Problem 46 - College Algebra

Question

Determine whether statement is true or false. \(-13<-5\)

Step-by-Step Solution

Answer

The statement \(-13 < -5\) is true.

Step 1: Identify the numbers

Identify the numbers given in the statement. In this case, the two numbers are \(-13\) and \(-5\).

Step 2: Apply the rule for negative numbers

For negative numbers, the one that is 'less negative' is actually greater. In this instance, \(-5\) is less negative than \(-13\).

Step 3: Compare the numbers

With the rule applied, we can see that \(-13\) is indeed less than \(-5\). Therefore the statement \(-13 < -5\) is true.

Key Concepts

Negative Numbers

Negative numbers are an essential part of the number system and can be a little tricky to understand at first. Unlike positive numbers, which increase as they move away from zero, negative numbers decrease.
This means the further away from zero, the 'more negative' a number is considered. For instance, while

• \(-3\) is closer to zero,
• \(-10\) is further away.

Because of this, \(-10\) is considered 'less' than \(-3\). Always remember, in terms of value, the more negative a number, the lower it is. Think of it like owing money: owing more is worse, so \(-13\) means more debt than \(-5\). This shift from positive to negative is crucial for understanding mathematical concepts like inequalities.

Comparison of Numbers

When comparing numbers, it's important to remember a few basic rules, especially when negative numbers are involved. Generally, with positive numbers:

• The larger the number, the greater its value.
• For example, \(7 > 3\) because 7 is greater than 3.

However, with negative numbers, it's the opposite. The number that appears larger, numerically, is actually less in value.

• \(-3\) is greater than \(-5\) because it is less negative and closer to zero.
• Every positive number is always greater than any negative number. For example, \(5 > -3\).

In our specific statement \(-13 < -5\), when comparing \(-13\) and \(-5\), we recognize that \(-13\) is more negative than \(-5\), thus it's smaller. Therefore, \(-13 < -5\) is true, adhering to the principles of negative number comparison.

True or False Statements

True or false statements are pivotal in math to determine the accuracy of an expression or inequality. In our example, whether the statement \(-13 < -5\) is true or false depends on the comparison of the values involved.
When examining the numbers:

• Recognize that negative numbers can confuse the basic intuition.
• The statement requires understanding of the rules of comparison specific to negative numbers.

Evaluating such statements relies heavily on understanding mathematical rules.
If a statement translates to a true condition as per those rules, it is labeled 'true'—in this case, since \(-13\) is indeed less than \(-5\), it's correct to declare the statement as true.
This process is fundamental for students, as it aids in building logical reasoning and helps in solving more complex mathematical problems in the future.