Problem 44 - College Algebra Essentials

Question

Determine whether each statement in Exercises 43–50 is true or false. $$-6>2$$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Understand the inequality symbols

First, understand what the symbol '>' means. It means 'greater than'. It's used to compare two numbers where one is larger (greater) than the other.

Step 2: Compare the values

Now, compare the numbers that are being given in the inequality. '-6' and '2'. A negative number, -6 here, is always less than a positive number, 2, since negative numbers lie to the left of 0 on a number line while positive numbers lie to the right.

Step 3: Make the conclusion

Since -6 is less than 2, the inequality '-6 > 2' is not true. So, the statement is false.

Key Concepts

Inequality Symbols

In mathematics, inequality symbols are tools that help us to compare the relationship between two values. There are several inequality symbols, but the two most common are:

• \( > \) : This means 'greater than'. It indicates the value on the left is larger than the value on the right.
• \( < \) : This means 'less than'. It indicates the value on the left is smaller than the value on the right.

The inequality symbols are crucial in determining which of two numbers is larger or smaller. They provide a clear indication of order between numbers, which is extremely useful for solving various mathematical problems. Remember, using the correct symbol is important, as it can entirely change the meaning of an expression. Think of these symbols as arrows pointing from the bigger number towards the smaller one.

Comparing Numbers

Comparing numbers involves determining whether one number is bigger, smaller, or equal to another. When comparing, it helps to visualize numbers on a number line:

• Numbers to the right on the number line are larger.
• Numbers to the left are smaller.

For example, to compare \(-6\) and \(2\), imagine the number line. The number \(-6\) is found well to the left of zero, while \(2\) is to the right. Thus, \(-6\) is less than \(2\), making any statement "\(-6 > 2\)" false.
When comparing numbers, especially in inequalities, always validate your understanding by checking the position and value of each number on a number line.

Negative and Positive Numbers

Understanding negative and positive numbers is fundamental in comparing and solving inequalities.

• Positive numbers are greater than zero and typically represent quantities greater than nothing.
• Negative numbers are less than zero and often represent quantities less than nothing or debt.

These numbers have different characteristics:

• Positive numbers are always greater than negative numbers.
• Negative numbers are smaller the further they are from zero. For example, \(-6\) is less than \(-1\).

To make this clearer, think about this scenario: when it comes to temperature, \(-10\text{°C}\) is much colder (hence, smaller) than \(5\text{°C}\). Understanding the relative size of these numbers is crucial for comparing and solving mathematical problems efficiently.