Problem 38 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$ 20-(-4)(3)(-2) $$

Step-by-Step Solution

Answer

The result is -4.

Step 1: Simplify Inside the Parentheses

In this expression, there are no numbers directly inside parentheses to simplify. We proceed by considering the operations involving the term \(-4)(3)(-2)\).

Step 2: Multiply the Negative and Positive Integers

First, multiply the negative and positive numbers from the expression \((-4) \times 3\). This results in \(-12\).

Step 3: Continue Multiplying with the Remaining Integer

Now, take the result from Step 2 and multiply it by the remaining negative integer: \(-12) \times (-2)\). The product of two negative numbers is positive, so \((-12) \times (-2) = 24\).

Step 4: Perform the Subtraction Operation

Finally, substitute the result of the multiplication back into the expression and perform the subtraction: \(20 - 24\). This results in \(-4\).

Key Concepts

Understanding Negative Numbers

Negative numbers can be a bit tricky at first, but with a little practice, they become easier to handle. A negative number is any number less than zero. It is represented with a minus sign (-) before the number. For example, -4 is a negative number. Negative numbers are often used to represent things like debt or below-average temperatures. They are the opposite of positive numbers, which are greater than zero.

It's essential to understand the rules when working with them, especially when they interact with positive numbers.

• If you add a negative number, it is the same as subtracting its absolute value.
• If you subtract a negative number, it is the same as adding its absolute value.

These rules will help you in simplifying expressions and performing arithmetic operations that involve negative numbers.

Multiplication of Integers

Multiplying integers, especially when they involve negative numbers, follows specific rules. Whether you're multiplying positives, negatives, or a mix, understanding these rules is crucial.

• Positive x Positive = Positive: For instance, multiplying 3 by 2 results in 6.
• Negative x Positive = Negative: When you multiply a negative integer by a positive one, like -4 times 3, the result is negative, such as -12.
• Negative x Negative = Positive: Interestingly, when multiplying two negative numbers, the product is positive. For example, -12 times -2 equals 24.

Always remember the sign rules for multiplication: the product of two numbers with different signs is negative, while the multiplication of two numbers with the same sign is positive. These sign rules simplify a lot of arithmetic problems involving integers.

Subtraction of Integers

Subtracting integers can seem complicated, especially when negative numbers are involved, but it follows clear rules that can simplify your calculations. Remember, subtraction can always be rewritten as adding the opposite.

• To subtract a positive number, simply move left on the number line.
• To subtract a negative number, convert it into adding a positive number. For example, subtracting -4 becomes adding +4.

In the example \(20 - 24\), you move 24 steps left from 20, landing at -4. By understanding these concepts, handling difficult integer subtractions becomes much easier. Be sure to utilize number lines or practice problems frequently, as they build a stronger foundation for working with all types of integer operations.