Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum. $$-25+[-|18+(-26)|]$$

Step-by-Step Solution

Answer

-33

Step 1: Perform Operation in Absolute Value

First, perform the operation within the absolute value, which is to add \(18\) and \(-26\). \(18 + (-26) = -8\)

Step 2: Apply Absolute Value

An absolute value makes a number positive. Thus, the absolute value of \(-8\) is \(8\). This then results into: \(-25 + [-|-8|\] = -25 + [-8]\)

Step 3: Finish Addition

Now that we know the result within the brackets is also a negative number, we can proceed with the addition. Add \(-25\) to \(-8\) to yield the final sum. \(-25 + (-8) = -33\)

Key Concepts

Integer Operations

Integer operations are fundamental in math as they involve basic calculations with whole numbers, including both positive and negative numbers. These operations include addition, subtraction, multiplication, and division of integers. Understanding how to handle these operations is crucial for solving many mathematical problems. When performing integer operations, it's important to remember the rules for adding or subtracting negative numbers, which can often be confusing. Successful integer operations rely on a clear understanding of the signs of the numbers involved and the operation being used. For example, when adding two negative numbers, the result is also negative. It's as if you are adding debts; combining them results in a greater debt. Similarly, subtracting a positive number is akin to adding a negative, thus increasing the debt further. These basic principles form the foundation of all integer operations.

Negative Numbers

Negative numbers indicate values less than zero and are often represented with a minus (-) sign. Understanding how to work with negative numbers is essential, as they frequently appear in various mathematical contexts, such as when representing losses or temperatures below zero. To solve problems involving negative numbers, remember a few key points:

• Negative multiplied by negative equals a positive.
• Negative multiplied by positive equals a negative.
• Negative added to a negative results in a larger negative.

When you're asked to find the absolute value of a negative number, you're essentially finding how far it is from zero on the number line, which is always positive. Hence, absolute value disregards the negative sign. In the original solution, calculating the absolute value of (-8) results in 8 after initially finding 18 + (-26). This key step helps in simplifying the expression.

Addition of Integers

Adding integers involves combining numbers, which might involve both positive and negative numbers. The process is straightforward if you follow the rules governing these operations. For positive integers, it is simply counting upward, while negative integers require careful attention to their rules. When adding integers of different signs, consider their absolute values. For instance:

• When adding a positive integer to a larger negative integer, the result is negative because the negative value dominates. For example, 5 + (-10) = -5.
• Adding two negative integers involves combining their absolute values under a negative sign. It's like increasing the amount you owe. For example, (-3) + (-4) = -7.

In the exercise provided, -25 is added to -8, resulting in -33, because the combination makes the negative value larger. By understanding and applying these rules, adding integers becomes a manageable and systematic task.