Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, write the set in interval notation. $$ \\{x \mid-1

Step-by-Step Solution

Answer

The set is \((-1, 3)\) in interval notation.

Step 1: Understand the Set Notation

The set \( \{x \mid -1 < x < 3\} \) represents all real numbers \( x \) that are greater than \(-1\) and less than \(3\). This is written in set-builder notation.

Step 2: Write in Interval Notation

To convert from set-builder notation to interval notation, identify the lower and upper bounds of the set. Here, the lower bound is \(-1\) (not included), and the upper bound is \(3\) (not included). In interval notation, this is represented as \((-1, 3)\), using parentheses to indicate that the endpoints are not included in the set.

Key Concepts

Set-builder Notation

Set-builder notation is a concise way to specify a set by describing the properties that its members must satisfy. In this example, the set is written as \( \{x \mid -1 < x < 3\} \). This notation tells us that the set includes all elements \( x \) that meet the condition \(-1 < x < 3\).
Set-builder notation commonly uses a vertical bar "\mid" or a colon ":" to mean "such that," and everything on the right specifies the condition. It provides a succinct way to write sets that could otherwise be complex or lengthy if listed explicitly.
Some general features include:

• The expression on the left, before the "\mid," represents an element of the set.
• The conditions or restrictions are specified on the right side.
• It's often used for sets of numbers like integers or real numbers.

Understanding set-builder notation is helpful for converting sets into other forms like interval notation.

Real Numbers

Real numbers include both rational numbers, such as fractions and integers, and irrational numbers, like the square root of 2 or pi. In mathematics, real numbers form an unbroken continuum which means they include all the decimal expansions.
The set described in the problem \( \{x \mid -1 < x < 3\} \), involves real numbers. This means that within the interval \((-1, 3)\), every point represents a real number. Real numbers are used in various scientific and engineering contexts, where continuous data range is essential.

• Rational numbers: These are numbers that can be expressed as a fraction \( \frac{a}{b} \) where both \( a \) and \( b \) are integers, and \( b eq 0 \).
• Irrational numbers: These numbers cannot be written as a simple fraction. Their decimal form is non-repeating and non-terminating.
• Every real number can correspond to a point on the number line.

Understanding real numbers is crucial for working with intervals and interpreting various mathematical problems.

Boundaries

Boundaries define the limits of a set or interval. They are important in determining the scope and focus of the set. In the given exercise's set-builder notation \( \{x \mid -1 < x < 3\} \), the boundaries are -1 and 3.
The boundaries in this case are known as open boundaries. This means that the boundary points themselves are not included in the set. The interval notation for this set is \((-1, 3)\), which clearly indicates the open boundaries using parentheses.
Key points to remember about boundaries in interval notation:

• Boundaries determine which end points are included or excluded from the set.
• Open boundaries use parentheses \((-1, 3)\).
• Closed boundaries use square brackets \([-1, 3]\).

Recognizing boundaries helps to understand other related concepts such as inclusivity and exclusivity, which are crucial in determining what values are part of the set.

Inclusive and Exclusive

When discussing intervals in mathematics, two essential concepts arise—inclusive and exclusive boundaries. These terms indicate whether the boundaries are part of the set (inclusive) or not (exclusive).
In the set \( \{x \mid -1 < x < 3\} \), both boundaries are exclusive, meaning -1 and 3 are not included in the set. This is represented as \((-1, 3)\) in interval notation, using parentheses.

• Inclusive indicates that the boundary value is part of the set. It uses square brackets, like \([-1, 3]\).
• Exclusive indicates that the boundary value is not part of the set. It uses parentheses, like \((-1, 3)\).

Understanding whether boundaries are inclusive or exclusive is crucial when interpreting and solving problems involving intervals. These concepts help specify exactly which values belong within the set or range described.