Problem 34 - Elementary Algebra

Question

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol \((<,>,=)\) in place of the \(*\) $$6 *-1$$

Step-by-Step Solution

Answer

Question: Determine the correct relation symbol between the real numbers 6 and -1, replacing the asterisk (*) with the appropriate symbol: \(6 * -1\). Answer: \(6 > -1\)

Step 1: Comparing the two numbers

Since 6 is a positive number and -1 is a negative number, we can conclude that 6 is greater than -1. Positive numbers are always greater than negative numbers.

Step 2: Replacing the asterisk with the correct relation symbol

As we concluded that 6 is greater than -1, we can replace the asterisk (*) with the greater than symbol (\(>\)): $$6 > -1$$

Key Concepts

Inequalities

When we compare two real numbers, we often use inequalities to show their relationship. These symbols—less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥)—are the language through which we express which number is bigger, smaller, or if they are equal.

For example, comparing 6 to -1, we observed that 6 is indeed greater than -1. Therefore, we state this relationship using the inequality symbol (>) to write the statement 6 > -1. It's really about understanding that inequalities help us rank numbers on the number line, giving us a visual interpretation of their order and magnitude.

• If a number 'a' is larger than 'b', we write a > b.
• If 'a' is smaller than 'b', then the statement is a < b.
• If 'a' is exactly the same as 'b', then we say a = b.

Positive and Negative Numbers

The use of positive and negative numbers is a foundational concept in algebra that helps distinguish between two types of numbers. Positive numbers are greater than zero, located to the right of zero on the number line. In contrast, negative numbers are less than zero and are found to the left.

To picture this, imagine a number line where zero is in the center, positive numbers extend to infinity on the right, and negative numbers stretch to infinity on the left. Each side's numbers are always increasing in their absolute value as you move further from zero. Hence, in any comparison between a positive and a negative number, the positive number is always greater. Therefore, 6 is greater than -1 because it is on the positive side of the number line, whereas -1 is on the negative side.

• Positive numbers are used to represent quantities that have a value more than nothing, such as distance traveled or profit.
• Negative numbers represent the absence of a quantity or a deficit, such as debt or a loss in temperature.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition, subtraction, multiplication, and division). The primary purpose of these expressions is to denote relationships between different quantities and to perform a variety of functions with these quantities.

In our exercise, although there are no variables present, understanding the roles numbers play within algebraic expressions is crucial. We are looking at a simplified expression comparing two numbers, but in more complex scenarios, algebraic expressions can equate to inequalities with variables. So, mastering how we compare numbers directly translates to more advanced algebra problems, where we can be dealing with expressions like 3x > 2y - 7. Remember, the foundational skills of comparing real numbers and understanding the signs will be immensely useful as problems become more complex.

• Expressions are made statements that allow us to solve for unknowns.
• They are the building blocks for forming equations and inequalities in algebra.