Problem 2 - Prealgebra and Introductory Algebra

Question

Fill in the blank with \(<\) or \(>\) a. On a number line, \(-1\) is to the right of \(-10,\) so \(-1\) \(_____\) -10 b. On a number line, \(-5\) is to the left of \(2,\) so \(-5\) \(______\) 2.

Step-by-Step Solution

Answer

a. -1 > -10\nb. -5 < 2

Step 1: Identifying the position of numbers for a

-1 is to the right of -10 on a number line. This means that -1 is greater than -10. So, the correct inequality symbol to use would be '>'

Step 2: Identifying the position of numbers for b

-5 is to the left of 2 on number line. This means that -5 is less than 2. So, the correct inequality symbol to use would be '<'

Key Concepts

Number Line Comparison

Understanding the concept of a number line is essential when learning about inequalities and ordering integers. A number line is a visual representation that helps us compare the size of numbers. Generally, on a horizontal number line, the further left a number is, the smaller its value. Conversely, the further right a number is, the larger its value. This is true for both positive and negative numbers.

For example, when comparing -1 and -10, we observe that on the number line, -1 is positioned to the right of -10. This indicates that -1 is greater than -10 because it is closer to zero, the central point on the number line. Thus, we can confidently say that -1 is greater than -10 by visualizing their positions on the number line.

Inequality Symbols

Inequality symbols are mathematical tools used to compare numbers and express their relational value. The two basic symbols we frequently encounter are '<' and '>'. The symbol '<' means 'less than', and it is used when the number on the left is smaller than the number on the right. The symbol '>' means 'greater than', and it is used when the number on the left is larger than the number on the right.

When filling in the blanks for exercises like those provided, we determine the correct symbol to use based on the positions of numbers on the number line. In our examples, -1 is greater than -10, therefore we fill the blank with a '>' symbol, written as -1 > -10. For -5 and 2, since -5 is to the left of 2 on the number line and therefore less than 2, we use the '<' symbol, written as -5 < 2.

Ordering Integers

Ordering integers involves arranging numbers from the smallest to largest (ascending order) or from largest to smallest (descending order). It is a fundamental concept that is built upon the understanding of the number line and inequality symbols. To correctly order integers, we look at their value and position on the number line.

Positive integers increase in value as we move to the right on the number line, while negative integers decrease in value as we move to the left. Zero is considered the middle point where positive and negative values are separated. When ordering integers, like -1, -10, -5, and 2, we'd arrange them as -10, -5, -1, 2 in ascending order based on their value and position on the number line, with the smallest integer on the far left and the largest on the far right.