Problem 20 - Algebra 1

Question

Find the difference. $$ -11-8 $$

Step-by-Step Solution

Answer

-19

Step 1: Understand the Problem

The task is to subtract 8 from -11. Note that subtracting a positive number is the same as adding a negative number.

Step 2: Perform the Subtraction

To calculate the difference, add the negative of 8 (which is -8) to -11 using integer addition. The sum of -11 and -8 is -19.

Key Concepts

Integer Addition

Adding integers, whether they are both positive, both negative, or one of each, is an essential skill in mathematics. When two positive integers are added together, the result is straightforward; for example, adding 5 and 3 gives 8. But when we involve negative numbers, things get a little trickier.

In the case of adding a positive and a negative integer or two negative integers, we need to pay attention to their signs. For instance, when we add -2 to 6, we actually move 2 steps to the left on a number line starting from 6, ending up at 4. Similarly, if we add -4 to -2, we move 4 steps further into the negative direction, landing on -6. The general rule of thumb is that if the signs are the same, you add the two numbers and keep the common sign. If the signs are different, you subtract the smaller absolute value from the larger one, taking the sign of the number with the larger absolute value.

Negative Numbers

Negative numbers are numbers with a value less than zero. They are written with a minus (-) sign in front of a positive number. The concept of negative numbers can be visualized on a number line, where they are positioned to the left of zero. Negative numbers are used to represent losses, temperature drops, below sea level depths, and many other situations where a value decreases.

Understanding negative numbers is crucial for grasping subtraction of integers. For example, starting with -11 on a number line and going backward (adding a negative) by 8 units lands us at -19. It is also important to note the relationship between negative numbers and positive numbers: they are opposites. This means that their absolute values are equal, but their signs are reversed. So, the opposite of -7 is 7, and vice versa. In arithmetic operations, if you have to add a negative number, you can think of it as moving backward or subtracting its absolute value.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They form the foundation of basic math and are used in everyday calculations.

Subtracting integers falls under these operations and can be thought of in terms of addition. To subtract a number is to add its opposite. For example, to solve problems such as \( -11 - 8 \), we add the opposite of 8, which is -8, to -11. This concept is particularly useful as it can simplify subtraction problems, especially when dealing with negative numbers. Remembering to 'add the opposite' when you see a subtraction sign before a number will help in quickly and accurately performing arithmetic operations involving integers.