Problem 60 - College Algebra Essentials

Question

Rewrite each expression without absolute value bars. $$||-5|-|-13||$$

Step-by-Step Solution

Answer

The final result of ||-5|-|-13||, when rewritten without using absolute values, is 8.

Step 1: Resolve Inner Most Absolute Value

The most embedded absolute value in the given problem is |-5|. Absolute value of -5 is 5, because the distance 5 units away from 0 on a number line is positive 5. So, ||-5|-|-13|| becomes |5 - |-13||.

Step 2: Resolve the Next Absolute Value

Now, the expression is |5|-|-13|. The absolute value of -13 is 13, turning the expression into |5|-13|. This is because the distance 13 units away from 0 on a number line is positive 13.

Step 3: Resolve the Remaining Absolute Value

The resulting expression, |5 - 13|, requires subtracting 13 from 5 and then taking the absolute value of the result. The subtraction gives -8 and then absolute value of -8 is 8, because the distance 8 units away from 0 on a number line is positive 8.

Key Concepts

Number Line

The number line is a simple and effective tool for visualizing numbers and their relationships with each other. Imagine it as a horizontal line that stretches infinitely in both directions. On this line, numbers increase as you move right and decrease as you move left. Zero is positioned in the middle, with positive numbers to its right and negative numbers to its left.

• Numbers on the number line can be whole numbers, fractions, or decimals.
• The importance of the number line is that it helps us understand the relationships between numbers.

By using the number line, we can easily see the distance a number is from zero, which is an essential concept when discussing the absolute value.

Distance from Zero

The distance from zero refers to how far away a number is from zero on the number line. This concept is crucial for understanding absolute value. Every number has a distance from zero, and in this context, distance is always considered as a non-negative quantity. For example, the number 7 has a distance of 7 units from zero, while -7 also has a distance of 7 units from zero. In mathematics, this distance is what we call the absolute value.

• Distances cannot be negative.
• Both positive and negative numbers have the same distance from zero if they are the same in magnitude but opposite in sign.

Understanding the distance from zero helps in solving equations involving absolute values.

Positive Integer

A positive integer is any whole number greater than zero. These numbers are found to the right of zero on the number line. Positive integers are the numbers we most commonly use when counting, such as 1, 2, 3, etc.

• Positive integers have no fraction or decimal part.
• They represent quantities or values above zero.

In the concept of absolute value, even the absolute value of a negative number will result in a positive integer, representing the number's distance from zero.

Absolute Value Properties

Absolute value has a few interesting properties that make it unique and useful in mathematics. First, the absolute value of a number is always non-negative. This is because it represents distance, which cannot be negative. Second, for any real number \(a\), the absolute value can be defined as:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)

These properties help ensure that absolute values handle both positive and negative numbers correctly by focusing on their magnitude or size, not their sign.Using absolute values simplifies many mathematical problems, as it allows us to focus purely on magnitude without concern for direction.