Problem 17 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$[-3,1]$$

Step-by-Step Solution

Answer

The interval [-3,1] can be represented as the inequality \(-3 \leq x \leq 1\). When representing it on a number line, simply mark the points at -3 and 1 with closed points and draw a line connecting them.

Step 1: Translate the Interval to an Inequality

Interval [-3,1] represents all values x that are between -3 and 1 inclusive. In terms of inequalities, this can be expressed as \(-3 \leq x \leq 1\). This inequality simply states that x is greater than or equal to -3 and less than or equal to 1, which is exactly what our interval states.

Step 2: Represent the Inequality on a Number Line

Draw a number line, and denote two points on it: -3 and 1. Both are closed points (filled circle), indicating that they are included in the interval, which corresponds with the interval notation. Draw a line connecting -3 and 1 as they represent the range of the interval \(-3 \leq x \leq 1\).

Key Concepts

Intervals

Intervals are a way to express a range of numbers between two endpoints. In mathematical terms, they define a set of numbers where each number within the set is greater than or equal to some value and less than or equal to another. There are different types of intervals: open, closed, and half-open (which also goes by "half-closed").

• A closed interval, denoted by square brackets like \([-3, 1]\), means that both endpoints are included in the set. In other words, -3 and 1 are part of the interval.
• An open interval, represented using parentheses like \((a, b)\), means neither endpoint is included.
• A half-open interval (also known as half-closed) has one endpoint included and one not, such as \([a, b)\) or \((a, b]\).

In the exercise example \([-3, 1]\), the square brackets indicate that the interval is closed, meaning all values between -3 and 1, including -3 and 1 themselves, are part of the solution set.

Number Line Graphing

Number line graphing is a visual way to demonstrate the solution set of an inequality. It helps to clearly show which numbers are included in an interval and which are not. To start, draw a horizontal line which will serve as your number line.

• Mark the relevant endpoints on your line, which in our example are -3 and 1.
• If the endpoints are included in the interval (which is the case in closed intervals), represent them with filled circles at these points. Otherwise, open circles would be used.
• Finally, connect these endpoint circles with a line to indicate that all numbers between -3 and 1—including those numbers themselves—are part of the set.

This method not only visualizes the range of possible values but also helps confirm the correct inequality representation from an interval.

Inclusive Inequality

In inequalities, "inclusive" means that the endpoints are included in the solution set. This is often represented using "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)) symbols. An inclusive inequality for the interval \([-3,1]\) is written as \(-3 \leq x \leq 1\).

• \(-3 \leq x\) indicates that the element \(x\) can be -3 or any number greater than -3.
• \(x \leq 1\) shows that \(x\) can be 1 or any number less than 1.

These combined expressions illustrate a scenario where \(x\) ranges between -3 and 1, including both those boundary numbers. This translates directly to our closed interval and mirrors the graphical representation on the number line. Understanding inclusive inequality is crucial for correctly interpreting and graphing intervals.