Problem 30 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$|8| \quad -2$$

Step-by-Step Solution

Answer

The correct symbol is \(>\), so \(|8| > -2\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, without considering direction. For any positive or zero number, the absolute value is the number itself. For negative numbers, it is the positive version of that number.

Step 2: Calculate |8|

Since 8 is a positive number, its absolute value is 8. Therefore, \(|8| = 8\).

Step 3: Compare the Numbers

Now we need to compare 8 and -2 to determine which is larger. Since 8 is positive and larger than any negative number, including -2, we have the comparison: \(8 > -2\).

Step 4: Insert the Correct Symbol

Based on the comparison in the previous step, place the greater-than symbol (\(>\)) between \(|8|\) and -2: \(|8| > -2\).

Key Concepts

Understanding Number Lines

A number line is a simple and effective visual tool that helps in understanding the order and distance between numbers. Imagine a straight line that extends infinitely in both directions; this represents all numbers, where zero is at the center.
Negative numbers, such as -1, -2, and -3, stretch to the left of 0, decreasing as they move further. Positive numbers like 1, 2, and 3 are to the right, increasing as they extend further out.

• Each step to the right on the number line represents an increase, while each step to the left signifies a decrease.
• The number line helps visualize not just the size of numbers but also their relationships to each other.

This tool makes it easier to see how far a number is from zero and effectively compare numbers, both negative and positive.

Positive and Negative Numbers

When dealing with numbers, it's essential to distinguish between positive and negative values. Positive numbers are greater than zero and often represent quantities like elevation levels or temperatures above freezing. Negative numbers, on the other hand, are less than zero and can describe debts or temperatures below freezing.

• Positive numbers have no negative sign in front of them, like +5 or simply 5.
• Negative numbers are marked with a minus sign, for example, -5.

Understanding this difference is crucial when arranging numbers or when performing operations such as addition and subtraction. One nifty feature is that any positive number is inherently greater than any negative number.

Making Comparisons

Comparing numbers involves determining which number is larger or smaller. This is easily done using the greater-than \((>)\) or less-than \( (<) \) symbols. To decide which symbol to use, look at the numbers on the number line.

• If a number is to the right of another on the line, it's greater.
• Conversely, if it's to the left, it's less.
• In cases involving absolute values, calculate the absolute value first, then compare.

For example, comparing \(|8|\) with -2, we find \(|8| = 8\), since 8 is positive. Clearly, 8 is further to the right on the number line than -2, making \(8 > -2\). This comparison helps decide the correct inequality symbol to use.