Problem 31 - Elementary Algebra

Question

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol \((<,>,=)\) in place of the \(*\) $$-5 *-1$$

Step-by-Step Solution

Answer

Answer: -5 < -1

Step 1: Compare the numbers

First, compare the given numbers -5 and -1. Since -5 is further to the left on the number line than -1, we can conclude that -5 is smaller than -1.

Step 2: Choose the appropriate symbol

Based on our comparison in step 1, we concluded that -5 is smaller than -1. Therefore, we will replace the * with the less than symbol (<) to indicate this relationship.

Step 3: Write the final expression

Now that we have chosen the appropriate symbol, we can replace the * with the less than symbol, and our final expression becomes: $$-5 <-1$$

Key Concepts

Inequalities

Understanding inequalities is essential for comparing values and making mathematical arguments. In an inequality, two values are compared to identify which one is greater, which is lesser, or to determine that they are unequal but without specifying by how much.

In the exercise, we compared two real numbers, \( -5 \) and \( -1 \). The relationship between these two numbers is an inequality because they are not equal, and we are tasked with finding the appropriate relational symbol. In this case, since \( -5 \) is to the left of \( -1 \) on the number line, it is considered 'less than' \( -1 \). Thus, the proper symbol to use is \( < \) leading to the solution \( -5 < -1 \).

Key Tips for Solving Inequalities

• Check the direction: Remember that numbers to the left on a number line are smaller than those to the right.
• Be mindful of negative numbers: Negative numbers can be tricky. The larger the negative number, the smaller its value.
• Use the number line: When in doubt, sketch a quick number line to visualize the relationship.
• Practice: The more inequalities you compare, the more intuitive it will become.

Number Line

A number line is a visual representation where real numbers are laid out in a straight, horizontal line. Every point on the line corresponds to a unique real number. In basic terms, it's like a map of numbers where the location of each number relative to others indicates its value relative to them.

In comparing \( -5 \) and \( -1 \), the number line helps us see that \( -5 \) is to the left of \( -1 \), indicating its lesser value. The further to the left a number is, the smaller its value. Similarly, numbers to the right have greater values.

Understanding Number Line Positioning

• Negative numbers: Are positioned to the left of zero.
• Positive numbers: Are found to the right of zero.
• Zero: Acts as the middle point, separating positive and negative numbers.

This concept is crucial because it provides a concrete way to understand and solve inequalities and other mathematical problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). They can represent real-world quantities and their relationships, and they are essential for expressing general mathematical rules.

In our exercise, the algebraic expression \( -5 * -1 \) required us to find the correct relational symbol to replace the asterisk. Algebraic thinking helps us understand that replacing the \( * \) with the right symbol completes the expression, which conveys the inequality relationship clearly.

Algebraic Expressions in Comparing Numbers

• Variables and symbols express relationships between quantities.
• Equality and inequality symbols (like \( =, <, > \)) are essential in showing how values relate to one another.
• Interpreting expressions correctly is important for solving equations and inequalities.

Through practice and application, algebra becomes a powerful tool for solving various types of mathematical problems.