Problem 35 - Beginning and Intermediate Algebra

Question

Perform the operation. See Example 3. Subtract \(-1\) from \(-6\)

Step-by-Step Solution

Answer

The result is -5.

Step 1: Understand the Problem

We are asked to subtract -1 from -6. This means we need to find the result of the expression -6 - (-1).

Step 2: Apply the Rule of Subtracting Negative Numbers

When we subtract a negative number, we can rewrite the expression by changing the subtraction of a negative to the addition of a positive. Therefore, -6 - (-1) becomes -6 + 1.

Step 3: Perform the Addition

Now, we need to calculate -6 + 1. The operation involves adding a positive number to a negative number, which results in moving to the right on a number line. Thus, -6 + 1 equals -5.

Step 4: Write the Final Result

Having calculated the sum in the previous step, we conclude that the result of subtracting -1 from -6 is -5.

Key Concepts

Integer Operations

When we discuss integer operations, we usually refer to three main types: addition, subtraction, and multiplication. Integer operations might seem challenging at first. But understanding how to work with positive and negative numbers is key to mastering them.

• **Adding Integers**: When adding together integers with the same sign, whether positive or negative, you simply add their absolute values and retain the sign.
• **Subtracting Integers**: To subtract an integer, you add its opposite. For example, to subtract \(-1\), you add \(+1\).
• **Multiplying Integers**: When two integers are multiplied, if their signs differ, the product is negative. If they share the same sign, the product is positive.

These operations are the foundation of arithmetic with integers. Understanding them can help you solve complex problems more efficiently.

Number Line

A number line is a visual tool that helps you understand and solve integer problems. It is a straight line with numbers placed at equal intervals along its length. Usually, it includes positive numbers, negative numbers, and zero right in the middle. Here is how a number line can be used effectively:

• **Locating Points**: Numbers are plotted on the number line to locate their positions. For instance, \(-6\) is plotted to the left of zero.
• **Adding and Subtracting**: You can perform operations visually on a number line by moving left or right. To add a positive number, move to the right. For subtracting (or adding a negative number), move to the left.

In our example, moving from \(-6\) to the right by one unit places you at \(-5\). The number line serves as a useful guide to visualize and solve integer operations.

Addition of Integers

The addition of integers is a crucial concept in mathematics, especially when negative numbers are involved. Adding integers effectively combines positive and negative numbers to reach a result. Here's how it works:

• **Positive + Positive**: Simply add the two numbers. E.g., \(5 + 3 = 8\).
• **Negative + Negative**: Add their absolute values and keep the negative sign. E.g., \(-4 + (-3) = -7\).
• **Positive + Negative (or vice versa)**: Subtract the smaller absolute value from the larger and use the sign of the integer with the larger absolute value.E.g., \(-6 + 1 = -5\).This is the scenario in our example, where \(-6 + 1\) results in \(-5\).

Mastering addition for both positive and negative integers is essential for effective mathematical problem-solving.