Problem 11 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ (-\infty, 3) $$

Step-by-Step Solution

Answer

The set-builder notation for the interval \(-\infty, 3\) is \(x: x<3\). On the number line, this interval extends indefinitely to the left from 3, with an open circle at 3 indicating that 3 is not included in the interval.

Step 1: Convert the interval to Set-Builder notation

The given interval is \(-\infty, 3\). This interval represents all numbers that are less than 3. So, in set-builder notation, this interval can be written as: \(x: x<3\).

Step 2: Graphing the interval on the number line

On a number line, an open circle is placed at 3, indicating that 3 is not included in the interval. All values to the left of 3 are part of the interval, so the line extends indefinitely to the left (representing \(-\infty\)).

Key Concepts

Understanding Interval Notation

Interval notation is a way of representing a range of numbers that fall between two endpoints. It can describe simple ranges, like those between two specific numbers, or it can stretch to infinity in either direction. Such as in the exercise, with the interval \( (-\infty, 3) \), we're looking at all the numbers that come before 3, but don't actually include the number 3 itself. Therefore, we use parentheses, \( ( \) or \() \) rather than brackets. Brackets, like \[ or \], imply that the endpoint is included in the interval, known as a closed interval. Whereas a parenthesis suggests that the endpoint is not included, creating what's called an open interval.

Let's exemplify: To denote all the numbers strictly less than 3 without including 3 itself, we use \( (-\infty, 3) \). Another interval might be \( [4, 7) \), which would include the number 4 but not the number 7. It's a mixed interval since it has a bracket and a parenthesis. Understanding how to read and write in this shorthand can simplify expressing ranges of numbers and the operations associated with them.

Visualizing with a Number Line

The number line is a visual representation of numbers along a horizontal line, allowing us to see the relative position and order of numbers at a glance. In the context of intervals, a number line can be exceptionally helpful. It provides a graphical representation of which numbers are included within an interval and whether the endpoints are part of the interval or not.

Consider the interval \( (-\infty, 3) \). To graph this on a number line, you'd draw a line with an open circle at 3 to show that it's not included in the set (since we used a parenthesis) and shade or draw a line to the left, extending towards infinity to express that all numbers less than 3 are included. Remember that when shading the interval, the direction you shade in represents all numbers that fall within the interval, and in this case, it goes backwards to indicate all numbers less than 3.

Inequalities and their Representation

Inequalities express the relation between two values when they are not equal. They help us determine the range of values that satisfy a particular condition. Inequalities often feature symbols like \( < \), \( > \), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to).

For the interval \( (-\infty, 3) \), the associated inequality is \( x < 3 \). This inequality states that \( x \) can be any number less than 3. The set-builder notation \( \{ x | x < 3 \} \) translates that same inequality into a format that defines a set of numbers, namely those that satisfy the condition \( x < 3 \). Inequalities and set-builder notation are often used in conjunction to give a complete picture of the solution set to a problem. When moving from interval notation to inequalities, it's important to correctly interpret the symbols to assure that the correct values are being described.