Problem 33 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$|-2| \quad|-7|$$

Step-by-Step Solution

Answer

\(|-2| < |-7|\)

Step 1: Calculate the Absolute Value of -2

To find the absolute value of a number, we disregard its sign and consider its magnitude. Thus, the absolute value of -2 is 2, written as \(|-2| = 2\).

Step 2: Calculate the Absolute Value of -7

Similarly, the absolute value of -7 is found by disregarding the sign, resulting in \(|-7| = 7\).

Step 3: Compare the Absolute Values

Now compare the absolute values obtained in Steps 1 and 2. We have \(2\) and \(7\). Since \(2 < 7\), we place a less than sign \(<\) between them.

Step 4: Write the Final Statement

Using the comparison from Step 3, the final statement with the correct inequality sign is \(|-2| < |-7|\), confirming the truth of the statement.

Key Concepts

Comparing Numbers

When we talk about comparing numbers, we're discussing how to determine which number is greater, lesser, or if they are equal. This concept is foundational in mathematics and helps us not only with basic arithmetic but also with more advanced concepts later on. Comparing absolute values is an interesting twist in this game because it focuses on the magnitude of numbers, ignoring their signs.
While comparing numbers, here’s what you typically do:

• If you have a positive and a negative number, the positive number is always greater.
• If both numbers are negative, the one closer to zero is considered greater since it has a smaller absolute value.
• If you are comparing absolute values, the magnitude sans sign is what counts, just as in our problem.

Comparing numbers in this way helps us make informed decisions, not only in math class but in real-world scenarios as well.

Understanding Inequalities

Inequalities are mathematical statements that describe the relative size or order of two objects. When we use symbols like < and >, we are expressing relationships between numbers. For example, in the case of | \(-2\) | < | \(-7\) |, we understand that the absolute value of \(-2\) is smaller than the absolute value of \(-7\) because 2 is less than 7.
Here are some fundamental points on inequalities:

• "Less than" is indicated by < and "greater than" by >.
• An inequality can be strict (using < or >) or it can be inclusive (using ≤ or ≥).
• When solving inequalities, you often perform similar steps as when solving equations. However, one key difference is that multiplying or dividing by a negative number will flip the inequality sign.

Understanding inequalities helps establish boundaries and conditions within math problems and everyday situations.

The Role of Mathematics Education

Mathematics education is fundamental in developing logical thinking and problem-solving skills. Understanding basic concepts such as comparing numbers and inequalities forms the building blocks for higher-level math. This education goes beyond the classroom, enhancing decision-making, critical thinking, and analytical skills.
Here’s how mathematics education equips learners:

• By fostering critical thinking, students learn to analyze situations and identify relationships between different elements.
• It encourages precision and attention to detail, essential for solving complex problems.
• Develops a strong foundation for future topics like calculus, statistics, and analytics.

Through consistent practice and application of these math concepts, students not only improve their math skills but also gain confidence in handling various tasks and challenges.