Problem 27 - Algebra 1: Concepts and Skills

Question

RULES OF ADDITION Find the sum.$$ 9+(-2) $$

Step-by-Step Solution

Answer

The sum of 9 and -2 is 7.

Step 1: Understanding the Numbers

We have two numbers here: 9 and -2. Here, 9 is a positive number and -2 is a negative number.

Step 2: Application of Addition Rule

In the given expression 9+(-2), it actually means we are subtracting 2 from 9. This is because adding a negative number is the same as subtracting the absolute value of that negative number.

Step 3: Perform the Operation

So now our expression becomes 9 - 2 which equals 7. So, 9 +(-2) = 7.

Key Concepts

Understanding Negative Numbers

Negative numbers can sometimes be tricky to understand at first, especially when it comes to addition. Let's take an intuitive approach. Negative numbers are those numbers that are less than zero. You can think of them as mirror reflections of positive numbers on the number line. While positive numbers move to the right, negative numbers shift to the left.

• For example: -1, -2, -3 are all negative numbers.
• They are just like using a thermometer to measure temperatures below freezing.
• Negative numbers are shown with a minus (-) sign.

When you add negative numbers to positive numbers, you might wonder what exactly is happening. Adding a negative number essentially means you are subtracting its absolute value from the positive number. As you can see, these concepts are closely linked to subtraction and understanding absolute values.

Exploring Absolute Value

Absolute value is a way of describing how far a number is from zero on a number line, regardless of direction. It focuses only on the magnitude and not the sign. You can say it "ignores" whether the number is positive or negative.

• For example, the absolute value of -3 is 3, noted as \(|-3| = 3\).
• This is because -3 is three steps away from zero on the number line.

Absolute value is essential when dealing with negative numbers, especially in addition and subtraction, because it simplifies the complexity by focusing only on the size of the number. So in our original exercise, where we have 9 + (-2), the absolute value of -2 is 2, which tells us we essentially need to subtract 2 from 9.
Understanding absolute value also helps us in comparisons, as it's easier to consider the magnitude alone when comparing distances or sizes.

Subtraction and Its Relation to Addition

Subtraction can be seen as the reverse of addition. When you subtract a number, you are essentially removing a certain amount from another number. This is evident when adding negative numbers because adding a negative is essentially the same as subtracting the absolute value.

• For example: 9 + (-2) is really just 9 - 2.
• Subtracting a positive number moves us to the left on the number line, similar to adding a negative number.
• By reversing the operation of addition, subtraction helps us understand the disappearance of quantity.

In our example, switching the addition of a negative: 9 + (-2) to subtraction as 9 - 2 makes calculation straightforward and results in 7.
Real-life applications of understanding subtraction include budgeting or finding out how much time you have left when time is running out, since you're always removing or reducing a quantity.