Problem 58 - Algebra 1: Concepts and Skills

Question

Find the difference. $$ 17-(-6) $$

Step-by-Step Solution

Answer

The difference is 23

Step 1: Change the operation

Since subtracting a negative number is the same as adding its absolute value, the equation can be re-written to \(17 + 6\).

Step 2: Perform the addition

Add 17 and 6 to get the result

Key Concepts

Understanding Absolute Values in Addition

When we talk about adding absolute values, we're discussing the fundamental notion that numbers have a magnitude that is independent of their direction on the number line. This concept is particularly important when dealing with negative numbers.

Absolute value is a number's distance from zero on the number line, regardless of direction. It's always a non-negative number. When you add the absolute value of a negative number to another number, you're essentially removing the 'negative' part and treating it as a positive contribution to the sum.

Example: Absolute Value in Action

In the problem \(17 - (-6)\), the absolute value of \-6\ is 6. Instead of thinking about taking away \-6\, we can think about what happens if we add 6. The original problem transforms into the simpler problem of \(17 + 6\), which readily yields the answer 23. This is because subtracting a negative number is equivalent to adding its positive counterpart.

Basic Algebra: The Importance of Understanding Signs

At the heart of basic algebra is the understanding of how numbers and operations relate to one another. Subtraction and addition are foundational operations, and their interactions with positive and negative numbers form a core algebraic principle.

Signs play a critical role in algebra. They determine the operations to be performed between terms. When a negative sign precedes a parenthesis, it indicates that the operation affects all terms inside. This means that a subtraction sign before a negative number flips its sign, turning the operation into an addition.

Turning Subtraction into Addition

Let's look at the operation \(17 - (-6)\). In this case, the -(-6) becomes +6 because two negatives make a positive. This turns our subtraction problem into an addition problem, simplifying the operation and making it easier to solve.

Performing Addition: Combining Positive and Negative Numbers

Performing addition with both positive and negative numbers is a common task in mathematics, and understanding how to navigate these combinations is essential.

To add numbers effectively, you must recognize that adding a negative number is equivalent to subtracting its positive value. Conversely, subtracting a negative is like adding a positive because the two negatives cancel out.

Technique for Easy Addition

In practice, if you encounter a term like \(17 - (-6)\), it helps to visualize what happens to the quantities. You start with 17 and instead of taking away 6 (which you would do if there were no negative sign in front of the 6), you're effectively moving 6 units in the positive direction on the number line, ending up with 23. This straightforward change in perspective simplifies the process and leads to quick and accurate results.