Problem 17 - Elementary Algebra

Question

Simplify the following problems. $$ (4)(-3) $$

Step-by-Step Solution

Answer

Question: Simplify the following multiplication operation: $(4)(-3)$. Answer: -12 Explanation: When multiplying a positive number (4) by a negative number (-3), the result is always negative. Multiply the magnitudes of the numbers (4 * 3 = 12), and then add a negative sign to the result, giving us -12.

Step 1: Identify the numbers and their signs

Here, we have two integers: 4 and -3. The number 4 is positive while the number -3 is negative.

Step 2: Apply the properties of integer multiplication

When multiplying a positive number by a negative number, the result is always a negative number. So, in this case, we are going to multiply the magnitudes (absolute values) of the numbers and then place a negative sign in front of the result. The magnitude of 4 is 4 and the magnitude of -3 is 3.

Step 3: Multiply the magnitudes

Now, we multiply the magnitudes of the numbers: 4 * 3 = 12.

Step 4: Apply the sign

As mentioned before, the result of multiplying a positive number and a negative number is negative. So, we place a negative sign in front of the result we found in step 3. This means the simplified expression is -12. So, the solution to the given problem is: $$(4)(-3)=-12$$

Key Concepts

Integer Multiplication

Multiplying integers might seem tricky at first, but it's actually quite straightforward once you know the basics. Integer multiplication is simply the process of adding an integer to itself a certain number of times. For instance, if we consider the expression \(4 \times -3\), it implies adding the number 4 to itself -3 times. But since adding a number a negative amount of times doesn’t make practical sense, we interpret this based on the rules of integer multiplication.

The key principle in integer multiplication, especially when mixing positive and negative numbers, is to determine the sign of the result first, which depends on the signs of the numbers being multiplied. If the signs are the same (both positive or both negative), the result is positive. If the signs are different (one positive and one negative), the result is negative. We then multiply the absolute values of the two numbers, disregarding the signs, and apply the determined sign to the final answer. This is precisely how we solve \(4)(-3)\), getting -12 as the final answer.

Properties of Integers

Integers include all whole numbers and their negatives, such as -3, -2, -1, 0, 1, 2, 3, and so on. Their properties are crucial for correctly carrying out operations like multiplication. Some properties to keep in mind are:

• Commutative Property: Changing the order of the numbers doesn't change the result (e.g., \(4 \times -3 = -3 \times 4\)).
• Associative Property: When multiplying three or more numbers, the way in which they are grouped doesn't affect the result (e.g., \( (2 \times 3) \times -4 = 2 \times (3 \times -4) \)).
• Distributive Property: This property comes handy, especially when dealing with expressions that have parentheses. It states \(a \times (b + c) = a \times b + a \times c\)).

Understanding these properties can help simplify complex multiplication problems and also check your work for possible errors.

Multiplying Positive and Negative Numbers

When we multiply positive and negative numbers, the rules are clear-cut: a positive number multiplied by a negative number gives a negative result, while a negative number multiplied by another negative number gives a positive result. It’s important to distinguish these rules clearly to avoid mistakes.

Let’s look at the exercise \(4)(-3)\) as an example. Here we multiply a positive number (4) by a negative one (-3). According to the rule, the result should carry a negative sign because the numbers have different signs. However, if we were multiplying \( -4 \times -3 \), both numbers being negative, the result would actually be positive (12), as multiplying two negatives results in a positive. Keeping these rules in mind will make tasks involving negative numbers far less daunting.