Problem 67 - Prealgebra

Question

Simplify the given expression. \(-6.4+|9.38-(-9.39)|\)

Step-by-Step Solution

Answer

The simplified expression is 12.37.

Step 1: Simplify the Inner Expression

The inner expression is \(9.38 - (-9.39)\). To simplify this, remember that subtracting a negative number is equivalent to addition. So, \(9.38 - (-9.39) = 9.38 + 9.39 = 18.77\).

Step 2: Evaluate the Absolute Value

Now, substitute 18.77 back into the expression to get \( |18.77| \). Since 18.77 is already positive, the absolute value is just 18.77.

Step 3: Simplify the Entire Expression

The entire expression now becomes \(-6.4 + 18.77\). By adding these numbers, we get the sum \(18.77 - 6.4 = 12.37\).

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics, particularly in prealgebra. It refers to the distance a number is from zero on the number line, regardless of direction. This means the absolute value of any number is always non-negative. For instance:

• The absolute value of 5 is 5, written as \(|5| = 5\).
• The absolute value of -8 is 8, written as \(|-8| = 8\).

In the provided exercise, we used the absolute value to find the magnitude of our result without considering whether it was obtained from positive or negative values. When we arrived at \(|18.77|\), it remained 18.77 since it is already positive.
Remember, absolute values are particularly useful when dealing with expressions because they simplify problems by focusing on size and ignoring positive or negative signs.

Addition of Integers and Subtraction of Negatives

When adding integers, particularly in expressions, it's essential to understand that subtraction of a negative is the same as adding its opposite. This can simplify many operations:

• If you have an expression like \(a - (-b)\), it becomes \(a + b\).
• Similarly, \(5 - (-3)\) equals \(5 + 3\), which is 8.

In our exercise, we applied this idea when simplifying \(9.38 - (-9.39)\). By converting the subtraction of a negative to an addition, it transformed into \(9.38 + 9.39\).
This simplification step is crucial, especially when combined with other operations like absolute values, to make sure the result is accurate and straightforward. Understanding these pairings not only helps in computation but also fosters a deeper grasp of number operations in algebra.

Simplifying Expressions

Simplifying expressions is all about making calculations as efficient and straightforward as possible. The goal is to reduce an expression to its simplest form through a series of logical and sequential steps:

• First, handle the operations inside any parentheses or absolute values, as we did when simplifying \(9.38 - (-9.39)\).
• Next, deal with the operations following the order of operations (PEMDAS/BODMAS).
• Combine like terms, such as similar constants or variables, by simple arithmetic.

In our exercise, simplifying the inner expression was essential before dealing with the outer terms. Once we resolved \(|9.38 - (-9.39)|\) to its absolute value, we then performed the arithmetic of adding \(-6.4 + 18.77\), resulting in the simplified expression of 12.37.
Simplifying ensures that anyone reading it finds the expression easier to interpret and the result more intuitive, reinforcing the importance of clarity in mathematical operations.