Problem 59 - Algebra 1: Concepts and Skills

Question

Find the difference. $$ |8|-13 $$

Step-by-Step Solution

Answer

-5

Step 1: Find the absolute value

The absolute value of a number refers to the distance of that number from zero, regardless of direction. Hence, the absolute value of 8, denoted by |8|, is 8. This is because 8 is 8 units away from zero.

Step 2: Subtract the number

Subtract 13 from the absolute value, like this: 8 - 13 = -5

Key Concepts

Integer Subtraction

Integer subtraction involves finding the difference between numbers. When dealing with integers, you subtract the second number from the first.
For example, in the expression \(8 - 13\), you start with 8 and take away 13 from it. This action goes beyond zero, resulting in a negative integer.

When performing integer subtraction, you can think of it as moving left on a number line, as each subtraction moves you further into the negative.
Here's a simple way to remember:

• Start at the first number on the number line.
• Move left by the number of steps equal to the number you are subtracting.

After completing the subtraction in this example, you arrive at -5, illustrating how subtraction can move you into the negatives.

Distance from Zero

The concept of 'distance from zero' is critical when discussing absolute value. Absolute value measures how far a number is from zero on a number line, without considering direction.
For instance, both 8 and -8 are 8 units away from zero. Their absolute values are equal, which are denoted by \(|8| = 8\) and \(|-8| = 8\).

This distance is crucial when solving problems involving absolute values since it helps simplify expressions by removing negative signs.
When calculating expressions like \(|8|-13\), you first find \(|8|\), then consider it's distance without any directional bias — this simplifies calculations and is a foundational step in solving such problems.

Negative Numbers

Negative numbers are values less than zero. They appear on the left side of zero on a number line.
Understanding negative numbers is important when performing operations like addition and subtraction.

Subtraction involving negative numbers can lead to larger negative values. For example:

• If you start with a positive number and subtract a larger positive number, the result is negative.
• In \(8 - 13 = -5\), 13 is greater than 8, so the result is a negative number.

Negative numbers are part of everyday math and physics, indicating things like debts or temperatures below zero. Recognizing how to handle these in operations is an essential mathematical skill.