Problem 10 - Algebra 1: Concepts and Skills

Question

Use the rules of addition to find the sum. $$ -4+5 $$

Step-by-Step Solution

Answer

The sum of -4 and 5 is 1.

Step 1: Understand the sign of the numbers

In this expression, -4 is a negative number while 5 is a positive number. When a positive number is added to a negative number, it becomes a subtraction problem where the value of the negative number is subtracted from the positive number.

Step 2: Apply the rule of subtraction of integers

The subtraction of the negative number from the positive number would therefore be \(5 - 4 = 1\)

Key Concepts

Adding Integers

Adding integers is a fundamental concept in mathematics, involving combining numbers to find their total or sum. When integers with the same sign are added together, their absolute values are simply summed. For instance, adding two positive numbers, such as 3 and 2, results in a sum of 5. Conversely, when adding two negative integers, like (-3) and (-2), you would also add their absolute values to get 5 but then include a negative sign to indicate the sum is negative, resulting in (-5).

When adding integers with different signs, such as a positive and a negative, essentially, you're engaging in a subtraction operation. The number with the larger absolute value dictates the sign of the result, and the smaller absolute value is subtracted from the larger one. As in our primary exercise (-4) + 5, the larger number is positive, so our result will also be positive.

• Identify the signs of the numbers.
• Add absolute values of like signs or subtract absolute values of unlike signs.
• The result takes the sign of the number with the larger absolute value if the signs are different.

Subtracting Integers

Subtracting integers is quite similar to adding integers but requires attentiveness to signs. To subtract an integer is to add its additive inverse, which essentially means for any number 'n,' its additive inverse is '-n,' and vice versa. Thus to subtract a number, we add its opposite.

Follow the Sign

For example, when we write 5 - (-3), we're really saying '5 plus the opposite of (-3)', which is 5 + 3, equaling 8. This principle comes in handy when dealing with integers of different signs. It's key to remember that subtracting a negative number actually increases the value of our original number.

• To subtract an integer, add the additive inverse of that number.
• Subtracting a negative number is the same as adding a positive number.
• The signs of the numbers will determine the starting number and the direction in which you're moving on the number line.

Negative Numbers

Negative numbers are an essential aspect of the number system, representing values less than zero. They appear to the left of zero on the number line and are denoted with a minus '-' sign. Negative numbers are used to express a lack of a quantity, below zero temperature, below sea level depth, or debt in financial accounting.

Opposites Attract

Understanding negative numbers is crucial for grasping operations like adding and subtracting integers. Each positive number has a negative counterpart, also known as its additive inverse. For instance, the additive inverse of 7 is (-7) because their sum equals zero (7 + (-7) = 0).
When dealing with negative numbers, remember that moving to the left on the number line decreases value, while moving to the right increases value. The rules of addition and subtraction change somewhat when negative numbers enter the mix, as seen earlier with (-4) + 5 turning into a subtraction problem leading to a sum of 1.

• Negative numbers represent values less than zero.
• Each positive number has a corresponding negative inverse.
• The sum of a number and its negative inverse is always zero.