Problem 4 - Beginning and Intermediate Algebra

Question

Fill in each blank with one of the following. positive,negative,0 If two negative numbers are multiplied and then their product is divided by a negative number, the result is ________

Step-by-Step Solution

Answer

negative

Step 1 - Understand the signs of the numbers

Identify the signs of the numbers involved. Here, we have negative numbers. According to the problem statement, if two negative numbers are multiplied, the product will be positive.

Step 2 - Multiply the two negative numbers

When multiplying two negative numbers, the result is positive. \(-a \times -b = ab\) where \(a\) and \(b\) are positive numbers

Step 3 - Divide the positive product by a negative number

Next, divide the positive product by another negative number. The result of dividing a positive number by a negative number is negative. \(\frac{ab}{-c}= - \frac{ab}{c}\) where \(a\), \(b\), and \(c\) are positive numbers.

Key Concepts

negative numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign (-) before the number, such as -1, -2, and -3. Understanding the behavior of negative numbers is key when performing operations like addition, subtraction, multiplication, and division.

- When you add a negative number, it is like subtracting the absolute value of that number.- When you subtract a negative number, it is like adding the absolute value of that number.

When it comes to multiplication and division:

- Multiplying two negative numbers together results in a positive product because the two negative signs cancel each other out.- For example, \(- (-a) \times (-b) = ab\)- Dividing a positive number by a negative number results in a negative quotient because one negative sign remains.

It's important to practice these rules to become more confident in working with negative numbers.

positive product

A positive product occurs when two negative numbers are multiplied together. This might seem counterintuitive at first, but remember that multiplying a negative by a negative results in a positive outcome.

For instance, consider the following multiplication problem:\(-3 \times -4 = 12\)- Here, both -3 and -4 are negative numbers.- Multiplying them together gives us 12, a positive number.

Understanding why this happens can be helped by thinking about the multiplication process as a series of repeated additions. Multiplying -3 by -4 can be viewed as adding -3 a total of 4 times in the negative direction, but since both numbers are negative, the operation turns back into the positive direction.

- Always remember: \((-a) \times (-b) = ab\)

division

Division with negative numbers follows specific rules, much like with multiplication. To understand it properly, consider the following points:

- Dividing a positive number by a negative number results in a negative quotient.- Dividing a negative number by a positive number also results in a negative quotient.- Dividing two negative numbers results in a positive quotient.

For example, let's see some scenarios:\(12 \frac{-3} = -4\)This shows how a positive divided by a negative results in a negative number.

\(-12 \frac{3} = -4\)This indicates a negative number divided by a positive also produces a negative outcome.

Finally,\(\frac{-12}{-3} = 4\), This indicates how two negative numbers divided by each other produce a positive outcome.

These guidelines will help ensure accurate results when performing division operations involving negative numbers.