Problem 16 - Algebra A Combined Function

Question

Determine whether each statement is true or false. See Examples 1 through 6 and 10. $$ -16>-17 $$

Step-by-Step Solution

Answer

The statement is true; \(-16 > -17\).

Step 1: Understand the Problem

We need to determine if the given inequality \(-16 > -17\) is true or false. This involves comparing two negative integers.

Step 2: Compare the Integers

When comparing two negative numbers, the one closer to zero is greater. Here, -16 is closer to zero than -17.

Step 3: Formulate the Conclusion

Since \(-16\) is closer to zero compared to \(-17\), \(-16\) is indeed greater than \(-17\). Therefore, the inequality is true.

Key Concepts

Negative Numbers

Negative numbers can often seem confusing because they behave a little differently from positive numbers. A negative number is any number that is less than zero, typically indicated by a minus (−) sign. Think of them as numbers that exist on the opposite side of zero from positive numbers on the number line.

To understand how negative numbers work, envision them on a number line:

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• The further left you go, the smaller—and thus less valuable—the number becomes.

These are essential for various real-world contexts. For example, temperatures below freezing, financial debts, or altitudes below sea level are expressed as negative numbers. While adding and subtracting negative numbers can seem tricky at first, viewing them on a number line can simplify these operations. Enhancing familiarity with their representation is vital to mastering comparisons and arithmetic involving negatives.

Integer Comparison

When comparing integers, it's essential to understand their placement relative to each other on the number line. Comparisons will tell us which of two integers is greater or if they are equal.

For negative numbers, the rule of thumb is:

• The number nearer to zero is the larger one because it is less negative.
• For example, -3 is greater than -6 because -3 is closer to zero.

This might seem counterintuitive if you're used to dealing with positive numbers, where the opposite is true. Looking at our exercise example, the comparison of -16 and -17 involves determining which one is closer to zero. Since -16 has a greater value than -17 in terms of proximity to zero, it confirms that -16 is greater.

Truth Value of Statements

The truth value of a statement determines whether it's correct. In mathematics, when we compare numbers using inequalities, those comparisons have a truth value; they can either be true or false.

Let's consider the inequality \(-16 > -17\). To check its truth value:

• By examining their positions on the number line, we see that -16 is indeed closer to zero than -17.
• Thus, the statement that \(-16\) is greater than \(-17\) is true.

It's essential to approach each statement systematically, ensuring that you compare the values accurately. This process helps reinforce the understanding and correct application of inequalities, making it easier to ascertain whether such statements are true or false.