Problem 76 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ |2| \quad \text{_____}\quad|-2| $$

Step-by-Step Solution

Answer

|2| = |-2|

Step 1: Evaluate the Absolute Value of 2

The absolute value of a number is the distance of that number from zero on the number line, without considering direction. For the number 2, the absolute value is simply 2, because that is its distance from zero. Thus, \(|2| = 2\).

Step 2: Evaluate the Absolute Value of -2

Similarly, we find the absolute value of -2 by taking its distance from zero, without considering the negative sign. So, the absolute value of -2 is also 2. That means \(|-2| = 2\).

Step 3: Compare the Absolute Values

Now that we have \(|2| = 2\) and \(|-2| = 2\), we can compare them. Since both absolute values are equal, we insert the symbol \(=\) between them.

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that show the relationship between two expressions, indicating whether one is greater than, less than, or equal to the other. You might see symbols like \(>\), \(<\), and \(=\) used in these kinds of problems.
When we talk about inequalities, we are often comparing two values to determine their magnitude. Here’s how you can understand each symbol:

• \(>\) means "greater than." This indicates that the value on the left is larger than the one on the right.
• \(<\) means "less than," showing that the value on the left is smaller than its counterpart on the right.
• \(=\) signifies that both values are the same.

In the case of absolute values, inequalities can sometimes seem tricky because absolute values reflect the size or magnitude of a number without considering its sign. It becomes important to compare these magnitudes regardless of whether the original values were positive or negative.

Using Number Line to Visualize Absolute Values

The number line is an incredibly helpful tool for understanding absolute values. It visually represents numbers in order, which aids in seeing how far a number is from zero, known as its absolute value.
Imagine standing on the number line. The number zero is your starting point. The absolute value tells you how many steps you'd take from zero, ignoring which direction you move in. Let’s look at examples:

• For \(2\), you’d move 2 steps to the right to reach it, so \(|2| = 2\).
• For \(-2\), even though you'd move 2 steps to the left, the absolute value still counts 2 steps, making \(|-2| = 2\).

By using the number line, you can easily grasp how each number relates to zero. It helps you understand why both \(2\) and \(-2\) have the same absolute value, because they are equidistant from zero on either side.

Evaluating Expressions with Absolute Values

Evaluating expressions with absolute values involves simplifying the expression to determine the absolute distance a number has from zero. When dealing with expressions involving absolute values:
1. **Identify the expression’s components**: Break down the parts. Isolate absolute value portions if needed.
2. **Calculate absolute values**: Convert each number into its respective absolute form, which simply means removing any negative signs to determine how far the number is from zero.
3. **Compare results**: Use your understanding of greater than, less than, or equal symbols to evaluate the relationship between these absolute numbers. For instance, in the expression \(|2|\) and \(|-2|\), both expressions evaluate to the same number, leading to \(\text{comparison } = \). By directly comparing these outcomes, we’re able to succinctly see that both numbers possess the same magnitude of 2, regardless of their initial positive or negative signs. Understanding these concepts is essential for handling more complex inequalities and expressions that you may encounter.