Problem 67 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ -51\quad-50 $$

Step-by-Step Solution

Answer

-51 < -50.

Step 1: Understand the Problem

We need to determine the relationship between the two numbers \(-51\) and \(-50\) using the symbols \(<\), \(>\), or \(=\).

Step 2: Identify the Order of Integers

Remember that in the number line, any number on the left is smaller than the number on the right. For negative numbers, the number with the higher absolute value is actually smaller.

Step 3: Compare the Numbers

Compare \(-51\) and \(-50\). Since \(-51\) is further to the left on the number line than \(-50\), it is smaller. Thus, \(-51 < -50\).

Step 4: Insert the Correct Symbol

Based on the comparison, insert the less than symbol \(<\) between \(-51\) and \(-50\): \(-51 < -50\).

Key Concepts

Number Line

The number line is a visual tool that helps us understand the order and relationships between numbers. It's like a ruler that extends endlessly in both directions. On a standard number line, zero is typically placed in the middle. Positive numbers increase as you move to the right, and negative numbers decrease as you go to the left.

When we use a number line to compare two numbers, the position on the line tells us which number is larger. If a number appears to the right of another number, it is greater. Conversely, if a number appears to the left, it is smaller. This straightforward visualization makes the number line an excellent aid for understanding integer comparison.

Negative Numbers

Negative numbers are numbers less than zero. They are located on the left side of the number line. When dealing with negative numbers, it's important to remember that the further left you go on the number line, the smaller the number becomes.

For instance,

• -1 is larger than -2 because -1 is closer to zero, and thus further to the right on the number line.
• Negative numbers play by different rules than positive numbers, particularly in terms of directionality on the number line and absolute values.

These properties of negative numbers are crucial when comparing them to each other.

Ordering Integers

Ordering integers involves arranging them from smallest to largest or vice versa. This can be particularly tricky when negative numbers are involved because our usual rules appear inverted. The integer with the higher absolute value is actually smaller when both are negative.

Here’s a simple trick: picture the number line. The integer further to the left is smaller, just like how

• -100 is smaller than -10.
• -3 is smaller than 0.

Ordering integers correctly is essential for understanding comparisons and sequences involving whole numbers.

Mathematical Symbols

Mathematical symbols help us express relationships between numbers effectively. The symbols you'll use most frequently in comparisons are

• \(<\): less than symbol, indicating the number on the left is smaller.
• \(>\): greater than symbol, indicating the number on the left is larger.
• \(=\): equals symbol, indicating the numbers on either side are the same.

In our exercise, the less than symbol (\(<\)) was used to signify that -51 is smaller than -50. Understanding these symbols and how to apply them is vital for solving mathematics problems and accurately communicating numerical relations.