Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$3-(-20)$$

Step-by-Step Solution

Answer

The result after performing the indicated subtraction is 23.

Step 1: Analyze the Operation

Observe the operation. When two minus symbols appear in a row, they effectively become a plus operation. So the equation \(3 - (-20)\) essentially becomes \(3 + 20\).

Step 2: Perform the Simple Addition

In this step, perform the addition operation which is now \(3 + 20 = 23\).

Key Concepts

Subtraction of Integers

Subtraction of integers can often be tricky, especially when negative numbers are involved. Understanding the basics of subtracting integers is important. When you see two minus signs in succession, such as in the expression \(3 - (-20)\), you can think of it as a mathematical rule where two negatives turn into a positive.
This means the problem can be restated as '3 plus 20.' Remember:

• If both numbers have the same sign, change the operation to addition.
• Subtracting a negative number is the same as adding its positive counterpart.

By recognizing this pattern, you can simplify complex-looking equations and make calculations easier.

Negative Numbers

Negative numbers represent values less than zero. They are used to express a deficit or a loss. In the context of subtraction, a negative number can change the operation altogether.
When you subtract a negative number, like in the expression \(3 - (-20)\), you're actually adding. The minus and negative signs cancel each other out, easily flipping the problem around.

• Negative numbers are like debts or holes – removing a debt (subtracting a negative) leaves you better off (adds a positive).
• Visualizing a number line may help: going in the opposite direction (subtracting) takes you up the line (addition) when negatives are involved.

Understanding this interaction between subtraction and negative numbers enhances problem-solving skills.

Addition of Integers

Addition of integers, especially when they combine with negative numbers, requires a clear understanding of sign rules. With equations like \(3 + 20\), the addition is straightforward since both numbers are positive. However, even if different signs were present, rules can guide the process.
If integers have the same sign, you simply add them together, keeping the sign. If the signs differ, subtract the smaller absolute value from the larger, and keep the sign of the larger number. Key points include:

• The sum of two positive integers is always positive.
• If adding a positive and negative integer, think "gains and losses" to determine the overall result.
• Addition with negatives follows its own straightforward rules making it simple once you get the hang of it.

Mastering integer addition enhances mathematical flexibility and helps solve real-world problems effortlessly.