Problem 60 - Beginning and Intermediate Algebra

Question

Find each absolute value. \(|9-4|\)

Step-by-Step Solution

Answer

The absolute value of \( |9-4| \) is 5.

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from 0 on a number line, regardless of direction. It is always positive or zero.

Step 2: Calculate the Expression Inside the Absolute Value

Solve the expression inside the absolute value: \( 9 - 4 = 5 \)

Step 3: Apply Absolute Value

The absolute value of a positive number is the number itself. Therefore, \( |5| = 5 \)

Key Concepts

Number Line

A number line is a straight, horizontal line that has numbers placed on it at equal intervals. It usually includes both positive and negative numbers, as well as zero.
Each point on the number line corresponds to a number. Zero is typically placed in the center.
To the right of zero, you find positive numbers like 1, 2, and 3.
To the left of zero, you find negative numbers like -1, -2, and -3.
When dealing with absolute value, the number line helps to visualize how far a number is from zero, regardless of its direction.
For example, both -4 and 4 are four units away from zero on the number line.

Positive Number

A positive number is any number greater than zero.
On a number line, positive numbers are found to the right of zero.
Examples include 1, 2, 3, and even numbers like 0.5 or 5.75.
Positive numbers are always greater than zero and signify values above zero in various contexts.
For absolute value calculations, if you are dealing with a positive number, it remains the same because its distance from zero is already positive.
For instance, in the solution above, 9 and 4 are both positive numbers, making the calculation straightforward.

Absolute Value Calculation

Absolute value is written using vertical bars like this: \( | | \). It represents the distance a number is from zero, without considering direction.
The absolute value of a positive number is the number itself. For instance, \( |5| = 5\).
The absolute value of a negative number is its positive counterpart. For example, \( |-5| = 5\).
In the given exercise, you first solve the expression inside the absolute value bars: \( 9 - 4 = 5 \).
Then, apply the absolute value: since 5 is already positive, \( |5| = 5 \).
So, no matter what the original number is, its absolute value is always non-negative.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols.
In algebra, letters like x and y are often used to represent numbers.
This allows us to create general formulas and solve equations.
Understanding algebra is crucial for solving many math problems, including those involving absolute value.
The given exercise demonstrates an application of algebra where you simplify an expression \( 9 - 4 \) before applying the absolute value.
Such steps are common in algebraic problems and help in breaking down complex problems into simpler, manageable parts.