Problem 46 - Introductory Algebra for College Students

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$4 \square -3$$

Step-by-Step Solution

Answer

The correct expression is \(4 > -3\).

Step 1: Understand the Problem

The task is to determine which symbol, less than (<) or greater than (>), should be placed between the numbers 4 and -3 to make the sentence true.

Step 2: Compare the Two Numbers

Place the numbers 4 and -3 on the number line. Remember, numbers on the right are larger than numbers placed on their left. The number 4 is to the right of -3 on the number line, so 4 is greater than -3.

Step 3: Insert the Correct Symbol

The correct symbol to insert between 4 and -3 is '>'. So, the correct expression is \(4 > -3\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that describe the relationship between two values, indicating that one is greater than or less than the other. Using symbols such as (less than) and (greater than), we can write these relationships in a concise form. Understanding inequalities is crucial because they are not only foundational in mathematics but also in real-life situations where comparison of quantities is necessary.

For example, considering the exercise comparing 4 and -3, we apply an inequality to demonstrate that 4 is larger than -3. This is also a fundamental part of algebra, where inequalities help us understand and solve a range of problems from simple comparisons to complex equations.

Number Line

A number line is a visual representation that helps us understand the ordering of numbers. It is a straight line with numbers placed at intervals, where each number's position is based on its value. The number line extends infinitely in both directions, but for comparison, we focus on the segment containing the numbers of interest. To compare numbers using a number line, we remember the simple rule: numbers to the right are greater than those to the left.

In the given exercise, when we place 4 and -3 on the number line, we immediately see that 4 is to the right of -3, indicating that 4 is indeed greater. This helps validate our understanding of inequalities in a tangible, visual manner, and reinforces the concept that higher numbers are placed further to the right.

Algebraic Reasoning

Algebraic reasoning is the process of using algebraic concepts to solve problems. It involves recognizing patterns, understanding the properties of numbers, and applying operations to manipulate algebraic expressions. This form of reasoning is fundamental when working with inequalities, as it requires an understanding of how a change in one part of an equation or inequality can affect the whole.

In our textbook exercise, algebraic reasoning leads us to conclude that 4 is indeed greater than -3 because we understand the properties of positive and negative numbers. We reason that a positive number is always greater than a negative number, therefore effectively solving the problem. This type of reasoning is a key skill in mathematics and grows more complex and powerful as one's study of algebra progresses.