Problem 55 - 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. $$-4.5 \quad\square\quad 3$$

Step-by-Step Solution

Answer

-4.5 < 3

Step 1: Understand Positive and Negative Numbers

On a number line, positive numbers are to the right of 0 and negative numbers are to the left of 0. Therefore, any positive number is greater than any negative number.

Step 2: Compare the Provided Numbers

When compared on a number line, -4.5 is to the left of 0 and 3 is to the right of 0. So, -4.5 is less than 3.

Step 3: Insert the Appropriate Symbol

The symbol to make the statement true is '<', so the statement becomes '-4.5 < 3'.

Key Concepts

Number Line

Imagine the number line as a ruler that stretches infinitely in both directions. In the center is zero. To the right, you find positive numbers stretching further away as you move right. The left side is home to negative numbers, which grow smaller as they move left.

On this line:

• Zero is the midpoint, neutral and neither positive nor negative.
• Positive numbers increase incrementally to 1, 2, 3, and beyond.
• Negative numbers decrease in value to -1, -2, -3, and so on.

Visualizing numbers on a line helps us comprehend how they relate to each other in terms of size and direction, making it easier to solve problems that involve inequalities like in the problem above with -4.5 and 3.

Positive and Negative Numbers

Numbers aren't just symbols; they tell stories. Positive numbers represent things we have or positions to the right of zero on the number line, like having 3 apples. Conversely, negative numbers show what we owe or positions to the left of zero, like owing 4.5 apples.

Key facts to remember:

• Any negative number is less than any positive number.
• Zero is greater than any negative number but less than any positive number.

In the given exercise, -4.5 is a negative number (left of zero), and 3 is a positive number (right of zero). This understanding is crucial as it shows us that -4.5 is automatically less than 3, without intense calculations.

Comparing Numbers

When comparing numbers, envision them on the number line.
Two essential concepts help us:

• **Greater than (>)**: This symbol shows a number further to the right on the number line compared to another number.
• **Less than (<)**: This symbol indicates the leftmost number on the number line is smaller than the one on the right.

In the given exercise, -4.5 is to the left of 3, meaning it is less than 3. Thus, you use the symbol '<', resulting in the inequality: -4.5 < 3.
Understanding these comparative signs on a number line simplifies the process of determining relationships, especially with mixed signs of positive and negative numbers.