Problem 68 - Prealgebra

Question

Simplify the given expression. \(-16.74+|16.64-2.6|\)

Step-by-Step Solution

Answer

The simplified expression is -2.7.

Step 1: Evaluate Inside the Absolute Value

The expression inside the absolute value is \(16.64 - 2.6\). To evaluate this, perform the subtraction: \(16.64 - 2.6 = 14.04\).

Step 2: Apply the Absolute Value

Next, apply the absolute value to the result from Step 1. The absolute value of a number is the distance from zero, which means it is always positive. Since \(14.04\) is already positive, \(|14.04| = 14.04\).

Step 3: Substitute Back Into the Original Expression

Now, replace the absolute value portion of the original expression with the evaluated value from Step 2. The expression becomes \(-16.74 + 14.04\).

Step 4: Perform the Addition/Subtraction

Finally, add \(14.04\) to \(-16.74\): \(-16.74 + 14.04 = -2.7\).

Key Concepts

Absolute Value

The absolute value is a fundamental concept in mathematics that refers to the distance of a number from zero on the number line. It is always non-negative. When calculating the absolute value, we focus solely on how far the number is from zero, ignoring its direction (positive or negative). For example, both 5 and -5 have an absolute value of 5, because both are 5 units away from zero.

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the positive version of that number.
• The absolute value symbol is represented by two vertical bars, like this: \(|x|\).

In our specific example, we encounter an expression inside the absolute value: \(16.64-2.6\). After performing the subtraction, we get 14.04. Since 14.04 is already positive, the absolute value remains the same: \(|14.04| = 14.04\). This process ensures we always work with non-negative values when absolute values are involved, simplifying our calculations.

Subtraction

Subtraction is the mathematical operation of taking one quantity away from another. It is a fundamental operation used in various aspects of mathematics and everyday tasks. Understanding how to subtract is crucial for simplifying expressions effectively. When subtracting two numbers, the result is called the 'difference.'

• To subtract, align the numbers by their decimal points if they are in decimal form.
• Perform the subtraction starting from the rightmost digits and move to the left.
• Borrow from the next left digit if necessary, similar to how we do in traditional subtraction techniques.

In the exercise above, we first subtract 2.6 from 16.64, resulting in 14.04. It is important to ensure that the subtraction is done accurately, especially with decimals, to get the correct answer when dealing with more complex expressions later on.

Addition and Subtraction of Decimals

Adding and subtracting decimals is pivotal when working with expressions involving numbers with a decimal point. The key here is precision and proper alignment. Understanding this process helps in achieving accurate results when simplifying expressions.

• Always align the numbers by their decimal points before performing the operation.
• Add or subtract the numbers column by column, starting with the smallest place value and moving to larger ones.
• After performing the operation, place the decimal point in the result directly below the other decimal points.

In the final step of our exercise, we need to add \(-16.74\) and \(14.04\). By carefully aligning the decimals and performing basic addition and subtraction techniques, we achieve a precise result: \(-2.7\). Ensuring each digit is correctly placed according to its value on the decimal scale is vital when performing these operations.