Problem 76 - Prealgebra

Question

Simplify the given expression. \(18 .4+|16.5-7.6|\)

Step-by-Step Solution

Answer

The simplified expression is 16.1.

Step 1: Solve Parentheses

First, address the expression inside the absolute value bars, which is \(16.5 - 7.6\). Perform the subtraction: \(16.5 - 7.6 = 8.9\).

Step 2: Apply Absolute Value

Now, apply the absolute value operation to \(8.9\). The absolute value of \(8.9\) is \(8.9\) since it's already positive.

Step 3: Perform Multiplication

Next, perform the multiplication \(18 \times 0.4\). Calculate this to get \(7.2\).

Step 4: Simplify Addition

Finally, add the two results from Steps 2 and 3: \(7.2 + 8.9\). Calculate this addition to get \(16.1\).

Key Concepts

Understanding Absolute Value

Absolute value refers to the distance a number is from zero on the number line. It is always expressed as a non-negative number. For instance, both -3 and 3 have an absolute value of 3, because each is three units away from zero.
In mathematical notation, the absolute value of a number \(x\) is written as \(|x|\). This concept is significant in various calculations to ensure the value is always non-negative.
For example, when we see \(|16.5 - 7.6|\), first we perform the operation inside the absolute value, which gets us 8.9. The absolute value of 8.9 is simply 8.9 itself, as it is not negative.

• Absolute values convert negative results to positive.
• They are used to find actual distances.
• It is crucial to simplify expressions accurately.

The Order of Operations

When simplifying expressions, following a specific order of operations is crucial to arrive at the correct answer. This order can be remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (like powers and square roots, etc.)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

When faced with the expression \(18 \times 0.4 + |16.5 - 7.6|\), we start by solving the operation within the absolute value. After that, multiplication is done before any addition. This ensures the expression is evaluated correctly and efficiently.
Ignoring this order might lead to incorrect solutions. Ensure each step is carefully followed depending on the components of the expression you are working with.

Simplification Strategies

Simplification involves reducing a mathematical expression to its simplest form. This process makes it easier to understand and solve problems.
Simplifying means performing operations such as addition, subtraction, multiplication, division, and using the properties of numbers such as the absolute value correctly.
Consider the expression from the exercise, \(18 \times 0.4 + |16.5 - 7.6|\).

• First, tackle the absolute value, which simplifies the subtraction within the bars.
• Next, perform the multiplication \(18 \times 0.4\).
• Finally, add the two results to achieve the simplest form.

The approach of breaking down operations and dealing with them step by step is essential. Simplification not only helps in finding the correct result but also improves understanding of the overall mathematical concept involved, making future problems easier to tackle.