresponding inequality would be \(2 \leq x < 7\). This inequality states that the variable \(x\) can take any value larger than or equal to 2, but strictly less than 7.Understanding inequalities is crucial for solving real-world problems, as they model constraints and thresholds in economics, engineering, and science.