Problem 104 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<,=,\) or \(>,\) between the two numbers. $$|-5| \quad|-2|$$

Step-by-Step Solution

Answer

So, \(|-5| > |-2|\). The symbol that should be placed between \(|-5|\) and \(|-2|\) to make a true statement is \(>\).

Step 1: Find the absolute value of the first number

The absolute value of a number is its numerical value without its sign. Therefore, the absolute value of -5 is 5. So, \(|-5|=5\).

Step 2: Find the absolute value of the second number

Proceeding in the same manner, the absolute value of -2 is 2. So, \(|-2|=2\).

Step 3: Compare the two absolute values

Now we know \(|-5|=5\) and \(|-2|=2\). It is clear that 5 is greater than 2.

Key Concepts

Integer Comparison

Comparing integers might seem simple, but when negative numbers are involved, it requires paying close attention. Integers are whole numbers that can be positive, negative, or zero. When comparing these numbers, the goal is to determine which is greater, or if they are equal. However, things change when we deal with the absolute values of these integers.
In this exercise, we begin by comparing the absolute values of two negative integers: -5 and -2. It is essential to remember that the individual numerical position of an integer on the number line dictates its size. When observing integers, a number further to the right on the number line is always greater. This fundamental understanding aids greatly when performing integer comparison tasks.

Inequalities

Inequality symbols ( <, =, > ) describe the size relationship between two values. Knowing how to use these symbols is essential in expressing mathematical sentences effectively.
Here are some pointers for better understanding:

• The symbol < means "less than." When you see this, it means the number on the left is smaller than the one on the right.
• The symbol > signifies "greater than." If this is the applicable symbol, it indicates the left number is larger than the right number.
• The symbol = shows that both sides are equal.

Applying this to absolute value comparison: Once you calculate the absolute values correctly, selecting the right inequality symbol becomes an easier task.
In our example, you compare 5 and 2 . Because 5 > 2 , we use the > symbol.

Numerical Values

Numerical values refer to the actual magnitude of numbers, regardless of whether they're positive or negative. In the case of absolute value, we are interested in these magnitudes without considering the signs.
For example, the rule for absolute value is straightforward:

• If you have a number like -5 , its absolute value |-5| is 5 .
• Similarly, the absolute value of -2 , which is |-2| , is 2 .

The purpose of dealing with absolute values is to simplify the task. Ignoring signs means we focus on the pure size of the numbers—this directly links to achieving the correct integer comparison and selecting the right inequality symbol.

Mathematical Symbols

Mathematical symbols such as | |, <, =, and > are tools that help represent mathematical scenarios succinctly. Understanding these symbols is key to tackling problems efficiently.
Here is a short breakdown:

• The bars, | |, indicate absolute value, which represents a number's distance from zero.
• The inequality symbols < and > compare values, signaling which number is bigger or smaller.
• The equal symbol = states that two values are exactly the same.

In this exercise, | | was used to transform negative integers into easier-to-compare positive values, before deciding that 5 > 2 , thus concluding that |-5| > |-2|. With a solid grasp of these symbols, solving related tasks becomes smooth and intuitive.