Problem 15 - Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) between each pair of numbers to form a true statement. See Example \(5 .\) $$ -16-17 $$

Step-by-Step Solution

Answer

-16 > -17

Step 1: Understand the numbers

We need to compare the two numbers: \ 1. \(-16\) 2. \(-17\). Our task is to determine the relationship between these two integers and decide whether to place \(<\), \(>\), or \(=\) between them to form a true statement.

Step 2: Recall the number line concept

On a number line, numbers are arranged in ascending order from left to right. Negative numbers are to the left of positive numbers, and the further a negative number is from zero, the smaller it is. Therefore, \(-17\) is further left on the number line than \(-16\).

Step 3: Determine the relationship

Since \(-16\) is closer to 0 than \(-17\), it is greater than \(-17\). Thus, the correct relationship between these two numbers is \(-16 > -17\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (−). They are the opposite of positive numbers and often come up when measuring temperatures below zero, debts, or elevations below sea level. One simple way to think of them is as numbers that "exist" on a line to the left of zero on a number line.
Negative numbers can seem tricky at first because they follow different rules compared to positive numbers:

• The smaller the numerical value of a negative number, the greater its actual value. For example, \(-1\) is greater than \(-5\).
• In addition and subtraction, combining two negative numbers yields a lower (or more negative) result. For example, \(-2 + (-3) = -5\).

Understanding these basic principles makes it easier to work with negative numbers in various mathematical computations.

Using the Number Line

The number line is an excellent tool for visualizing and understanding numbers and their relationships. It is a long horizontal line with numbers placed at regular intervals. Zero is usually placed in the center, with positive numbers to its right and negative numbers to its left.

• When comparing integers, visualize them on the number line. The number further to the right is always greater.
• The close proximity of a number to zero signifies that it is larger compared to numbers further away in the negative direction.

In the example of \(-16\) and \(-17\), using the number line makes clear why \(-16\) is greater: it is closer to zero than \(-17\). The visualization helps clarify these sorts of comparisons and eliminates confusion.

Exploring Integer Relationships

Integers include both positive and negative whole numbers as well as zero. Understanding the relationships between integers is crucial in mathematics, especially when determining which number is greater, lesser, or equal.
There are simple rules that dictate integer relationships:

• Any positive integer is always greater than a negative integer.
• Between two negative integers, the one closer to zero is greater.
• Zero itself is greater than any negative integer but lesser than any positive integer.

Applying these rules helps make quick and accurate comparisons. In our solved problem, these principles make determining that \(-16 > -17\) an intuitive process. Understanding integer relationships is foundational, whether you are solving basic math problems or more complex arithmetic calculations.