Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-5.5 \quad\square\quad 2.5$$

Step-by-Step Solution

Answer

\(-5.5 < 2.5\)

Step 1: Understand the symbol meanings

The symbols for less than and greater than are \(<\) and \(>\), respectively. When you see these symbols, the small end always points to the smaller number, and the wider end points to the larger number.

Step 2: Compare the numbers

\(-5.5\) is a negative number and \(2.5\) is a positive number. Remember that any positive number is always greater than any negative number.

Step 3: Insert the Correct Symbol

Since \(-5.5\) is less than \(2.5\), the correct symbol to place in between them is \(<\). So, the correct statement is \(-5.5 < 2.5\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They have a minus sign (-) in front of them, indicating they are on the opposite side of the number line from positive numbers.

These numbers are crucial in many real-world applications, such as measuring temperature (e.g., -5°C), finances (e.g., -$20 for debt), and altitudes below sea level.

• Negative numbers decrease as they move further away from zero. For example, -3 is less than -1 because it is further left on the number line.
• All negative numbers are smaller than zero and any positive number.

Understanding negative numbers helps in comparing them with positive numbers and determining their relative magnitudes.

Understanding Positive Numbers

Positive numbers are those greater than zero. They can be written without a plus sign but often display a "+" for emphasis.

These numbers include natural numbers (1, 2, 3,...) and fractions or decimals greater than zero (0.5, 1.75, etc.). Positive numbers typically represent quantities or measures we deal with daily, such as money, distance, or volume.

• Positive numbers increase as they move further away from zero on the number line. For instance, 5 is greater than 2 because it is further right.
• Every positive number is greater than zero and any negative number, making them always larger in such comparisons.

Recognizing the properties of positive numbers is essential for easily comparing them to other types of numbers.

Using Inequality Symbols

Inequality symbols are used for comparing two numbers to show their relationship. The main symbols are:

• \(<\) means "less than" – when the first number is smaller.
• \(>\) means "greater than" – when the first number is larger.

When using these symbols, remember:- The symbol always opens towards the larger number. For instance, in a statement like \(a < b\), \(a\) is less than \(b\). - This helps in ordering numbers and understanding equations where sizes vary.For example, comparing - If you have to compare \(-5.5\) and \(2.5\), note that \(-5.5\) being a negative and \(2.5\) a positive number ensures that \(-5.5 < 2.5\).Mastering these symbols helps confidently express mathematical relationships.