Problem 65 - Prealgebra

Question

In Exercises 65-80, simplify the given expression. \(-12.05-|17.83-(-17.16)|\)

Step-by-Step Solution

Answer

The expression simplifies to -47.04.

Step 1: Simplify the innermost expression

Start by dealing with the innermost expression within the absolute value. Calculate the operation inside the absolute value: \( 17.83 - (-17.16) \). This simplifies to \( 17.83 + 17.16 \), because subtracting a negative is the same as adding a positive.

Step 2: Perform the addition

Now, add the numbers together: \[ 17.83 + 17.16 = 34.99 \]. This is the value inside the absolute value bars.

Step 3: Apply the absolute value

The expression inside the absolute value bars \(|\cdot|\) results in a non-negative number, so\[ |34.99| = 34.99 \].The calculation inside the absolute value is now complete.

Step 4: Simplify the entire expression

Now that we have simplified the absolute value, substitute it back into the original expression:\(-12.05 - 34.99\).Proceed to subtract these values.

Step 5: Perform the subtraction

Subtract \( 34.99 \) from \( -12.05 \): \[ -12.05 - 34.99 = -47.04 \].Hence, this is the simplified result of the given expression.

Key Concepts

Absolute Value Operation

In mathematics, dealing with absolute values is important to understand as it is a common operation in simplification problems. An absolute value refers to the non-negative value of a number without regard to its sign. It is denoted by vertical bars, like this: \(|x|\). If you see \(|x|\), it implies the distance of the number 'x' from zero on the number line. Therefore, the absolute value of a negative number becomes positive, and the absolute value of a positive number remains the same.

Here's a quick recap of how absolute values work:

• The absolute value of a positive number is the number itself. For example, \( |5| = 5\).
• The absolute value of a negative number is the positive version of that number. For instance, \( |-7| = 7\).
• The absolute value of zero is zero. That is, \( |0| = 0\).

In the context of our operation, we first calculated the underlying subtraction (inside the absolute value), \( 17.83 - (-17.16) \), which simplifies to \( 34.99 \.\) Since absolute values always yield non-negative numbers, \( |34.99| \) simply remains \( 34.99 \). This helps preserve the correct value for further calculations.

Integer Subtraction

Subtracting integers can be tricky, especially when dealing with negative signs. However, a solid understanding of how integer subtraction works makes operations easier and helps avoid mistakes. Essentially, subtraction takes a number away from another. When you subtract a negative number, it is equivalent to adding its positive counterpart.

Here's a simple way to understand integer subtraction:

• Subtracting a positive integer involves moving left on the number line. For example, \(-2 - 3 = -5\).
• Subtracting a negative integer is equivalent to adding the same positive integer. For instance, \(-2 - (-3)\) simplifies to \(-2 + 3 = 1\).

In the given example, starting with the expression \( 17.83 - (-17.16)\), you're essentially adding \( 17.16\). This simplifies the operation and leads to an easier calculation: \( 17.83 + 17.16 = 34.99\). Recognizing these shifts when subtracting negatives is key to simplifying complex expressions.

Arithmetical Operations

Arithmetic operations form the basis of prealgebra and provide the foundational skills necessary for understanding more complex mathematical concepts. These include addition, subtraction, multiplication, and division. They are the building blocks for simplifying expressions and solving equations.

• Addition: Combining numbers to get a total, such as \( a + b \).
• Subtraction: Taking one number away from another, such as \( a - b \).
• Multiplication: Finding the total of one number repeated a certain number of times, such as \( a \cdot b \).
• Division: Splitting a number into equal parts, such as \( a / b \).

In our example, after handling the absolute value and integer subtraction, we performed subtraction: \(-12.05 - 34.99 = -47.04\). Understanding this operation is critical because it combines the results of previous calculations into a final simplified form. Each arithmetic operation has particular rules that, when mastered, make solving problems straightforward and logical.