Problem 41 - Elementary Algebra

Question

Find the product of -6 and 9 .

Step-by-Step Solution

Answer

The product of -6 and 9 is -54.

Step 1: Identify the Numbers

We have two numbers to multiply: -6 and 9. The task is to find their product.

Step 2: Understand the Sign Rule

When multiplying integers, if the signs of the two numbers are different (one positive and one negative), the result is negative. Since -6 is negative and 9 is positive, their product will be negative.

Step 3: Multiply the Absolute Values

Now, multiply the absolute values of the numbers, which means ignore the signs for now. Multiply 6 by 9. \( 6 \times 9 = 54 \).

Step 4: Apply the Sign to the Product

Since the rule from Step 2 tells us the product should be negative, add a negative sign to the product calculated in Step 3. Thus, the product of -6 and 9 is \(-54\).

Key Concepts

Sign Rule in Multiplication

When multiplying two integers, the sign rule helps us determine the sign of the final product.
If both numbers have the same sign (either both positive or both negative), their product is positive.

• For example, multiplying two positive numbers, like 3 and 5, yields a positive product: 3 x 5 = 15.
• Similarly, multiplying two negative numbers, like -4 and -7, also results in a positive product: (-4) x (-7) = 28.

Conversely, if the numbers have different signs (one positive and one negative), the product is negative.
For instance, in the original exercise, we multiply -6 (negative) and 9 (positive). So, the result is negative.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, without considering its sign.
It's like focusing on the number part itself, ignoring whether it's positive or negative.

• The absolute value of -6 is 6, written as |-6| = 6.
• Similarly, the absolute value of 9 is 9, written as |9| = 9.

In multiplication, finding the absolute values of numbers simplifies calculations because you temporarily disregard their signs.
After determining the result, you apply the appropriate sign following the sign rule.

Negative Numbers

Negative numbers are numbers less than zero. We represent them with a minus sign (-) in front of the number.
They lie on the left side of zero on the number line.

• For instance, -6 is six units to the left of zero.
• Negative numbers often represent values below a standard or baseline, such as debts or temperatures below freezing.

In arithmetic, dealing with negative numbers - especially in operations like multiplication - requires understanding how their signs affect the result.
Knowing how to apply the sign rule correctly helps in getting accurate answers.

Product of Numbers

The product of numbers is the result of multiplying them together.
It gives the total value when one number is grouped repeatedly a certain number of times.

• For instance, 6 multiplied by 9 gives a product of 54, because 6 is taken nine times.
• In the case of -6 and 9, we first find the product of their absolute values, which is 54.

Then, apply the determined sign from the sign rule, making the final product negative: -54.
Understanding how to combine both positive and negative integers to find the product can transform how you solve such problems efficiently.