Problem 26 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -18+(-26) $$

Step-by-Step Solution

Answer

-44

Step 1: Understand the Problem

We need to add two negative numbers: \(-18\) and \(-26\). When adding two negative numbers, the sum is also negative.

Step 2: Add Absolute Values

To find the sum of \(-18\) and \(-26\), add their absolute values: \[ |-18| = 18 \]\[ |-26| = 26 \]Now, add these absolute values: \[ 18 + 26 = 44 \]

Step 3: Apply Sign

Since both numbers being added are negative, the result of their addition will also be negative. Hence, the sum of \(-18\) and \(-26\) is: \[ -44 \]

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line. It is always a positive quantity, regardless of the sign of the original number. For example, the absolute value of both \(-18\) and \(18\) is \(18\). Similarly, the absolute value of \(-26\) is \(26\).

• The absolute value is represented with vertical bars: \(|-18| = 18\).
• It helps in simplifying calculations, especially involving negative numbers.

When you add two numbers, and both are negative, converting them to their absolute values allows you to easily calculate their sum without worrying about their signs initially.
For example, in the problem \(-18 + (-26)\), you would first find the absolute values of both numbers and then add them: \(18 + 26 = 44\).
The absolute value approach simplifies complex arithmetic and lets you focus on pure addition before considering directions on the number line.

Integer Addition

Integer addition involves combining whole numbers, which can be positive, negative, or zero. The rules of addition vary slightly depending on the signs of the integers:

• Adding two positive integers or two negative integers gives a result with the same sign as the numbers being added.
• When adding a positive and a negative integer, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.

In our exercise, we are adding two negative integers: \(-18\) and \(-26\).

• The step involves finding the absolute values of both numbers: \(18\) and \(26\).
• Add these absolute values: \(18 + 26 = 44\).

Since both integers are negative, the sum will be negative too. The final answer is \(-44\). Understanding how to handle signs in integer addition is crucial for solving problems involving any types of integers.

Negative Numbers Concept

Negative numbers represent amounts below zero and can be found on the left side of the number line. They have intriguing properties that differentiate them from positive numbers.

• When added together, multiple negative numbers result in a larger negative number.
• The further left a number is on the number line, the smaller it is.

Adding negative numbers generally means combining losses or reductions.
In the problem \(-18 + (-26)\), both numbers are negative, which means you are adding more to the already negative value, resulting in \(-44\).

• Think of negative numbers as movements or changes in the opposite direction.
• The key is to focus on the absolute magnitudes first and then apply the negative sign.

This concept simplifies working with negative numbers by treating their addition as combining distances or shifts on a number line, emphasizing the directional nature of negative values.