Problem 18 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$2 \quad -13$$

Step-by-Step Solution

Answer

The correct statement is: 2 > -13.

Step 1: Understand the Problem Statement

The task is to compare two numbers: 2 and -13. We need to determine which number is greater or smaller to place either `<` or `>` symbol between them to make the statement true.

Step 2: Compare the Numbers

Analyze the two numbers given: 2 and -13. Since 2 is a positive number and -13 is a negative number, 2 is greater than -13. In other words, any positive number is greater than any negative number.

Step 3: Write the Correct Inequality

Now that we know 2 is greater than -13, we should place the `>` symbol between them. Thus, the correct inequality is: \[ 2 > -13 \]

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is fundamental in comparing inequalities. Positive numbers are all the numbers greater than zero. They stretch to the right of zero on the number line and include 1, 2, 3, and so on. Every number that doesn't have a minus (-) sign in front of it is a positive number, even zero, which is neither positive nor negative.

Negative numbers, on the other hand, are numbers less than zero. They appear to the left of zero on the number line. You can recognize them easily because they always carry a minus sign before them, like -1, -2, and -3.

When comparing positive and negative numbers, a key rule to remember is that positive numbers are always greater than negative numbers. Even the smallest positive number is bigger than the largest negative number. So, in any situation, if you have to compare a positive number with a negative number, the positive number will always be greater.

Inequality Symbols

Inequality symbols are used to compare two values or expressions. These symbols tell us whether the value on the left is less than, greater than, or equal to the one on the right.

Here are the basic inequality symbols to know:

• \(<\): Less than
• \(>\): Greater than
• \(\leq\): Less than or equal to
• \(\geq\): Greater than or equal to
• \(=\): Equal to (not technically an inequality but useful in comparisons)

In the context of inequalities, we use these symbols to convey the relationship between two numbers, just as we do in the exercise provided where we determined that \(2 > -13\). This statement uses the 'greater than' symbol \(>\) to express that 2 is larger than -13.

Number Line

The number line is a visual tool that helps us understand and compare numbers, including inequalities. Imagine a straight, horizontal line. This line extends infinitely in both directions. Zero is located at the center of the number line.

On the number line:

• Numbers to the right of zero are positive, increasing from 1, 2, 3, and beyond.
• Numbers to the left of zero are negative, decreasing as you move left (-1, -2, -3, etc.).

To compare two numbers using a number line, identify their positions. The number farther to the right is always the greater number. So, in the example of 2 and -13, 2 is to the right of -13 on the number line, making it larger.

This visualization helps make sense of the inequality 2 > -13, showing where each number lies in relation to zero and each other.