Problem 58 - Algebra A Combined Function

Question

Find the sum of -44 and 16 .

Step-by-Step Solution

Answer

The sum of -44 and 16 is -28.

Step 1: Identify the Problem

We need to find the sum of -44 and 16. This is an addition problem involving two integers, one negative and one positive.

Step 2: Understand Integer Addition

When adding integers, if the signs are different, subtract the smaller absolute value from the larger absolute value, and keep the sign of the integer with the larger absolute value.

Step 3: Determine Absolute Values

Calculate the absolute values of the numbers: the absolute value of -44 is 44, and the absolute value of 16 is 16.

Step 4: Subtract the Absolute Values

Subtract the smaller absolute value (16) from the larger absolute value (44): 44 - 16 = 28.

Step 5: Determine the Sign of the Result

Since the larger absolute value was from the negative number, -44, the result will be negative.

Step 6: Combine the Results

Combine the numerical result with the sign determined: -28.

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that often helps simplify the process of integer addition and subtraction. It represents how far a number is from zero on the number line, irrespective of its direction. For instance, the absolute value of -44 is 44 because it is 44 units away from zero, just as 44 is 44 units away from zero. When we write the absolute value of any number, we use vertical bars, like this: \(|-44| = 44\).

Understanding absolute values makes it easier to perform arithmetic operations, especially when dealing with negative numbers. For example:

• The absolute value of 8 is written as \(|8| = 8\).
• Similarly, the absolute value of -16 is \(|-16| = 16\).

By focusing on absolute values when adding or subtracting numbers, one removes the complexity of negative signs temporarily, simplifying the calculation.

Negative Integers

Negative integers are numbers that are less than zero and are represented with a minus sign, like -44 in the exercise. They can sometimes seem tricky to handle, especially when involved in addition with positive numbers, but they follow logical rules. Understanding these can make solving problems much more manageable.

When working with negative integers, keep these points in mind:

• They are always to the left of zero on the number line.
• They indicate the opposite direction of their positive counterparts.
• Adding two negative integers results in a negative number, as you are moving further away from zero.

For instance, adding -2 and -3 gives you -5. Understanding how negative integers work is crucial when determining the overall result of an addition problem involving both positive and negative numbers.

Addition of Positive and Negative Numbers

Adding positive and negative numbers might seem complicated at first, but it simply requires understanding how these two types of numbers interact. The rule to follow is simple: if the signs are different, subtract the smaller absolute value from the larger absolute value, and adopt the sign of the larger absolute value.

For example, in the exercise where we add -44 and 16, we follow these steps:

• First, determine the absolute values: \(|-44| = 44\) and \(|16| = 16\).
• Next, subtract the smaller absolute value from the larger: 44 - 16 = 28.
• Finally, since 44 is larger than 16 and its original sign was negative, the result needs to be negative. Therefore, the sum of -44 and 16 is -28.

This approach helps by turning addition into a simpler subtraction problem when the integer signs are different. Once understood, this makes handling these sorts of problems quite straightforward.