Problem 69 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. See Examples 12 and \(13 .\) $$ |-5| \quad-4 $$

Step-by-Step Solution

Answer

|-5| > -4

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. So, the absolute value is always non-negative. For the number \(-5\), its absolute value is \(|-5| = 5\).

Step 2: Evaluate Absolute Value Expression

Find the absolute value of \(-5\). We have \(|-5| = 5\). This means the left side of the expression is 5.

Step 3: Compare the Values

Now, compare the values we have: 5 and -4. Since 5 is greater than -4, we place '\(>\)' in between them.

Step 4: Final Answer

Insert the correct symbol in the space: \(|-5| > -4\).

Key Concepts

Understanding the Number Line

Imagine a straight line that stretches infinitely in both directions. This is the number line. It is a tool we use to understand the position of numbers and how they relate to each other.
On this line, zero is usually placed at the center, positive numbers extend to the right, and negative numbers to the left. Each step right from zero increases in value, while each step left decreases.
Absolute value ties directly into the number line because it measures how far a number is from zero, regardless of direction. When you say the absolute value of \(-5\), you are looking at the number line and measuring 5 units away from zero, without considering whether you went left or right.

Inequality Symbols Explained

Inequality symbols are like little math tools that help us compare numbers.
The most common are \(<\), \(>\), and \(=\):

• \(<\) means "less than" and shows when one number is smaller than another.
• \(>\) means "greater than" and indicates when one number is larger.
• \(=\) means "equal to" and is used when two numbers are the same.

These symbols are crucial when working with inequalities because they describe how numbers relate to each other on the number line.
In our example, because \(|-5| > -4\), the symbol \(>\) tells us that the positive distance 5 is more than -4. This concept helps us visually and mathematically confirm which number takes the larger value.

Comparing Numbers

Comparing numbers can be simplified into a series of basic steps. Whether positive, negative, or zero, understanding how numbers stack against each other is foundational.
First, consider the value of each number, which is where absolute values can become handy, especially when negatives are involved. Neglecting the sign for an instant, only the distance from zero counts, which is how absolute value works.
When comparing \(|-5|\) and \(-4\), we converted \(-5\) into \(5\) using its absolute value, which made it straightforward to compare 5 and -4. We see that 5, being positive and larger than any negative number, easily outclasses -4.
In mathematics, converting numbers to a common visibility, like positive terms, often makes comparing them much easier, revealing mathematical truths succinctly and clearly. Understanding this can help you solve more complex problems with ease.