Problem 7 - Beginning and Intermediate Algebra

Question

The additive inverse of -5 is _________, while the additive inverse of the absolute value of -5 is _________.

Step-by-Step Solution

Answer

The additive inverse of -5 is 5, and the additive inverse of the absolute value of -5 is -5.

Step 1: Understanding the Additive Inverse

The additive inverse of a number is what you add to the original number to get a sum of zero. For any number 'a', its additive inverse is '-a'.

Step 1: Find the Additive Inverse of -5

To find the additive inverse of -5, think of what number you would add to -5 to get zero. Since -5 + 5 = 0, the additive inverse of -5 is 5.

Step 3: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. The absolute value of any number 'a' is written as \(|a|\).

Step 2: Find the Absolute Value of -5

The absolute value of -5 is 5, as \(|-5| = 5\).

Step 3: Find the Additive Inverse of the Absolute Value of -5

Now, we need the additive inverse of \(|-5|\), which is -5 because \(|-5| = 5\) and the additive inverse of 5 is -5.

Key Concepts

Absolute Value

The absolute value of a number reflects how far it is from zero on the number line, regardless of its direction. To find the absolute value of a number, simply ignore the sign. For example, the absolute value of -5 is 5 because its distance from zero is 5. This can be represented mathematically as \(|-5| = 5\).
Absolute values are always non-negative.

• For a positive number like 7, \(|7| = 7\).
• For a negative number like -7, \(|-7| = 7\).
• For zero, \(|0| = 0\).

Understanding absolute values helps in many mathematical operations, especially those involving distances and magnitudes.

Number Line

A number line is a visual representation of real numbers placed at equal intervals along a straight line. It allows us to see the order and spacing of numbers, making many numerical concepts easier to understand. Positive numbers are placed to the right of zero, while negative numbers are placed to the left.
Using the number line, we can easily find absolute values. The distance from zero to a number is the number's absolute value, whether the number is positive or negative. For instance, both +5 and -5 are five units away from zero on the number line, so their absolute values are 5.
The number line is a helpful tool for understanding addition, subtraction, and the concept of opposites or 'additive inverses.'

Sum of Zero

The sum of zero is achieved when two numbers are additive inverses of each other. The additive inverse of a number is what you need to add to it to get zero. For example, the additive inverse of -5 is 5 because \(-5 + 5 = 0\).

• If you start with a number and its additive inverse, their sum will always be zero.
• This concept simplifies problem-solving and checks for balance in equations.

When considering the additive inverse of the absolute value of a number, be mindful of the sign. If \(|-5| = 5\), its additive inverse is -5 because \(5 + (-5) = 0\). Understanding these relationships helps solve many algebraic problems.