Problem 59 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -9 \quad-8 $$

Step-by-Step Solution

Answer

-9 < -8

Step 1: Understanding the problem

We need to compare two numbers: -9 and -8, and determine the correct inequality sign (either \(<\) or \(>\)) that makes a true statement.

Step 1: Analyzing Negative Numbers

When comparing negative numbers, remember that the larger number is the one that is less negative. This means it has a value closer to zero.

Step 2: Compare -9 and -8

Since -9 is further from zero than -8, -9 is actually less than -8. So we state this as \(-9 < -8\).

Step 3: Conclusion

The inequality \(-9 < -8\) correctly represents the relationship between the two numbers.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front of them. For instance, -9 is a negative number. It's important to understand that the more negative a number is, the further away it is from zero. This is in contrast to positive numbers which increase in value as they move away from zero.

A useful tip when thinking about negative numbers is to imagine them on a temperature scale. Consider -9 as being colder than -8, as it is lower down the thermometer. This perspective can help you remember that as negative numbers decrease, they become more negative.

When comparing negative numbers, keep in mind:

• The number closer to zero is the larger number.
• The further from zero, the smaller the number.
• -8 is closer to zero than -9, so -8 is larger than -9.

Inequalities

Inequalities are mathematical expressions used to show how two numbers, expressions, or quantities relate to each other. The basic inequality symbols are:

• \(<\) meaning "less than"
• \(>\) meaning "greater than"

These symbols help indicate the relative size of numbers.

In the context of negative numbers, inequalities are a bit counterintuitive. With positive numbers, you might usually say 9 > 8 because 9 is larger. With negatives, however, you must remember that numbers decrease as their absolute value increases. Hence, for negative numbers, -9 < -8 because -9 is more negative than -8 and therefore smaller. Inequalities play a crucial role in ordering numbers and understanding number relationships.

Number Line Position

A number line is a straight, horizontal line with numbers placed at intervals along the length of the line. Numbers increase as you move right and decrease as you move left.

On a number line, you can clearly see the position of -9 and -8. -9 is to the left of -8, demonstrating that it is smaller than -8. The closer a number is to the left end of the line, the smaller it is. Conversely, numbers on the right side are larger.

Using a number line, you can

• Visually compare two numbers.
• Determine which number is larger or smaller quickly.
• Understand the distance from zero, providing an idea of the number's size.

Using number lines is a simple and effective way to master inequalities, especially when dealing with negative numbers.