Problem 75 - Elementary Algebra

Question

Replace the \(*\) with the appropriate relation symbol \((<,>) .-7 *-3\).

Step-by-Step Solution

Answer

Answer: <

Step 1: Compare the numbers

In order to determine the correct relation symbol, compare -7 with -3. Since -7 is smaller than -3 (because it has a higher absolute value), the relation symbol should be '<'.

Step 2: Replace the '*' symbol

Now that we have determined that -7 is smaller than -3, we can replace the '*' in the expression with the relation symbol '<'.

Step 3: Write the final expression

With the relation symbol, the final expression is \(-7<-3\).

Key Concepts

Comparison of Numbers

When comparing numbers, especially negative ones, it's crucial to understand how the number line works. The further a number is to the left on the number line, the smaller it is. For example, when comparing -7 and -3, although both numbers are negative, -7 is further to the left on the number line compared to -3.
To judge which number is smaller or larger, you can imagine the number line:

• Numbers increase in value as you move to the right.
• Numbers decrease in value as you move to the left.

Thus, for two negative numbers such as -7 and -3, -7 is the smaller number because it is to the left of -3 on the number line. Understanding this helps in accurately comparing any set of numbers.

Relation Symbols in Mathematics

In mathematics, relation symbols are used to indicate the relationship between two numbers. The most common symbols are:

• "\(<\)" which means "less than"
• "\(>\)" which means "greater than"
• "\(=\)" which means "equal to"

Using these symbols provides an easy way to summarize the relationship between numbers. For example, with the expression \(-7 < -3\), the symbol "\(<\)" indicates that -7 is less than -3.
It's typically straightforward to use these symbols once you determine the numerical relationship. If you're ever unsure, visualize or draw a number line to see which number lies where.
Remember:

• Use "\(<\)" when the first number is smaller than the second.
• Use "\(>\)" when the first number is larger than the second.
• Use "\(=\)" when both numbers are the same.

Absolute Values

Absolute value can be a helpful tool when considering inequalities, especially with negative numbers. The absolute value of a number is its distance from zero on the number line, without considering direction. In mathematical terms, the absolute value of \(x\) is denoted as \(|x|\).
For example:

• The absolute value of 7 is \(|7| = 7\).
• The absolute value of -7 is \(|-7| = 7\).

Notice that both 7 and -7 have the same absolute value, demonstrating that absolute value removes any negative sign.
When dealing with inequalities and comparing negative numbers, realizing that \(-7 < -3\) based on their absolute values sets a clear understanding. However, absolute values themselves are not helpful in strictly determining which of two numbers is larger or smaller—just how far a number is from zero. Always combine the concept of absolute values with the number line analysis to fully understand inequalities.