Problem 47 - College Algebra

Question

Write the interval in set-builder notation. $$ (-\infty, 6) $$

Step-by-Step Solution

Answer

\( \{ x \mid x < 6 \} \)

Step 1: Understanding Intervals and Set-builder Notation

The interval \((-\infty, 6)\) represents all real numbers that are less than 6. "Set-builder" notation is a way to describe a set by stating the properties its members must satisfy. We are going to express these properties in the form of conditions that define the set.

Step 2: Defining the Set Condition

We need to translate the interval \((-\infty, 6)\) into a set by defining the property of the numbers. In this case, the numbers \(x\) in our set satisfy the condition \(x < 6\).

Step 3: Writing the Set-builder Notation

The set-builder notation encapsulates the condition from the previous step using curly braces and a vertical bar which means "such that." The notation in this particular example is: \[ \{ x \mid x < 6 \} \]This reads as: "the set of all \(x\) such that \(x\) is less than 6."

Key Concepts

Interval Notation

Interval notation provides a concise way to describe a set of numbers along a number line. It uses parentheses and brackets to represent whether particular endpoints are included or excluded from the set.
In interval notation:

• Parentheses, \((\text{ and } )\), indicate that an endpoint is not included in the interval. For example, \((a, b)\) means all numbers greater than \(a\) and less than \(b\), excluding \(a\) and \(b\) themselves.
• Brackets, \([\text{ and } ]\), mean that an endpoint is included. Thus, \([a, b]\) includes \(a, b\), and all numbers between these endpoints.
• Infinity symbols are always accompanied by parentheses, as infinity is a concept rather than a tangible number we can include.

For instance, \(( -\infty , 6)\) describes all numbers less than 6. Infinity has no bound, so we use a parenthesis to denote that we never actually "reach" infinity in the interval.

Inequalities

Inequalities are mathematical expressions that describe the relativity between two values. They tell us whether one value is smaller, larger, or simply different from another.
The following symbols signify different types of inequalities:

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

These symbols are useful for expressing ranges of numbers and are pivotal in set-builder notation. For example, the expression \(x < 6\) illustrates all numbers \(x\) less than 6. Even though inequalities do not require solving like equations, they allow us to define a set. When expressing a set with inequalities, you establish a condition for all elements within the set.

Real Numbers

Real numbers comprise all the numbers that can be found on the number line. This includes integers, rational numbers (fractions), and irrational numbers (non-repeating decimals). Real numbers are fundamental in mathematics as they represent continuous quantities unlike discrete numbers such as whole numbers.
Here are a few key characteristics of real numbers:

• They include positive, negative, and zero values.
• Rational numbers like 1/2 can be expressed as fractions of two integers.
• Irrational numbers such as \(\pi\) or \(\sqrt{2}\) cannot be expressed as exact fractions as they have non-terminating, non-repeating decimal expansions.

In mathematics, real numbers are crucial to expressing both integer and fractional values, which are essential when defining intervals or inequalities. The concept of real numbers ensures understanding of both the whole and continuous nature of numerical sets.