Problem 63 - Intermediate Algebra

Question

In the following exercises, simplify. $$ |15-7|-|14-6| $$

Step-by-Step Solution

Answer

0

Step 1: Evaluate the first absolute value expression

Calculate the expression inside the first absolute value bars: |15-7| 15-7=8Therefore, |15-7| = |8| = 8

Step 2: Evaluate the second absolute value expression

Calculate the expression inside the second absolute value bars: |14-6| 14-6=8Therefore, |14-6| = |8| = 8

Step 3: Subtract the evaluated absolute values

Subtract the values found in Step 1 and Step 2: 8-8=0

Key Concepts

absolute value

Absolute value is a core concept in algebra. It refers to the distance of a number from zero on the number line, regardless of direction.

In mathematical notation, we use vertical bars to denote absolute value: \(|a|\).
Here's the main point to remember:

• If \(a\) is positive or zero, then \(|a| = a\).
• If \(a\) is negative, then \(|a| = -a\) (which results in a positive number).

This simply means that absolute value is always positive. Understanding this helps to simplify complex expressions.

simplification

Simplification is the process of reducing expressions to their simplest form. To simplify an absolute value expression:
1. First, calculate the value inside the absolute value bars.
2. Apply the absolute value rules to get a positive result.
Taking the expression \(|15-7|\), follow these steps:

• Calculate inside the bars: \(15-7=8\)
• Apply absolute value: \(|8| = 8\)

Repeat these steps for other parts of the expression. Simplification makes the problem easier to solve.

subtraction

Subtraction is a fundamental arithmetic operation that involves taking one number away from another. This is crucial when simplifying expressions with absolute values.
Here's a quick revisitation using our example:

• First, simplify each absolute value expression: \(|15-7| = 8\) and \(|14-6| = 8\).
• Then, subtract the results: \(8-8 = 0\).

Remember, simplifying each part of the expression first makes final operations, like subtraction, much clearer and easier.
Using these steps ensures clarity and accuracy in solving such exercises.