Problem 18 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$[-2,5]$$

Step-by-Step Solution

Answer

The interval \([-2,5]\) can be written as the inequality -2 ≤ x ≤ 5.

Step 1: Understand the Interval

An interval \([-2,5]\) in mathematics is a set of real numbers with the property that any number that lies between -2 and 5 is also included in the set. The square brackets mean that the interval is closed at the endpoints -2 and 5, meaning these numbers are included in the interval.

Step 2: Express as an Inequality

The interval \([-2,5]\) can be written as the inequality -2 ≤ x ≤ 5. This captures the idea that x is any number between -2 and 5, inclusive.

Step 3: Draw the number line

To represent this interval on a number line, mark the points -2 and 5 on the number line. Since the interval includes -2 and 5, darken these points on the number line. Then, draw a line segment connecting -2 and 5 to show that every number between -2 and 5 is included.

Key Concepts

Interval Notation

Interval notation is a way of describing a set of numbers along a number line. It's compact and combines inequalities into a simple format.
For example, the interval

• \([-2,5]\) means that the set includes all the numbers between -2 and 5.
• The use of square brackets \([ \ ]\) indicates that the endpoints -2 and 5 are part of the interval.

Interval notation is ideal for describing continuous sets and is widely used in calculus and analysis. Instead of writing complex inequalities, it simplifies understanding by packaging everything into one neat expression.

Closed Interval

A closed interval, like \([-2, 5]\), includes its endpoints. This means both -2 and 5 are part of the interval, and any number in between is also included.

Here are some key points about closed intervals:

• In mathematical notation, square brackets \([, ]\) are used to denote a closed interval.
• The inequality representation of a closed interval \([-2, 5]\) is -2 ≤ x ≤ 5, showing that x can be -2, 5, or any number between them.
• Closed intervals are perfect for scenarios where the exact endpoints matter, like when calculating actual boundary values.

Understanding closed intervals helps in better grasping the concept of inclusiveness in sets.

Number Line Graphing

Number line graphing provides a visual way to represent intervals, making the concept tangible. It helps in understanding which numbers belong to the interval and which don't.

Here's how you can graph \([-2, 5]\) on a number line:

• Start by drawing a horizontal line with evenly spaced marks. These are the number line segments, representing numbers.
• Mark the points -2 and 5.
• Since the interval is closed, you’ll darken or fill in the marks on both -2 and 5. This signifies inclusion.
• Then, draw a solid line segment between -2 and 5 to show that every number between them is included in the interval.

Number line graphs are an excellent tool for visual learners and provide clarity by illustrating abstract mathematical concepts.