Problem 68 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 27-|19+4(3-1)-7| $$

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Simplify inside the absolute value

First, simplify the expression inside the absolute value: \( 19 + 4(3 - 1) - 7 \). Start by simplifying inside the parentheses: \( 3 - 1 = 2 \). Next, multiply by 4: \( 4 \times 2 = 8 \). Now, substitute back into the expression: \( 19 + 8 - 7 \).

Step 2: Perform the addition and subtraction

Now, perform the addition and subtraction: \( 19 + 8 = 27 \) and then \( 27 - 7 = 20 \). So, the expression inside the absolute value becomes \( |20| \).

Step 3: Apply the absolute value

The absolute value of 20 is 20: \( |20| = 20 \).

Step 4: Perform the final subtraction

Now, perform the final subtraction: \( 27 - 20 \).

Step 5: Simplify the final result

Simplify the subtraction: \( 27 - 20 = 7 \). Therefore, the simplified expression is 7.

Key Concepts

absolute value

The concept of absolute value is crucial in algebra. Absolute value refers to the distance of a number from zero on the number line, without considering direction. This is always a non-negative number. For example, the absolute value of 20 is 20, and the absolute value of -20 is also 20.
When dealing with absolute values in an expression, you first simplify the expression inside the absolute value bars before applying the absolute value.
In the given exercise:
1. First, simplify inside the absolute value: |19 + 4(3 - 1) - 7|
2. Find the result, which is 20. Apply the absolute value: |20| = 20
Knowing this, you can confidently handle any absolute value operation you encounter.

order of operations

The order of operations is a fundamental concept in algebra that ensures expressions are solved correctly. The order can be remembered as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
This means you first resolve any expressions inside parentheses, then move to exponents, followed by multiplication and division, and finally addition and subtraction.
Let's break down the exercise using the order of operations:
1. Simplify inside the parentheses: 3 - 1 = 2
2. Then perform multiplication: 4 × 2 = 8
3. Substitute back into the original expression: 19 + 8 - 7
4. Perform the addition and subtraction: 19 + 8 = 27 , then 27 - 7 = 20
By following these steps in order, you avoid mistakes and arrive at the correct solution.

algebraic simplification

Algebraic simplification involves reducing expressions to their simplest form, making them easier to work with. This often includes combining like terms and performing arithmetic operations.
Let's see the process in our example:
1. Start with the expression: 27 - |19 + 4(3 - 1) - 7|
2. Simplify inside the absolute value step by step: 19 + 4(2) - 7

• Simplify parentheses: 3 - 1 = 2
• Multiply results: 4 × 2 = 8
• Perform addition and subtraction: 19 + 8 - 7 = 20

3. Apply the absolute value: |20| = 20
4. Finally, subtract from 27: 27 - 20 = 7
Understanding these steps ensures you can simplify complex expressions correctly and efficiently.