Problem 14 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ (-\infty, 3.5] $$

Step-by-Step Solution

Answer

Set-builder notation for \(-\infty, 3.5]\) is \{x | x \(\leq\) 3.5\}. The graph on a number line starts at 3.5 (included) and extends towards the left all the way.

Step 1: Convert Interval Notation to Set-Builder Notation

The interval \(-\infty, 3.5]\) can be written in set-builder notation as \{x | x \(\leq\) 3.5\}. This notation describes a set of all numbers 'x', such that 'x' is less than or equal to 3.5.

Step 2: Graph the Interval on a Number Line

This interval includes every number up to and including 3.5, and doesn't include numbers greater than 3.5. Put a solid dot at 3.5 to represent that it's included in the set and draw a line to the left to represent all the numbers less than 3.5. The line should proceed to the end of the number line, representing \(-\infty\).

Key Concepts

Interval Notation

When we talk about interval notation, we are referring to a way of describing the set of all numbers between two endpoints. It's a type of shorthand for writing intervals and is very helpful in mathematics. Interval notation uses brackets and parentheses to show which numbers are included or excluded in a set.

• Square brackets [ ] mean that the endpoint is included in the interval. For example, \[ (5, 10] \] means from 5 to 10, where 10 is included but 5 is not.
• Parentheses ( ) mean that the endpoint is not included, just like in \[ (5, 10) \], which means all numbers greater than 5 and less than 10, without including the endpoints.

This is particularly useful when dealing with infinite sets. For example, \[ (-∞, 3.5] \] includes all numbers from negative infinity up to and including 3.5. Understanding interval notation makes it easier to read and interpret mathematical sets of numbers.

Number Line Graph

Number line graphs are visual representations of intervals. They can help us understand the set of numbers described by interval notation. We begin with a straight horizontal line, which we think of as extending in both directions. On this line, we mark numbers to show the position of any specific points of interest.
To represent an interval on a number line:

• Identify the numbers included in the interval.
• Use a solid dot to show that a number is included in the interval. If a number is not included, use an open dot.
• Draw a line or arrow to show all numbers that are part of the interval.

For instance, the interval \[ (-∞, 3.5] \] has a solid dot on 3.5 and an arrow pointing left, toward negative infinity, which shows that all numbers less than or equal to 3.5 are included in the set.

Inequalities

Inequalities describe relationships between two expressions or numbers. They tell us if one value is smaller, larger, or equal to another and can be used to describe numerical ranges or conditions within a set.
Common inequality symbols include:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

When we write \( x \leq 3.5 \), it means 'x' can be any value that is less than or equal to 3.5. This inequality is a concise way of describing the interval \[ (-∞, 3.5] \] and includes the number 3.5 itself, but not any number greater than 3.5.

Mathematical Sets

A set is a collection of distinct objects or numbers. In mathematics, we use sets to group numbers together based on certain rules or conditions.
Set-builder notation is a powerful tool that defines a set by stating a property that its members must satisfy. It's written in the form \( \{ x | x \text{ condition} \} \).
For example, the set described by the notation \( \{ x | x \leq 3.5 \} \) includes all numbers x that are 3.5 or less. It's directly connected with interval notation and inequalities and provides another way to express the same idea. This helps mathematicians and students work with complex concepts in simplified terms, ensuring clarity when solving problems or communicating ideas.