Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$15-(-15)$$

Step-by-Step Solution

Answer

The result of the operation \(15-(-15)\) is \(30\).

Step 1 Title: Identify the operation

The problem is asking to subtract a negative number from a positive number. The expression given is \(15 - (-15)\).

Step 2 Title: Apply the rule for subtracting negatives

The rule for subtracting a negative number is to add the absolute value of that number. So the given problem simplifies to \(15 + 15\).

Step 3 Title: Perform the operation

Adding \(15\) and \(15\) will give us \(30\).

Key Concepts

Understanding Absolute Value

When working with numbers, understanding absolute value is a key concept. The absolute value of a number refers to its distance from zero on the number line, without considering its direction. In simpler terms, it’s the "positive" version of any given number. For instance, both \(15\) and \(-15\) have an absolute value of 15. The symbol for absolute value is a pair of vertical bars, like so: \(|-15| = 15\). This concept becomes especially important when dealing with operations like subtraction—specifically, subtracting negative numbers. When we see \(15 - (-15)\), the absolute value helps us understand that we are essentially doubling the distance from zero, leading to an increased value. It helps convert complex-looking expressions into more manageable ones by allowing us to "flip" negative signs when necessary.

Strategies for Addition of Integers

Adding integers might seem trivial at times, but it's crucial when performing operations like subtracting negative numbers. In the expression \(15 - (-15)\), subtracting a negative is equivalent to adding its positive counterpart. This is due to the property of subtraction involving negative numbers. With the rule \(a - (-b) = a + b\), the subtraction of a negative number changes into addition. So in our example, it changes to \(15 + 15\). This intuitive switch from subtraction to addition ensures that we are summing the two absolute values, making calculations straightforward. Remember:

• Positive + Positive = Positive
• Negative + Negative = The sum becomes more negative
• Positive + Negative = The sum depends on which absolute value is greater

Keep practicing with various integers, and soon working with them will become second nature.

Simplifying Expressions Effectively

Simplifying expressions is about finding the most streamlined way to solve problems. It's not just about reducing the number of steps in a calculation, but also about finding clarity in the process. Consider the example of \(15 - (-15)\). Initially, this might appear complicated due to its format. However, by recognizing the rules for subtracting negatives, we simplify it to \(15 + 15\). Simplifying expressions involves understanding the operation’s fundamental properties and applying them accurately. This skill is crucial not just in math, but in many real-world applications where we face complex data. To do this effectively:

• Identify unnecessary operations and eliminate them.
• Look for quick substitutions, like changing subtraction of a negative to addition.
• Break down large problems into smaller, easier parts.

Mastering the art of simplification can boost your efficiency and accuracy in problem-solving.