Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. \(-21-(-3)\)

Step-by-Step Solution

Answer

The result of the operation \(-21-(-3)\) is -18.

Step 1: Identify the Numbers

The two numbers that are involved in this operation are -21 and -3.

Step 2: Understand the Double Negative

A double negative, --, can be replaced with a + symbol. Therefore, the expression \(-21 - (-3)\) becomes \(-21 + 3\).

Step 3: Perform the Operation

Perform the operation with these two numbers: \(-21 + 3 = -18\).

Key Concepts

Negative Numbers

Negative numbers are a fundamental concept in mathematics, representing values less than zero. They are the opposite of positive numbers and are typically written with a minus sign (−) in front. Negative numbers are essential for showing loss, debt, decline, or any context where a decrease from a reference point is necessary.

When working with negative numbers, it is important to recognize that they follow specific arithmetic rules. For example:

• Adding a negative number is like subtracting the corresponding positive number. For instance, adding \(-3\) to a number is the same as subtracting 3.
• Subtracting a negative number can be compared to adding its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive number, while doing so with one negative and one positive number results in a negative number.

Understanding these rules helps with performing arithmetic operations involving negative numbers, such as subtraction, as demonstrated in exercises like \(-21 - (-3)\).

Double Negative

The double negative is a concept that can initially seem confusing but is straightforward with practice. In mathematics, when you encounter two negative signs in a row, such as in subtraction operations, they can be transformed into a positive operation. This is because subtracting a negative number is equivalent to adding the opposite.

• If you see \(-(-x)\), it changes to \(+x\).
• This transformation makes complex expressions simpler and easier to solve. For example, \(-21 - (-3)\) simplifies to \(-21 + 3\).

Understanding and applying the concept of a double negative is crucial for solving mathematical problems accurately and efficiently.

Arithmetic Operations

Arithmetic operations encompass the basic mathematical procedures used for calculations: addition, subtraction, multiplication, and division. Each of these operations follows specific rules, particularly when negative numbers are involved.

• For subtraction, it's essential to remember the effect of removing values. Hence, subtracting a larger number from a smaller one results in a negative outcome.
• Addition, conversely, often involves combining quantities or values. When adding a smaller positive number to a larger negative number, the result will still be negative, but closer to zero.

In the exercise \(-21 - (-3)\), we initially deal with subtraction, but the manipulation of the double negative turns it into an addition problem: \(-21 + 3\). Calculating further, we find that the result of subtracting these values actually involves simple addition due to the double negative rule, yielding \(-18\). Understanding these basic operations and their rules allows for easier and more accurate problem-solving.