Problem 32 - Elementary Algebra

Question

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol \((<,>,=)\) in place of the \(*\) $$-3 * 0$$

Step-by-Step Solution

Answer

Answer: The correct relation symbol for the relationship between -3 and 0 is <, since -3 is less than 0.

Step 1: Understand the given numbers

In this exercise, we have two real numbers: \(-3\) and \(0\). We'll need to compare these numbers to determine their relationship in terms of the relation symbols \((<,>,=)\).

Step 2: Compare the numbers

Since \(-3\) is a negative number and \(0\) is neither positive nor negative, we can already infer that \(-3\) is less than \(0\).

Step 3: Identify the correct relation symbol

As determined in Step 2, the relationship between \(-3\) and \(0\) is such that \(-3\) is less than \(0\). Therefore, we can replace the \(*\) symbol with the \(<\) symbol.

Step 4: Write the final answer

Now that we have found the correct relation symbol, we can rewrite the expression. The final answer is: $$-3 < 0$$

Key Concepts

Real Numbers

Real numbers include all the numbers on the number line that we can think of:

• Positive numbers like 1, 2, and 3
• Negative numbers like -1, -2, and -3
• Zero
• Fractions like \( \frac{1}{2} \) and \(-\frac{2}{3} \)
• Irrational numbers like \( \pi \) and the square root of 2

These numbers are used to measure, count, and size things in everyday life. Understanding them allows us to perform operations like addition, subtraction, multiplication, and division with ease.
In many mathematical problems, we'll be comparing different real numbers to identify patterns and relationships.

Negative Numbers

Negative numbers are a fundamental part of real numbers. They are less than zero and are usually represented with a minus sign (-) in front.
Negative numbers can be confusing at first because they represent values below zero. But they are incredibly useful in mathematics, helping to describe things like debts in finances or temperatures below freezing.
For example, in the original exercise, a negative number, (-3), is compared to zero. Understanding that negative numbers are always less than zero is crucial when comparing numbers.

Relation Symbols

Relation symbols include \(<, >, =\). They help us express the relationship between two numbers.

• \(<\) means "less than"
• \(>\) means "greater than"
• \(=\) means "equal to"

These symbols are handy in mathematics to convey information at a glance without words.
Whenever we see two numbers with a symbol in between, those symbols allow us to easily identify which number is larger, smaller, or if they are the same.
In our example, the symbol \(<\) is used because -3 is less than 0.

Comparing Numbers

When comparing numbers, the goal is to determine which is bigger, smaller, or if they are equal. This comparison often involves real numbers, like what we saw in our example.

• Start by aligning the numbers on a mental or physical number line.
• Identify if one number lies to the left or right of the other.
• Remember, numbers on the left are always smaller.

By applying this method, comparing numbers becomes straightforward.
For instance, -3 is to the left of 0 on the number line, making -3 smaller, which results in -3 < 0. This simple insight helps us use relation symbols accurately in mathematical expressions.