Problem 24 - Beginning and Intermediate Algebra

Question

Multiply. $$ 5(-9) $$

Step-by-Step Solution

Answer

The product is -45.

Step 1: Understand the Problem

We are given the multiplication of two numbers: 5 and -9. The task is to calculate the product of these two numbers.

Step 2: Identify the Signs

Identify the signs of the numbers we are multiplying. We have a positive number (5) and a negative number (-9).

Step 3: Apply Multiplication Rule

When multiplying a positive number by a negative number, the product is negative. Thus, multiplying 5 and -9 will result in a negative product.

Step 4: Multiply the Absolute Values

Calculate the product of the absolute values: \[ 5 \times 9 = 45 \]

Step 5: Assign the Negative Sign to the Product

Since the rule states that a positive number times a negative number is negative, the result will be: \[ -45 \]

Step 6: Verify the Solution

Review the steps to make sure they follow correctly from multiplication rules: multiplying the numbers and assigning the correct sign leads to -45.

Key Concepts

absolute value

Absolute value is a way of representing how far a number is from zero on the number line, disregarding its direction.
This means it does not consider whether a number is positive or negative; it only measures the "distance" of that number from zero.
For example, the absolute value of both 5 and -5 is 5, as both numbers are 5 units away from zero.

• If a number is positive, its absolute value is the same as the original number.
• If a number is negative, its absolute value is the positive version of that number.

When solving multiplication exercises, we often first calculate using absolute values to simplify math operations.
Then, we apply the correct sign according to the multiplication rules.
In the original problem, we first multiply the absolute values of 5 and -9, which are 5 and 9 respectively, resulting in 45.

negative numbers

Negative numbers are numbers less than zero. They are often used to represent values below a certain reference point, such as temperatures below freezing or years before a certain historical event.
They have unique multiplication and addition rules compared to positive numbers:

• Adding a negative number is the same as subtracting its absolute value.
• Subtracting a negative number is equivalent to adding its absolute value.

For multiplication, one needs to remember that multiplying two negative numbers gives a positive result,
whereas multiplying a positive with a negative yields a negative result.
In our exercise with 5 and -9, because 5 is positive and -9 is negative, their product results in a negative number.

multiplication rules

Understanding multiplication rules helps in solving problems efficiently, especially when dealing with negative numbers.
These are fundamental rules critical for calculations:

• Multiplying two positive numbers always results in a positive product.
• Multiplying two negative numbers results in a positive product.
• Multiplying a positive number by a negative number results in a negative product.

The direction of the result (positive or negative) can be determined simply by counting the number of negative numbers being multiplied:

• An even number of negative factors results in a positive product.
• An odd number of negative factors results in a negative product.

This rule is crucial when interpreting the final result.
In the provided example, 5 (positive) and -9 (negative) multiplied together result in -45, consistent with these rules.
This simple guideline helps in ensuring that the signs of the results are accurate.