Problem 32 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$20 \quad|-6|$$

Step-by-Step Solution

Answer

20 > | -6 |

Step 1: Find Absolute Value

Calculate the absolute value of \(-6\), which is the distance of \(-6\) from zero, without considering the sign. The absolute value of a number is always positive or zero.\[ |-6| = 6\]

Step 2: Compare the Values

Compare the two numbers: \(20\) and \(6\) (the absolute value of \(-6\)). We compare them to determine which number is larger.

Step 3: Determine the Correct Inequality Symbol

Since \(20\) is greater than \(6\), the correct inequality symbol to use is \(>\). Therefore, the completed statement is \(20 > | -6 |\).

Key Concepts

Absolute Value

When we talk about absolute value, we're discussing how far a number is from zero on the number line. It's all about distance and not direction. This means that absolute value is always a non-negative number.
For example:

• The absolute value of \(-6\) is \(6\) because \(-6\) is 6 units away from zero.
• Similarly, the absolute value of \(6\) is also \(6\).
• For zero, the absolute value is \(0\) because it's already at zero!

Think of absolute value as if you're measuring how far you've traveled regardless of your direction. So, whether you're moving forward or backward, your distance in terms of 'how far' is what matters.

Comparing Numbers

Comparing numbers is a basic math skill where we determine which number is bigger or smaller. This is crucial when you need to solve inequalities or place numbers in order.
To compare numbers like \(20\) and \(6\):

• Start by looking at the two numbers. \(20\) is bigger than \(6\).
• Imagine these numbers on a number line; the one further to the right (higher) is the larger number.

This skill becomes especially useful when dealing with larger numbers or when you're asked to compare absolute values in situations, just like our example with the number \(-6\). Regardless of how big numbers are, these steps will help in identifying which is greater or smaller.

Inequality Symbols

Inequality symbols are used to show the relationship between numbers regarding their size. They tell us which number is bigger, smaller, or if they're equal.

• The symbol \(>\) means 'greater than.' So, when you see \(a > b\), it means \(a\) is bigger than \(b\).
• The symbol \(<\) means 'less than,' indicating that \(a < b\) means \(a\) is smaller than \(b\).
• These symbols are crucial in math because they help us understand and express relationships between different numbers.

For example, when we determined that \(20 > | -6 |\), we used the \(>\) symbol because 20 is greater than the absolute value of \(-6\), which is \(6\). This way, inequality symbols are perfect tools for expressing mathematical statements clearly.