Problem 19 - Introductory Algebra for College Students

Question

Express the solution set of each inequality in interval notation and graph the interval. \(x<4\)

Step-by-Step Solution

Answer

The solution to the inequality \(x<4\) in interval notation is \((-∞, 4)\).

Step 1: Solve the Inequality

The inequality given is \(x<4\). As there is no other variable or constant with x, it's already solved. So, the solution for this is all real numbers less than 4.

Step 2: Write in Interval Notation

Interval notation is a simplified way to write the solution set for an inequality. In this case, the solution to our inequality \(x<4\) is all real numbers less than 4. This is represented in interval notation as \((-∞, 4)\). The parentheses represent that 4 is not included in the solution. The symbol \(-∞\) represents the negative infinite, which says the solution goes all the way to negative infinity.

Step 3: Graph the Interval

Mark 4 on the number line. As the inequality is \(x<4\) (and not \(x≤4\)), we use an open circle at 4 that indicates 4 is not a part of solution set. Draw a line starting from 4 and going towards the left, indicating the values the x can take, which are all the real numbers less than 4.

Key Concepts

Inequalities

Inequalities are mathematical statements that compare two values or expressions. They tell us about the relative sizes or orders of the expressions. An inequality like \(x < 4\) means "x is any number less than 4." Inequalities are different from equations, as they allow for numerous possible solutions rather than a single number. Here are some common symbols used in inequalities:

• \(<\) means "less than"
• \(>\) means "greater than"
• \(≤\) means "less than or equal to"
• \(≥\) means "greater than or equal to"

It's important to remember that inequalities describe a range of solutions. For an inequality like \(x < 4\), this means any number less than 4 is a valid answer. Understanding how to express and interpret these mathematical relationships is crucial for solving problems in algebra and calculus.

Graphing Intervals

Graphing intervals is a visual way to represent solutions to inequalities on a number line. It helps us see the range of numbers that satisfy the inequality. To graph an interval like \((-∞, 4)\), start by marking the number 4 on the number line.Because the inequality \(x < 4\) does not include 4 itself, we use an open circle at 4. This shows that 4 is not part of the solution. If the inequality were \(x ≤ 4\), a closed circle would be used to indicate inclusion.Next, draw a line extending from the open circle at 4 to the left towards negative infinity. This line signifies that all numbers to the left of 4 satisfy the inequality \(x < 4\). Graphing intervals is a great way to visualize mathematical concepts and see how solutions stretch across the number line.

Real Numbers

Real numbers include all the numbers we typically think of: whole numbers, fractions, and decimals. They form a continuous set of values that lie on an infinite number line. Real numbers can be positive, negative, or even zero.When we talk about solutions to inequalities like \(x < 4\), we're usually referring to real numbers. Real numbers can either be rational (like 2 or \(\frac{1}{2}\)) or irrational (like \(\sqrt{2}\) or \(\pi\)). What makes real numbers "real" is their completeness, meaning between any two real numbers, there are infinitely many other real numbers.In interval notation, we're expressing a part of the real number line. For instance, the interval \((-∞, 4)\) expresses all real numbers that are less than 4. Understanding real numbers and how they work is foundational for grasping more complex mathematical concepts and solutions.