Problem 109 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. \(-24 \quad-20 \text { (Section } 1.3, \text { Example } 6)\)

Step-by-Step Solution

Answer

The correct filled statement is \(-24 < -20\).

Step 1: Analyzing Negative Numbers

Negative numbers can feel tricky to comprehend. On the number line, -24 is positioned to the left of -20 which implies it is less than -20.

Step 2: Applying Comparison Operator

After understanding the positioning of -24 and -20 on the number line, the correct operator is chosen to form a true statement. As -24 is less than -20, the '<' operator is used.

Step 3: Writing the Final Answer

The correct operator that makes a true statement is inserted between the numbers, leading to the final expression -24 < -20.

Key Concepts

Number Line

A number line is an immensely useful tool for visualizing and understanding the relationship between numbers. It is a straight, horizontal line on which every point corresponds to a unique number. In discussing negative numbers, the number line extends to the left from 0, with numbers getting smaller as they move further away from zero.

For instance, looking at the negative numbers -24 and -20 on the number line, we see that -24 would be placed to the left of -20. This visual cue instantly informs us that -24 is less than -20. It's like a visual expression of smaller values being positioned further left. The number line provides a clear and definitive way to compare these numbers and understand the concept of 'less than' or 'greater than' at a glance.

Using the number line to compare negative numbers is essential for building foundational skills in mathematics as students progress towards more complex topics. It simplifies the concept of negative value comparison by giving a spatial representation, which can be more intuitive than abstract concepts.

Inequalities

Inequalities are mathematical expressions that relate two values or expressions, showing that one is lesser (<), or greater (>) than the other. They are a cornerstone for understanding how different numbers relate to each other.

When comparing negative numbers, inequalities help us express the relationship explicitly. For instance, if we take two numbers, like -24 and -20, we can say that -24 is less than -20, which we write as -24 < -20. This is not just a statement of fact, it's an inequality that mathematically represents the ordering of these two numbers.

Inequalities are not just about numbers; they also apply to variables and algebraic expressions. This mathematical concept is not only used for simple calculations but also forms the basis of solving problems, understanding functions, and much more. While they might seem straightforward, inequalities require a good grasp of number sense and position on the number line, especially when negative numbers are involved.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (such as x or y), and operations (like addition, subtraction, multiplication, and division). They are the building blocks of algebra, allowing us to represent real-world situations and mathematical concepts in a simple and concise way.

When we compare negative numbers using algebraic expressions, we express a relationship between these numbers. For example, if we want to represent that a certain number 'a' is greater than -24 and less than -20, we can write this as -24 < a < -20. This expression succinctly conveys that 'a' sits between -24 and -20 on the number line.

Algebraic expressions become particularly powerful when we combine them with inequalities. They can then be used to solve for unknowns and to understand how changing one value affects others within an equation. While they can initially appear intimidating, algebraic expressions are just a very effective shorthand for conveying mathematical ideas and relationships.