Problem 41 - Elementary Algebra

Question

For the following problems, write the appropriate relation symbol \((=,<,>)\). $$ \begin{array}{ll} -3 & 0 \end{array} $$

Step-by-Step Solution

Answer

Answer: The correct relation symbol to be placed between -3 and 0 is the "less than" symbol, <, because -3 is a negative number, and 0 is greater than any negative number. The inequality can be written as $$-3 < 0$$.

Step 1: Identify the numbers to be compared

In this exercise, the numbers to be compared are: -3 and 0

Step 2: Understand the properties of negative numbers

Remember, any positive number is greater than any negative number. Zero is neither positive nor negative, and it is greater than any negative number.

Step 3: Compare the numbers

Now that we understand the properties of negative numbers, we can compare -3 and 0. As -3 is a negative number and 0 is greater than any negative number, the correct relation symbol is the "less than" symbol, <.

Step 4: Write the final inequality

So, the correct inequality to write for this problem is: $$-3 < 0$$

Key Concepts

Negative Numbers: Understanding Their Behavior

Negative numbers are numbers that are less than zero, and they are represented with a minus sign (-) in front of them, such as

• -1
• -3
• -10

These numbers are quite distinct from positive numbers, which are numbers greater than zero and usually don't have any sign, like 1, 3, or 10. One of the key characteristics of negative numbers is that they are always less than zero.
Another important point about negative numbers is that they get smaller as they move further away from zero. For example, -1 is greater than -3, because -1 is closer to zero on the number line.
Negative numbers also follow certain rules when performing arithmetic operations. For example, multiplying two negative numbers gives a positive result. In addition, when you add a negative and a positive number, you're essentially subtracting the negative number's absolute value from the positive number.
Understanding these principles will aid in comparing negative numbers with zero and positive numbers effectively.

Properties of Zero: The Neutral Ground

Zero is a unique number. It is neither positive nor negative. It acts as a dividing line on the number line, separating positive numbers from negative ones.
One of the primary properties of zero is that any negative number is less than zero, and zero itself is less than any positive number. Therefore, zero is greater than any negative number and less than any positive number.

• For instance, 0 is greater than -4.
• However, 3 is greater than 0.

In arithmetic, zero has some unique properties. For example, adding or subtracting zero from any number does not change the number. Thus, it is often referred to as the additive identity. Multiplying any number by zero results in zero, highlighting its role as a pivotal point in multiplication. Understanding when and how to use zero appropriately is crucial for effective arithmetic comparison.

Arithmetical Comparison: Evaluating and Using Symbols

Arithmetical comparison involves evaluating numbers and determining their relationships using specific symbols such as

• = (equals)
• < (less than)
• > (greater than)

Each symbol is used to illustrate the relationship between two numbers.
To compare -3 and 0, you first need to understand the properties of negative numbers and zero. As discussed, -3 is a negative number and zero is greater than any negative number.
Therefore, by knowing these basics, you can easily conclude that -3 is less than 0, and represent this comparison with the "less than" symbol, resulting in the inequality \(-3 < 0\).
Arithmetical comparisons help in solving equations, understanding number ordering, and performing calculations more effectively. They form a foundation for recognizing patterns in numbers and applying logical reasoning.