Problem 9 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -16-(-18) $$

Step-by-Step Solution

Answer

-16 - (-18) = 2.

Step 1: Understand the Problem

The question asks us to subtract \(-16\) from \(-18\). This means we have to find the difference between the two numbers. The formula to apply is \(a - b = a + (-b)\), which means subtracting is the same as adding the opposite.

Step 2: Apply the Subtraction Rule

Given \(-16 - (-18)\), apply the rule \(a - (-b) = a + b\). This simplifies to \(-16 + 18\).

Step 3: Perform the Addition

Calculate \(-16 + 18\): Start at \(-16\) on the number line, and move 18 steps to the right (since we are adding a positive number). This brings us to \(2\).

Step 4: Verify the Result

Double check by considering the properties of numbers: Adding \(18\) to \(-16\) results in \(2\) since \(18 - 16 = 2\).

Key Concepts

Integer Operations

Integer operations are actions performed on integers, which include positive and negative whole numbers. In mathematics, dealing with integers requires understanding their unique properties. Here, subtraction is key, and you can see it as adding the opposite.
This is because when you subtract one integer from another, you are essentially adding its additive inverse. For example, subtracting \( -18 \) means you add \( 18 \).

• Positive integers are greater than zero.
• Negative integers are less than zero.
• Zero is considered neutral, neither positive nor negative.

Understanding these properties is crucial for solving problems accurately.

Number Line

A number line is a visual aid for understanding number operations, including addition and subtraction. It involves placing numbers along a straight line, which helps visualize their positions.
When using a number line to solve the problem \( -16 - (-18) \), start at the position for \( -16 \).

• Negative numbers are to the left of zero.
• Positive numbers are to the right of zero.
• Each step to the right typically increases the value.

In this case, after switching subtraction to addition, you'll move 18 steps to the right because you are adding 18. This motion helps clarify why \( -16 + 18 = 2 \) makes sense.

Addition

Addition is one of the basic arithmetic operations and is generally the process of combining numbers. Here, when dealing with integers, it's vital to apply the concept of adding the opposite.
In the expression \( -16 - (-18) \), subtraction becomes addition, turning the task into \( -16 + 18 \).

• Start from the first number, which is often your base point.
• Adding a negative number involves moving left on the number line.
• Adding a positive number involves moving right.

Understanding these movements ensures you find the right answer, demonstrating addition's broader role across multiple contexts.

Algebra Basics

Algebra basics include understanding how to manipulate expressions and equations. This requires knowing the rules that govern the operations among integers.
When simplifying expressions like \( -16 - (-18) \), using the formula \( a - (-b) = a + b \) is vital. It reflects the broader algebraic principle of operations on numbers.

• Recognize that subtraction can be turned into addition.
• Identify the opposite of numbers (negative/positive).
• Recall basic algebraic formulas and principles.

These fundamentals are the building blocks upon which complex algebraic operations are structured, making them indispensable in problem-solving.