Problem 54 - 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. $$0\quad\square-\frac{1}{2}$$

Step-by-Step Solution

Answer

The correct symbol to insert is \(>\), so the true statement is \(0 > -1/2\).

Step 1: Perform Numerical Comparison

Compare the two numbers 0 and -1/2. We know 0 is greater than any negative number.

Step 2: Insert the Appropriate Symbol

Based on Step 1 result, insert the greater than (>) symbol between 0 and -1/2 to make a true statement.

Key Concepts

Numerical Comparison

When we talk about numerical comparison, we are simply trying to identify the relationship between two numbers. This usually involves deciding which number is larger or smaller. Comparing numbers helps us understand important mathematical concepts and solve real-world problems.

To compare numbers, you can think of the number line. Numbers on the right side of the number line are always greater than those on the left. Here are some practical steps:

• Locate both numbers on the number line.
• Determine which number is further to the right.
• The number on the right is greater than the number on the left.

In our exercise, comparing 0 and -1/2 is straightforward because 0 appears to the right of all negative numbers.

Negative Numbers

Negative numbers are numbers less than zero and are usually preceded by a minus sign (-). They show a decrease or represent a deficit. Understanding negative numbers is crucial not only for mathematics but also for everyday life situations, like measuring temperature below zero.

Key points about negative numbers:

• They are located to the left of zero on the number line.
• Any positive number is greater than any negative number.
• The closer a negative number is to zero, the greater its value. For instance, -1 is greater than -5.

In the given problem, we deal with -1/2. As a negative number, it is automatically smaller than 0, which is a neutral point on the number line.

Greater Than Symbol

The greater than symbol (\(>\)) is used to show numerical relationships. It points towards the smaller value while opening towards the larger number, symbolizing openness to 'more'. This symbol helps in expressing inequalities clearly.

How to use the greater than symbol:

• Ensure you understand which number is larger or smaller using comparison methods.
• The symbol \(>\) is placed between two numbers, with the open side facing the larger number.
• Read the inequality as "greater than", such as "0 is greater than -1/2".

In our exercise, we establish that 0 is greater than -1/2, so we insert the \(>\) symbol between them to accurately portray their relationship.