Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Express the solution set of each inequality in interval notation and graph the interval. $$x \leq 1$$

Step-by-Step Solution

Answer

The solution to \(x \leq 1\) is \((-\infty, 1]\). The graphical representation includes every number on the number line that is less than or equal to 1.

Step 1: Understand the inequality

The inequality given is \(x \leq 1\). This means that x is less than or equal to 1.

Step 2: Translate the inequality to interval notation

In interval notation, \(x \leq 1\) is expressed as \((-\infty, 1]\). This means that x takes all the values from negative infinity up to and including 1.

Step 3: Graph the solution

On a number line, plot a filled dot at the point corresponding to 1. This indicates that 1 is included in the solution set. Then draw a line extending to the left from 1, towards negative infinity. This graphical representation shows that every number that is less than or equal to 1 is part of the solution set.

Key Concepts

Interval Notation

When dealing with inequalities, interval notation provides a concise way to describe a set of numbers that satisfy the inequality. For the inequality \(x \leq 1\), interval notation is especially handy.
To write this in interval notation, you identify the range of values that satisfy the inequality.

• The inequality \(x \leq 1\) includes all numbers less than or equal to 1. Since there is no lower limit, we start from negative infinity.
• In interval notation, we represent this as \(( -\infty, 1 ]\), where the parenthesis \(( -\infty \)) indicates that negative infinity is not a specific number and cannot be included.
• The bracket \([ 1 ]\) shows that 1 is included in the set.

Remember, parentheses \(()\) are used for values that are not included, such as infinity or when the inequalities are strict (like < or >). Brackets \([]\) are used for values that are part of the set (like ≤ or ≥).
Thus, interval notation offers a neat method to express an unending range of numbers concisely.

Graphing Inequalities

Graphing inequalities on a number line helps visually interpret the solution set. Let's take the inequality \(x \leq 1\) as an example. This inequality shows that x includes all numbers less than or equal to 1.
To graph this, follow these steps:

• Begin by drawing a number line where you can mark relevant points, especially the endpoint of the inequality.
• For the point where \(x = 1\), use a filled circle or dot. This indicates that the value 1 is included in the solution set because of the "equal to" component.
• From this point, draw a line or arrow extending to the left, signifying all numbers less than 1 are included. This arrow shows that the series of solutions continues indefinitely toward negative infinity.

By graphing, we can easily see and understand the full range of solutions for inequalities like \(x \leq 1\). The visual representation makes it clear how much of the number line falls under the solution set.

Number Line Representation

A number line representation is a visual tool that aids in understanding mathematical concepts like inequalities. It simplifies the process of identifying which numbers are part of a solution set for an inequality.
For \(x \leq 1\), the representation is straightforward.

• The number line is a straight endless line with numbers marked on it to show a range.
• You start by marking 1 on this line since it's a crucial boundary for our inequality.
• Place a filled dot or circle at 1 to indicate it's part of our solution due to the 'equal to' component in \(\leq\).
• Shade the section of the number line that stretches left from 1, showing the inclusion of all numbers less than 1.

This representation not only helps in solving single inequalities but also when evaluating compound inequalities. By using a number line, we gain a clear view of which values fulfill the criteria set by an inequality like \(x \leq 1\). It’s an essential skill for students to master in order to learn how to visually analyze mathematical solutions.