Problem 24 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$(-\infty, 2)$$

Step-by-Step Solution

Answer

The inequality corresponding to the interval \(-\infty, 2\) is \(x < 2\) and the graph on the number line has an open circle at 2 with a line extending to the left of 2.

Step 1: Understand the Interval Notation

An interval notation uses parentheses and/or brackets to show which numbers are included in an interval. The given interval is \(-\infty, 2\). The round bracket or parenthesis \((\ or\ )\) excludes the number besides it from the interval, while the square bracket \([ \ or\ ]\) includes the number. Here -\infty essentially means that the interval extends to all numbers less than 2.

Step 2: Convert the Interval to Inequality

To express this interval as an inequality, look at the direction of the interval for help. Since the interval \(-\infty, 2\) includes numbers less than 2 but not 2 itself, the inequality corresponding to this interval is \(x < 2\). This inequality states that x can be any real number that's less than 2.

Step 3: Graph the Interval on the Number Line

To graph the interval on a number line, put a small open circle at 2 on the number line, to show that 2 itself is not included in the solution set and draw a line extending to the left from 2 to represent all real numbers less than 2.

Key Concepts

Inequality Notation

Inequality notation is a way to describe a range or set of numbers using symbols. It helps us understand whether specific numbers are included or excluded from that set. For instance, in the example of the interval \[(-\infty, 2)\] the inequality form would be \(x < 2\). This inequality means that "x" can be any real number that is smaller than 2.

• "<" means "less than," indicating no equality with 2.
• In contrast, "≤" would include the boundary number, like 2.

Recognizing inequality notation helps you interpret mathematical expressions quickly and effectively, and it’s crucial when transferring information from interval notation.

Number Line

The number line is a visual tool that helps you understand the relationship between numbers. It is a straight line that represents all real numbers, extending infinitely in both directions. When working with intervals, a number line gives a clear picture of which numbers are included and which are not.For our example interval \((-\infty, 2)\), we use the number line to show:

• The open circle at 2 indicates that 2 is not included.
• The shaded line extending to the left represents all numbers less than 2.

Using a number line can simplify complex inequalities and make them more understandable at a glance. It's a fundamental tool in visualizing intervals in mathematics.

Graphical Representation

Graphical representation brings abstract concepts to life through visualization. When representing an interval on a number line, it becomes easier to see what is happening.For the interval \((-\infty, 2)\), follow these steps:

• Draw a horizontal line marked with numbers.
• Place a small open circle at 2 to show it’s not included.
• Shade the line to the left of 2 to depict all numbers less than 2.

This method allows anyone to quickly grasp the extent of the interval and whether specific values like 2 are included (in this case, they are not). It's a powerful tool for interpreting inequalities and can be used to solve, verify, and understand mathematical problems.

Open Interval

An open interval is a specific type of interval notation indicating that the boundary numbers are not included. The open interval \((-\infty, 2)\) uses parentheses to show exclusion.

• "(" before -∞ suggests it goes infinitely in the negative direction.
• ")" after 2 means the interval approaches 2 but doesn't include it.

Understanding open intervals is crucial as they frequently appear in calculus and algebra. They signify continuity up to, but not including, their boundary values. Recognizing this concept can aid in solving problems related to limits, derivatives, and integral calculus.