Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Express the solution set of each inequality in interval notation and graph the interval. $$x \leq 3$$

Step-by-Step Solution

Answer

The solution of the inequality \(x \leq 3\) in interval notation is \((-\infty, 3]\) and the graph extends from 3 to negative infinity on a number line.

Step 1: Understand the inequality

The inequality \(x \leq 3\) means that x can be any number that is less than or equal to 3.

Step 2: Express in interval notation

Interval notation is another way of expressing which numbers make an inequality true. The symbol '[' is used to include the number and the symbol '(' is used to exclude the number. For the inequality \(x \leq 3\), the solution set is \((-\infty, 3]\) where '-\infty' represents negative infinity meaning x can be any number up to and including 3.

Step 3: Graph the interval

To graph this interval on a number line, a solid circle is drawn at 3 indicating that 3 is included in the solution, and a line is drawn from 3 extending to negative infinity. The line must extend left, indicating that all numbers less than 3 are part of the solution.

Key Concepts

Inequalities

In the world of algebra, inequalities are statements that demonstrate a non-equal relationship between expressions. They often show that one expression is bigger, smaller, or at most equal to another. For instance, the inequality represented by x ≤ 3 tells us that the variable x can take any value that is less than or equal to 3.

Understanding inequalities is crucial because they delineate the range of values that variables can take in real-life situations, such as setting a budget (x must be less than or equal to the amount of money available) or filling a container with water (the volume of water v must be less than or equal to the container's capacity).

It's important to note that there are different types of inequality symbols, each with a specific meaning:

Number Line Graphing

Graphing on a number line provides a visual representation of a range of numbers that satisfy an inequality. It's a clear way to see all possible values of x that make the inequality true. In our example x ≤ 3, we graph this by drawing a solid circle at 3 on the number line. The solid circle means 3 is included in the possible values because the inequality is 'less than or equal to'.

If it were 'less than' but not equal to, we'd use an open circle to show that the number itself is not a solution. For x ≤ 3, the line is drawn extending leftwards from 3 because all numbers less than 3 also satisfy the inequality. It's this visual aid that can be especially helpful to students when trying to grasp the concept of ranges and inequalities.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and involve variables, numbers, and operations such as addition, subtraction, multiplication, and division. In an inequality like x ≤ 3, x represents an algebraic expression, although it is simply a variable in this case.

In more complex scenarios, you might see something like 2x + 4 ≤ 10, where 2x + 4 is the algebraic expression. Understanding how to manipulate these expressions is key to solving inequalities.

When working with algebraic expressions, it's important to maintain balance. If you perform an operation on one side of the inequality, you must do the same to the other side to maintain the inequality's truth. This balance principle is the bedrock for solving algebraic expressions embedded within inequalities.