Problem 29 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-85+(-42)$$

Step-by-Step Solution

Answer

The result of adding \(-85\) and \(-42\) is \(-127\).

Step 1: Understand the Problem

We are asked to combine two numbers: \(-85\) and \(-42\). Both numbers are negative. Our goal is to find their sum by applying the rule for addition of positive and negative numbers.

Step 2: Apply the Rule for Addition of Negatives

When adding two negative numbers, we add their absolute values and then apply the negative sign to the sum. The absolute value of \(-85\) is 85, and the absolute value of \(-42\) is 42.

Step 3: Calculate the Absolute Value Sum

Calculate the sum of the absolute values: \[ 85 + 42 = 127 \]

Step 4: Apply the Negative Sign

Since both original numbers were negative, we apply a negative sign to the result of our addition: \(-127\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept that many find challenging at first, but it's quite simple once you get the hang of it. The absolute value of a number is its distance from zero on the number line, ignoring any direction. This means it's always a non-negative number.

Here's how it works:

• The absolute value of a positive number, say 5, is simply 5.
• For a negative number like -5, its absolute value is also 5.

This concept is incredibly useful when dealing with problems involving negative numbers. It allows us to focus on the size of the number rather than its sign.

In our problem, we looked at the absolute values of -85 and -42. These are 85 and 42, respectively. Once you understand absolute values, adding, subtracting, or comparing numbers regardless of their sign becomes much easier.

Handling Negative Numbers

Negative numbers are numbers that are less than zero, and they can often trip people up. They're not just numbers with a minus sign; they require a different way of thinking.

When working with negative numbers, it's crucial to remember that:

• Moving to the left on the number line decreases a number's value.
• Adding two negative numbers results in a negative number.
• Subtracting a negative number is like adding a positive one.

In the problem given, we add two negative numbers, -85 and -42. Since both numbers are moving left on the number line, their combined value becomes even more negative, resulting in -127. Understanding how negative numbers interact helps simplify a lot of mathematical operations.

Mastering Integer Operations

Integer operations are fundamental in mathematics and involve handling both positive and negative whole numbers. Understanding how to add, subtract, multiply, and divide integers is crucial.

The operation involved in this problem is addition. Here's what you should keep in mind when performing integer additions:

• If you're adding two positive integers, the result is positive.
• If you're adding two negative integers, add their absolute values and give the result a negative sign.
• If you're adding a positive and a negative integer, subtract the smaller absolute value from the larger one, and the result takes the sign of the larger absolute value.

In our specific exercise, -85 + (-42), we'll perform the following steps:

• Take the absolute values of both numbers: 85 and 42.
• Add them together: 85 + 42 = 127.
• Apply the negative sign because both were negative: -127.

Understanding these rules ensures you can apply the right method to any integer operation you encounter.