Problem 77 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -44-44 $$

Step-by-Step Solution

Answer

The result of the operation is -88.

Step 1: Identify the Operation

The operation we need to perform is subtraction. We have the expression \(-44 - 44\). Both terms involved are negative numbers.

Step 2: Write the Expression

Let's set up the expression properly: \[-44 - 44\]. This involves subtracting \(44\) from \(-44\).

Step 3: Simplify the Expression

In subtraction, subtracting a positive number from a negative number can be thought of as adding the absolute values and keeping the sign negative.Therefore, the expression becomes: \[ -44 - 44 = -(44 + 44) \].

Step 4: Calculate the Sum

Now, calculate the sum of the absolute values: \[ 44 + 44 \]. This equals \(88\).

Step 5: Apply the Negative Sign

Since the original operation involved subtracting a positive from a negative, our result is the negative of the sum calculated:\[-(88) = -88\].

Key Concepts

Subtraction of Integers

Subtraction of integers can sometimes be tricky, especially when negative numbers are involved. When you subtract one integer from another, it is helpful to think of subtraction as the addition of a negative number. For example, subtracting 44 from -44 can be rewritten as

• \(-44 + (-44)\)

translating subtraction into the addition of the opposite. But when two numbers are both negative and you are subtracting them, picture moving further in the negative direction on the number line. As a result, negative numbers combine to give a larger negative value. In the exercise

• \(-44 - 44\)

you essentially add absolute values and keep the negative sign. So, the logic is

• Negative number minus a positive number equals a negative result.

This is vital in solving any integer subtraction problem successfully.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign

• (-)

in front of them. These numbers are usually encountered when dealing with debts, temperatures below freezing, or dives below sea level. Negative numbers are always smaller than zero and positive numbers. For instance, -44 is lower than -2, despite -2 seeming smaller. Keep this in mind when dealing with subtraction involving negative numbers. They can be visualized on a number line, where the left side of zero houses all negative numbers, and they decrease further as you move left.
Negative numbers can sometimes feel daunting, especially in calculations, because they represent values in the opposite direction to positive numbers. Getting accustomed to these means getting comfortable with the idea of declining or moving backwards in quantity. Once the concept is grasped, using negative numbers becomes as intuitive as using positive numbers.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering which direction you have moved. It always results in a non-negative value. The absolute value of both positive and negative numbers is always spoken about as a positive number. For example,

• the absolute value of -44, denoted as \(|-44|\), is 44,
• and the absolute value of 44, \(|44|\), is also 44.

No matter what, absolute values convert each integer to its non-directional magnitude. This becomes crucial in operations like addition and subtraction, where the magnitude dictates the outcome but may not affect the sign. Understanding absolute value helps simplify operations involving negative numbers, as one only needs to focus on the size of the number rather than its sign. In this way, it brings clarity to more complex integer operations by removing contemplation over which number's influence worsens the sum in subtraction problems.