Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, write the interval in set-builder notation. [-3,5)

Step-by-Step Solution

Answer

\( \{ x \, | \, -3 \leq x < 5 \} \)

Step 1: Identify the Interval Type

The interval given is \([-3, 5)\). A closed bracket \([-3]\) means it includes -3, while an open bracket \((5)\) means it does not include 5.

Step 2: Define Set-Builder Notation

Set-builder notation describes a set by specifying the properties that its members must satisfy. It often takes the form \( \{ x \, | \, \text{condition} \} \), where \( x \) is a member of the set if it satisfies the condition.

Step 3: Translate Interval to Condition

For the interval \([-3, 5)\), the condition for set-builder notation would be \(-3 \leq x < 5\), meaning \( x \) is greater than or equal to -3 and less than 5.

Step 4: Write in Set-Builder Notation

Based on the condition derived, the set-builder notation for the interval is: \( \{ x \, | \, -3 \leq x < 5 \} \). This describes all \( x \) values that satisfy the condition of being between -3 and 5, including -3 but excluding 5.

Key Concepts

Interval Notation

Interval notation offers a way to represent a range of numbers using brackets and parentheses. This method helps identify which numbers are included and excluded in the interval.

When using interval notation:

• A square bracket entails inclusion of the number it accompanies.
• A round parenthesis indicates exclusion.

For example, the interval \([-3, 5)\) includes all numbers starting from -3 up to, but not including, 5. Thus, -3 is part of the set, whereas 5 is not.
Next, when handling infinity, always use round parentheses. Since infinity represents a concept rather than a number, it cannot be included. Interval notation is concise and convenient for describing continuous ranges.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Set theory provides a foundational framework for various mathematical concepts.

Sets are typically denoted by curly braces, and an important feature is how we define what is in a set.
A way to specify a set is through set-builder notation, such as \( \{ x \, | \, \text{condition} \} \).

This form lists a variable (usually \( x \)) and a condition that \( x \) must satisfy. Set-builder notation is incredibly useful for describing complex sets succinctly by focusing on the properties of elements that belong to the set.

Inequalities

Inequalities describe the relative size or order of two values. They express that one quantity is larger or smaller than another. Understanding inequalities is crucial for translating interval notations into conditions for set-builder notation.

Several symbols are commonly used:

• \( < \) denotes 'less than'
• \( \leq \) signifies 'less than or equal to'
• \( > \) means 'greater than'
• \( \geq \) indicates 'greater than or equal to'

Consider the interval \([-3, 5)\).
The corresponding set-builder notation is \( \{ x \, | \, -3 \leq x < 5 \} \), where \(-3 \leq x\) implies \( x \) can equal -3, but \(x < 5 \) means it cannot equal 5.
Employing inequalities allows us to convey precise conditions that define which elements belong to a set.