Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

\( \{ x \mid -3 \leq x < 5 \} \)

Step 1: Understand the Interval Notation

The interval \([-3, 5)\) represents all the numbers between \(-3\) and \(5\), including \(-3\) but excluding \(5\).

Step 2: Convert to Set-Builder Notation

In set-builder notation, we describe the elements of the set using a condition. For the interval \([-3, 5)\), this can be written as \( \{ x \mid -3 \leq x < 5 \} \). This notation means 'the set of all \(x\) such that \(x\) is greater than or equal to \(-3\) and less than 5.'

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way of expressing a set by specifying a property that its members must satisfy. It is particularly useful in mathematics for describing intervals, ranges, or specific conditions. In set-builder notation, a general set is represented as \( \{ x \mid \text{condition} \} \). Here, \(x\) represents the elements, and "condition" specifies the property they must satisfy.
The vertical bar is read as "such that." Hence, when we look at the set-builder notation for the interval \([-3, 5)\), written as \( \{ x \mid -3 \leq x < 5 \} \), it tells us that our set includes all numbers \(x\) meeting the criterion of being equal to or greater than \(-3\), and less than 5.
This chosen notation succinctly captures all values that fit the criteria without having to list them individually, offering a clean and efficient representation.

Inequalities

Inequalities are mathematical expressions that show the relative size or order of values. They are used to denote that two values are not equal and instead, one is larger, smaller, or within a specific range compared to another. Common symbols include:

• \(<\): Less than
• \(>\): Greater than
• \(\leq\): Less than or equal to
• \(\geq\): Greater than or equal to

When we examine the interval \([-3, 5)\), it involves two inequalities: \(-3 \leq x\) and \(x < 5\). These indicate that \(x\) can be any value starting from \(-3\) and extending up to, but not including, 5.
Using inequalities in set-builder notation helps specify the range of numbers clearly and allows for quick manipulation or alteration of conditions, which can be highly useful in various mathematical contexts.

Mathematical Notation

Mathematical notation is a system of symbolic representations of mathematical objects, concepts, and their relationships. It allows complex ideas to be conveyed efficiently with precision.
Understanding the common forms of mathematical notation, such as interval and set-builder notation, boosts problem-solving efficiency in mathematical expressions and algebraic operations.
In the context of describing intervals such as \([-3, 5)\), mathematical notation simplifies expressing these ideas without the need for wordy descriptions. Intervals represent continuous sets of numbers, and understanding their notation is crucial in fields like calculus and algebra.
As you deepen your knowledge in mathematics, becoming familiar with different types of notation becomes key to communicating complex ideas succinctly and correctly across diverse mathematical fields.