Problem 39 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ -11-(-4) $$

Step-by-Step Solution

Answer

Answer: The result of -11 - (-4) is -7.

Step 1: Rewrite the subtraction as an addition

Since we're subtracting a negative number, we can rewrite the expression as an addition: $$ -11 - (-4) = -11 + 4 $$

Step 2: Add the numbers

Now, we can add -11 and 4 together: $$ -11 + 4 = -(11 - 4) = -7 $$ So, -11 - (-4) is equal to -7.

Key Concepts

Addition and Subtraction of Integers

Working with integers, which include both positive and negative whole numbers, can be a bit tricky, but there's a way to simplify the process. When subtracting integers, it's helpful to remember that subtracting a negative number is the same as adding its positive counterpart.

Let's look at our example: \( -11 - (-4) \). To navigate this, you perform the subtraction by adding the opposite. The 'minus' sign in front of the 4 turns into a 'plus' sign, and the operation changes accordingly to \( -11 + 4 \). Why does this work? It's rooted in the idea that subtracting a debt actually increases your wealth, just as subtracting a negative number increases the value of your original number.

In this specific case, you start with -11 and move 4 steps closer to zero on the number line, landing you at -7. It’s helpful to visualize this with a number line, where you can see each step from the starting integer.

Negative Number Operations

Dealing with negative numbers can often be counterintuitive, but there's logic to it. The rules governing negative number operations are designed to maintain consistency across the number system.

When you subtract a negative number, you're essentially moving in the positive direction. For example, in the operation \( -11 - (-4) \), you are moving 4 places to the right from -11 on the number line, because subtracting a negative is akin to adding a positive. The subtraction of negative numbers can sometimes be confusing, but remembering that two negatives make a positive––as in negative times negative equals positive––can help anchor your understanding of these operations.

Additionally, using this rule, whenever you encounter two negative signs next to each other, you can replace them with a plus sign, simplifying the operation considerably.

Basic Algebra

The subtraction problem \( -11 - (-4) \) is a straightforward example of basic algebra. Algebra often requires you to find equivalent expressions that are easier to work with. In this case, we convert the operation from subtraction to addition.

By rewriting the expression to \( -11 + 4 \), the problem becomes more manageable and resembles a typical single-variable equation that you might encounter in algebra. This skill of recognizing when to simplify an expression by changing the operation is fundamental in algebra. It allows you to tackle more complex problems by working through them step by step, transforming them into operations you are comfortable with.

Remember that in algebra, the goal is often to isolate the 'unknown' or to simplify the expression to something more familiar. While our example doesn't have an unknown variable, the concept of simplifying to a more familiar operation is aligned with one of algebra's core principles.