Problem 51 - Elementary Algebra

Question

Determine each of the values, \(|5|-|-2|\)

Step-by-Step Solution

Answer

Question: Evaluate the expression \(|5|-|-2|\). Answer: The value of the expression \(|5|-|-2|\) is 3.

Step 1: Calculate the absolute values

To begin, calculate the absolute value of 5 and -2: \(|5| = 5\) because 5 is already positive, \(|-2| = 2\) as the absolute value of a negative number is its positive counterpart.

Step 2: Perform the subtraction

Now, substitute the absolute values back into the expression and perform the subtraction: \(|5|-|-2| = 5 - 2\) So, our expression is simplified to: \(5 - 2\)

Step 3: Final calculation

Finally, subtract 2 from 5: \(5 - 2 = 3\) Hence, the value of the expression \(|5|-|-2|\) is equal to 3.

Key Concepts

Subtraction

Subtraction is a basic arithmetic operation that we use to find the difference between numbers. When we perform subtraction, we start with a number, known as the minuend, and take away another number, called the subtrahend. The result we get is known as the difference. For instance, in the expression \(5 - 2\), 5 is the minuend, 2 is the subtrahend, and the outcome 3 is the difference.

When dealing with the subtraction of absolute values, as in \(|5| - |-2|\), it's important to first calculate each number's absolute value. The absolute value function always returns the positive version of the number you input. So, the subtraction that follows is based on these positive values, making it a straightforward difference calculation.

Positive and Negative Numbers

Understanding positive and negative numbers is key in arithmetic. Numbers can be positive, negative, or zero. Positive numbers are greater than zero and appear without a plus sign or simply as is (e.g., 5), while negative numbers are less than zero and have a minus sign in front (e.g., -2). Zero itself is neither positive nor negative.

The absolute value of a number, represented by vertical bars (e.g., \(|x|\)), is the number's distance from zero on the number line. This concept is vital when you deal with both positive and negative inputs because the absolute value simplifies all numbers to a non-negative form, enhancing ease of operations like subtraction. For instance, regardless of whether you start with \(5\) or \(-2\), absolute values convert them to \(5\) and \(2\) respectively.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, making them easier to work with or understand. In mathematics, simplification can include removing brackets, combining like terms, or performing arithmetic operations.

Back to our exercise with \(|5| - |-2|\), the first step in simplification is finding the absolute values, which swiftly converts to a simpler form: \(5 - 2\). Each step taken reduces complexity until you reach the simplest outcome, which, in this case, is 3.

• Identify elements needing simplification, such as absolute values or brackets.
• Perform basic operations like subtraction to streamline the expression.
• Confirm each calculation step to ensure accuracy in obtaining the simplified expression.

Following these processes ensures clarity and precision in solving expressions.