Problem 15 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. 3 \(\quad\) 7

Step-by-Step Solution

Answer

3 < 7

Step 1: Identify the Given Numbers

We have two numbers to compare: 3 and 7.

Step 2: Understand the Symbols

We need to place either the symbol < (less than) or > (greater than) between the numbers to form a true statement.

Step 3: Compare the Numbers

Since 3 is less than 7, the correct symbol to use between them is <.

Step 4: Write the Conclusion

Place the symbol < between the numbers to get the statement: 3 < 7, which is true.

Key Concepts

Understanding the Less Than Symbol

The less than symbol is a fundamental part of mathematics used to compare numbers. It is represented by the symbol \(<\). When you see this symbol between two numbers, it means that the number on the left is smaller than the number on the right. For example:

• 3 < 7 means "3 is less than 7"
• 2 < 5 means "2 is less than 5"

The less than symbol is not just limited to numbers. It can also be used in comparing variables and expressions, such as \(x < y\) or \(2x + 3 < 7\). Whenever you encounter the less than symbol, ask yourself: Is the number on the left smaller than the number on the right? If yes, then your statement is true.

How to Use the Greater Than Symbol

The greater than symbol, denoted by \(>\), is used when a number is larger than another number. When you see this symbol between two numbers, it indicates that the first number is greater. For instance:

• 7 > 3 means "7 is greater than 3"
• 10 > 5 means "10 is greater than 5"

Much like the less than symbol, it is not restricted to numbers alone. You can use it for variables and expressions, such as \(x > y\) or \(3x + 2 > 8\). Remember, when using the greater than symbol, verify: Is the number before the symbol larger than the one after? If yes, then the statement is valid.

Crafting True Statements in Mathematics

To form true mathematical statements, it is crucial to properly use the less than and greater than symbols. A true statement accurately represents the relationship between numbers or expressions. Consider these examples:

• For 4 and 9, since 4 is smaller, the true statement is 4 < 9.
• With 15 and 8, as 15 is larger, the correct statement is 15 < 8.

True statements can extend beyond numbers to involve algebraic expressions and inequalities, like \(2x + 5 < 10\) — verifying these often requires solving for variables. Understanding how to compare numbers and expressions accurately allows you to make valid conclusions in mathematics. Always double-check your comparisons to ensure they reflect the true relationship among the elements involved.