Problem 36 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$ 8(-3)-4(-5) $$

Step-by-Step Solution

Answer

The result is -4.

Step 1: Multiply the First Pair

Begin by performing the multiplication within the first set of parentheses. Multiply 8 by -3. \[8 \times (-3) = -24\]

Step 2: Multiply the Second Pair

Perform the multiplication within the second set of parentheses. Multiply -4 by -5. \[-4 \times (-5) = 20\]

Step 3: Combine the Results

Combine the results from Steps 1 and 2. \[-24 + 20\]

Step 4: Simplify the Expression

Perform the addition to simplify the expression. \[-24 + 20 = -4\]

Key Concepts

Multiplication

Let's dive into the concept of multiplication, especially when it involves integers. Multiplication at its core is essentially repeated addition. When you multiply a number by another number, you are adding the first number to itself a specified amount of times. For instance, 3 multiplied by 4 (or \(3 \times 4\)) means you add 3 a total of four times (3 + 3 + 3 + 3 = 12). However, when working with negative numbers, the rules of multiplication receive an interesting twist.

Signs play a crucial role in determining the result:

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers also results in a positive product. This might seem counter-intuitive, but think of it as reversing direction twice.
• Multiplying a positive number by a negative number, or vice-versa, will yield a negative product. It’s like stepping back when you initially intended to move forward.

In the exercise, multiplying 8 by -3 gives us -24 because a positive number and a negative number result in a negative value. Similarly, multiplying -4 by -5 provides a positive result, 20, since two negatives make a positive.

Addition

Addition is the process of combining numbers to find their total. When working with just positive numbers, addition is quite straightforward. Yet, combining positive and negative numbers brings about new challenges.

Think of positive numbers as steps forward and negative numbers as steps backward on a number line. Adding a negative number is like moving in the opposite direction.

• Addition of two positive numbers is simple and straightforward, resulting in a positive value.
• Adding a positive number and a negative number involves finding their difference and taking the sign of the larger absolute value.
• If you add negative numbers together, you continue in the negative direction, augmenting the negativity of the sum.

In the exercise, after performing the multiplications, we combined -24 and 20. Since 24 is larger, but negative, the result is a negative number: -4. The difference of 24 and 20 (which is 4) respects the sign of the larger number, resulting in a negative 4.

Negative Numbers

Negative numbers represent values less than zero and have unique properties when used in arithmetic operations. They are essential to fully understanding problems involving integer operations.

Negative numbers can be thought of as positions on a number line to the left of zero. This has implications for both multiplication and addition:

• Negative numbers are often the result of subtracting a larger number from a smaller one or when working with debts or temperatures below freezing.
• When adding a negative number, picture it as subtracting the absolute value of that number.
• As touched on before, multiplying or adding two negative numbers can create results that differ from multiplying or adding two positive numbers.

In our example, recognizing how negative numbers affect calculations was crucial. By understanding these core properties, you can manage operations involving negative numbers more effectively, yielding results like those in the exercise (-24 + 20 = -4) with confidence!