Problem 6 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ \\{x | x \leq-3\\} $$

Step-by-Step Solution

Answer

The interval notation is \((-\infty, -3]\).

Step 1: Understand the Set Notation

The given set is described using set-builder notation: \( \{x | x \leq -3\} \). This notation specifies that we're interested in all the real numbers \( x \) such that \( x \) is less than or equal to \(-3\).

Step 2: Identify the Boundary of the Interval

The boundary of this interval is \(-3\), as we are looking for all numbers less than or equal to \(-3\). There is no upper boundary since the values can extend infinitely in the negative direction.

Step 3: Convert to Interval Notation

In interval notation, an inequality of the form \( x \leq a \) is represented as \((-\infty, a]\) where the closing bracket ']' indicates that \( a \) is included in the interval (closed interval at \( a \)). Thus, the interval notation for the given set \( \{x | x \leq -3\} \) is \((-\infty, -3]\).

Key Concepts

Set-Builder Notation

Set-builder notation is a powerful mathematical way to define a set by stating properties that its members must satisfy. It is often used to specify a set of numbers that meet a given condition, such as inequalities. For example, \( \{x \mid x \leq -3\} \) describes a set of all real numbers \( x \) that are less than or equal to \(-3\). The vertical bar "|" can be read as "such that," and it separates the elements from the condition they must satisfy.

• Basics of Set-Builder Notation: It's a concise way to express large sets by defining a rule for inclusion rather than listing all elements individually.
• Use in Mathematics: Useful for defining intervals, especially when dealing with inequalities.

Understanding set-builder notation is key in moving from a descriptive representation to a graphical or interval representation of sets.

Real Numbers

Real numbers are the backbone of mathematics and include all the numbers on the number line. This includes all the rational numbers, such as fractions and integers, and all the irrational numbers, which can't be expressed as simple fractions. Real numbers can be positive, negative, or zero, and they make up the set of numbers we use most often in everyday life and math classes.

• Understanding Real Numbers: Real numbers occupy the entire continuous number line with no gaps.
  This completeness allows for easy representation of intervals.
• Real Numbers in Intervals: Interval notation often includes real numbers to depict all numbers within certain bounds.

When working with inequalities, it's helpful to remember that real numbers allow us to include decimals and fractions, ensuring precision and flexibility in mathematical expressions.

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. They are like equations but instead of an "=" sign, they use signs like "<", ">", "≤", or "≥" to show that one value is greater than, less than, or simply not equal to another. In the expression \( x \leq -3 \), it tells us that \( x \) can be any number that's equal to or less than \(-3\).

• Types of Inequalities: "Greater than" (\(>\)), "less than" (\(<\)), "greater than or equal to" (\(≥\)), and "less than or equal to" (\(≤\)).
• Graphical Representation: Inequalities define regions in graphical plots, indicating where a particular solution set exists.
• Conversion to Interval Notation: An inequality like \( x \leq -3 \) translates to interval notation as \((−∞, -3]\), where the bracket indicates inclusion of \(-3\).

Mastering inequalities helps in understanding and solving a variety of mathematical problems, where the solutions are not always a single number but rather a range of possible values.