Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

The inequality representation of the interval \([-5, 2)\) is -5 ≤ x < 2.

Step 1: Understanding the interval

The interval \([-5,2)\) contains all the real numbers from -5 to 2 (including -5 but not including 2). To include -5 in the interval, a square bracket [ is used and to not include 2 in the interval, a parenthesis ) is used.

Step 2: Representing the interval as an inequality

The interval \([-5,2)\) can be written as an inequality -5 \(\leq\) x < 2 where x is any real number between -5 and 2.

Step 3: Graphing the interval on a number line

To graph the interval on a number line, a line is drawn, marking the endpoints of -5 and 2. Since -5 is included in the interval, a filled circle is used at -5 and an open circle (representing exclusion) is used at 2. All values between the two points are part of the interval.

Key Concepts

Inequality Representation

Representing intervals as inequalities is a core concept in understanding mathematical expressions involving ranges. In interval notation, brackets and parentheses indicate whether endpoints are inclusive or exclusive.
For example, the interval \([-5, 2)\) includes all numbers from \(-5\) to \(2\). The square bracket \([\) means \(-5\) is included, and the parenthesis \()\) means \(2\) is not included.
This can be expressed as an inequality: \(-5 \leq x < 2\). This tells us that \(x\) can be any real number such that it is greater than or equal to \(-5\) and less than \(2\).
Using inequalities helps you visualize and understand the scope or limit of the values that \(x\) can take within the defined boundaries.

Number Line Graphing

Graphing intervals on a number line is a visual way to understand the set of numbers within a given interval. Let's break down how to graph the interval \([-5, 2)\).
First, draw a horizontal line to represent the number line. Mark the important points, \(-5\) and \(2\), on this line. At \(-5\), since the interval includes this value, use a filled circle.
Next, at \(2\), because it is not included, represent this with an open circle. This tells anyone looking at the graph that while \(-5\) is part of the set, \(2\) is not.
The segment between these two points is shaded or a solid line is drawn to show that all numbers between \(-5\) and \(2\) are included in the interval.

Real Numbers Range

Understanding the range of real numbers in an interval is essential for interpreting and working with numerical data. The term real numbers encompasses all numbers on the number line, including positives, negatives, and zero.
An interval such as \([-5, 2)\) defines a specific range within the set of real numbers. It includes all numbers greater than or equal to \(-5\) and less than \(2\).
Recognizing how brackets and parentheses define the range is crucial. A square bracket \([\) indicates inclusion of an endpoint in the range, while a parenthesis \()\) indicates exclusion.
By mastering these concepts, students can better understand mathematical statements and solve related problems with greater confidence.