Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, write the set in interval notation. $$ \\{x \mid x<4\\} $$

Step-by-Step Solution

Answer

The set is \((-\infty, 4)\) in interval notation.

Step 1: Understand the Set Notation

The given set notation \( \{x \mid x < 4 \} \) describes the set of all real numbers \( x \) such that \( x \) is less than 4. This is known as set-builder notation.

Step 2: Identify Type of Inequality

The inequality \( x < 4 \) indicates all numbers smaller than 4. Notice that 4 is not included in the set because the inequality is strict (i.e., does not include equal to).

Step 3: Translate to Interval Notation

In interval notation, if a set includes all numbers less than a certain value (but not the value itself), it is written using a parenthesis to indicate exclusion. Therefore, the interval for \( x < 4 \) is written as \((-\infty, 4)\).

Step 4: Interpret the Interval Notation

The interval \((-\infty, 4)\) starts from negative infinity, which is not a real number, hence always has a parenthesis, up to 4, where 4 is not included (denoted by a parenthesis). This means the set includes all real numbers less than 4.

Key Concepts

Set-builder notation

Set-builder notation is a method of describing a set by stating the properties that its members must satisfy. It is a concise way of defining a set by a condition. In the expression \( \{x \mid x<4\} \),

• \(x\) represents a typical element of the set.
• The vertical bar \(\mid\) can be read as "such that."
• The condition \(x<4\) specifies that the set includes all values of \(x\) that are less than 4.

This notation is very powerful when working with sets, especially when dealing with large or even infinite sets. For instance, writing out all numbers less than 4 would be impossible, but using set-builder notation, it's clear and easy to understand. This efficient form of representation is particularly useful in mathematics for defining complex sets without having to list out every single element.

Inequalities

Inequalities are mathematical expressions involving the symbols like \(<\), \(\leq\), \(>\), or \(\geq\), which indicate the relative size or order of values. When you see \(x < 4\), you know:

• \(x\) is any number less than 4.
• 4 is not part of this set, emphasized by the use of \(<\) instead of \(\leq\).

Understanding inequalities is essential because they allow us to describe a range of values rather than a single number. Often, you'll encounter strict inequalities, such as \(<\) or \(>\), which do not include the boundary value (like 4 in this case). In contrast, non-strict inequalities, denoted with \(\leq\) or \(\geq\), do include the boundary value. It's crucial to pay attention to these symbols, as they significantly impact the solution sets you're working with.

Real numbers

Real numbers form the set of numbers that include all the rational and irrational numbers, essentially encompassing everything you can find on the number line. In the context of set-builder notation and inequalities:

• The expression \(\{x \mid x < 4\}\) refers to all real numbers to the left of 4 on the number line.
• This set involves infinite real numbers since real numbers are continuous, meaning between any two real numbers, there are infinitely many others.

Real numbers include integers, fractions, and decimals that extend infinitely in both the positive and negative directions. Understanding real numbers is fundamental because they provide a complete picture of the values that variables can take in many mathematical contexts. They make it possible to work with all possible numbers that can be found on a number line, a concept depicted whenever we use interval or set-builder notation with inequalities.