Problem 57 - Algebra 1: Concepts and Skills

Question

Find the product. $$ 11(-7) $$

Step-by-Step Solution

Answer

-77

Step 1: Identify the sign of each number

The first number is positive and the second number is negative.

Step 2: Multiply the absolute values

Ignore the signs for now and just multiply the absolute values i.e. \(11 \times 7 = 77\).

Step 3: Apply the rule of signs

Since the numbers have unlike signs, their product will be negative. So, the final answer is -77.

Key Concepts

Absolute Value

When we work with integers, especially in multiplication, understanding the concept of absolute value is crucial. Absolute value refers to the distance of a number from zero on the number line, regardless of direction.
For example, the absolute value of both 7 and -7 is 7, because both are 7 units away from zero. This is represented as \(|7| = 7\) and \(|-7| = 7\). Absolute value essentially strips the number of its sign.
When multiplying integers, we first consider their absolute values, just as if we are working with positive numbers. This means that when you have to multiply -7 and 11, you would first multiply 7 and 11, which equals 77, and then consider the signs to decide on the final product.

Rules of Signs

The rules of signs help us determine the sign of the product when multiplying integers. These rules are straightforward and easy to remember.

• When you multiply two numbers with the same sign, the result is positive. This means multiplying two positive numbers or two negative numbers will always give you a positive result.
• When you multiply two numbers with different signs, the result is negative. This means if one number is positive and the other is negative, their product will be negative.

In our given exercise, we have 11 and -7. Since one number is positive and the other is negative, according to the rules of signs, their product will be negative. Hence, 11 multiplied by -7 gives us -77.

Negative Numbers

Negative numbers represent values less than zero. They are the opposite of positive numbers and have a minus sign in front of them to indicate their direction.
Negative numbers are essential in multiplication, particularly because they influence the sign of the result.
When dealing with negative numbers:

• If you see '(-)' in front of a number, it means you're dealing with a negative. For instance, -7 is a negative number.
• Understanding the impact of a negative number involves knowing how it affects calculations, such as changing the sign of the final product when multiplied with a positive number.
• Practicing operations with negative numbers across different arithmetic functions can strengthen understanding.

In the exercise provided, -7 is considered, and its role is to change the result of the multiplication with 11 to a negative product, namely -77.