Problem 69 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -1,903-(-1,732) $$

Step-by-Step Solution

Answer

The result of the operation is -171.

Step 1: Understand the Problem

The operation we need to perform is subtraction with negative numbers: \(-1,903 - (-1,732)\). To tackle this, remember that subtracting a negative number is equivalent to adding the opposite of that number.

Step 2: Convert Subtraction to Addition

Rewrite the expression by converting the subtraction of a negative number into addition. So, \(-1,903 - (-1,732)\) becomes \(-1,903 + 1,732\).

Step 3: Perform the Addition

Calculate the result of the addition \(-1,903 + 1,732\). Perform the operation:\(-1,903 + 1,732 = -171\).

Step 4: Provide Final Answer

The result of the expression \(-1,903 - (-1,732)\) is \(-171\), meaning that after performing the operations, we end up with \(-171\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (-) in front of them. Negative numbers came from the need to extend the concept of numbers to include values that represent a lack of something, such as debt or temperature below zero brands. Using negative numbers, we can handle a variety of real-life situations.

• Examples of negative numbers include -1, -12, -193, and -1,000,000.
• When using a number line, negative numbers are found to the left of zero.
• Negative numbers are an important part of the modern number system used in mathematics.

Understanding how negative numbers work is crucial, especially when performing operations like addition and subtraction, which often involve turning situations involving negative figures into more solvable math problems.

Subtraction as Addition

When dealing with subtraction, especially with negative numbers, it's helpful to remember the concept of subtraction as addition. This might sound surprising, but subtracting a negative number can actually be seen as adding a positive number.
This can be understood because subtracting is just like taking away, and when you subtract a negative number, you essentially move in the opposite direction on the number line. Imagine you start at zero and move to the right if you're adding, or move to the left if you're subtracting. When you're subtracting a negative, the act of turning around and moving to the right makes it feel like you're adding.
Here's a quick process to adopt this concept:

• Identify the operation and the numbers involved, e.g., \(-a - (-b)\).
• Convert to addition by negating the negative sign, e.g., rewrite as \(-a + b\).
• Perform the operation as an addition problem.

Realizing subtraction as addition opens new doors in solving mathematical problems effectively, reducing potential errors.

Step-by-Step Problem Solving

Step-by-step problem solving is an effective way to tackle mathematical problems. By breaking down the process into manageable parts, it makes complex problems easier to understand and solve. This approach is especially beneficial with operations involving integers like negative numbers and subtraction.
Here is a brief overview of how to apply step-by-step problem solving:

• Understand the Problem: Clearly grasp what is being asked. In the example given, we need to handle a subtraction involving negatives.
• Break it Down: Identify each part of the problem, such as whether you need to tackle negative numbers or convert an operation to one you find simpler. Converting subtraction of negatives to addition is a helpful strategy.
• Perform Calculations: Follow through the steps to solve the problem, step by step as converted, i.e., solve two steps combined into one by converting \(-a - (-b)\) to \(-a + b\) and then solving.
• Review: Always double-check your work for accuracy. Make sure each transition and calculation is logical.

Step-by-step problem solving not only aids in understanding the procedure but also strengthens overall mathematical skills, making it easier to tackle similar problems in the future.