Problem 46 - Beginning and Intermediate Algebra

Question

Find (a) the additive inverse and (b) the absolute value. -4

Step-by-Step Solution

Answer

The additive inverse of -4 is 4 and the absolute value of -4 is 4.

Step 1 - Understanding the additive inverse

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

Step 2 - Find the additive inverse of -4

To find the additive inverse of -4, change its sign. Hence, the additive inverse of -4 is 4 because (-4) + 4 = 0.

Step 3 - Understanding the absolute value

The absolute value of a number is its distance from zero on the number line, irrespective of direction. It is always non-negative. For any number 'a', the absolute value is denoted as \(|a|\).

Step 4 - Find the absolute value of -4

To find the absolute value of -4, disregard the sign and take the positive value. Hence, the absolute value of -4 is 4.

Key Concepts

Additive Inverse

The concept of the additive inverse is essential in mathematics, particularly when dealing with equations and algebraic expressions.
The additive inverse of a number is what you add to that number to get zero. For any number 'a', its additive inverse is '-a'. This concept helps to balance equations and solve for unknowns.
For example, the additive inverse of -4 is 4 because when you add them together, you get zero: \( -4 + 4 = 0 \).
Simply change the sign of the number to find its additive inverse, whether it's positive or negative.

Absolute Value

Understanding the absolute value of a number is crucial for grasping more complex mathematical principles.
The absolute value of a number is its distance from zero on the number line, irrespective of the direction.
This means it is always a non-negative number. The absolute value is denoted as \( |a| \).
For example, the absolute value of -4 is 4 because distance cannot be negative. To find the absolute value, simply disregard the sign of the number and take its positive value, no matter whether the original number was positive or negative.

Number Line

A number line serves as a visual representation of numbers where each point corresponds to a number. It's helpful for understanding concepts like additive inverse and absolute value.
On a number line, zero is the central point, with positive numbers extending to the right and negative numbers to the left.
The distance between each point represents the value of the numbers.
Using a number line:

• Additive Inverse: Locate a number and find the same distance on the opposite side of zero.
• Absolute Value: Find the distance from zero, ignoring direction.

This visual aid simplifies understanding how numbers relate to each other and their properties.