Problem 11 - Prealgebra and Introductory Algebra

Question

Multiply. $$5(-4)$$

Step-by-Step Solution

Answer

The result is -20

Step 1: Analyze the problem

Identify the type of problem and select the appropriate mathematical approach.

Step 2: Solve step by step

The result is -20.

Step 3: Verify

Check the solution by substitution or alternative method.

Key Concepts

Absolute Values

Absolute value is a math concept that simplifies the process of multiplying integers. The absolute value of a number is the distance of that number from zero on a number line, and it is always a non-negative number. This concept helps simplify multiplication, especially when negatives are involved.

• The absolute value of a positive integer remains the same. For instance, the absolute value of 5 is 5, denoted as \(|5| = 5\).


• The absolute value of a negative integer is its opposite. For example, the absolute value of -4 is 4, represented as \(|-4| = 4\).

When multiplying integers like in the example \(5(-4)\), we first find the absolute values: \(|5| = 5\) and \(|-4| = 4\). By multiplying these absolute values, we get \(5 \times 4 = 20\), making it easier to proceed with the next steps, such as determining the sign of the final product.

Negative Numbers

Negative numbers can make multiplication seem tricky and need careful handling of their signs. A negative sign indicates that the number is on the opposite side of zero compared to positive numbers on the number line.

• When a positive number is multiplied by a negative number, the product is always negative. This is demonstrated when multiplying 5 by -4 to get -20.


• Multiplying two negative numbers results in a positive product. This occurs because two negative signs "cancel out," in a way.


• The multiplication rules for negative numbers are consistent but may initially seem confusing. Remember, one negative sign ends in a negative result; two lead to a positive outcome.

If you encounter negative numbers in multiplication, rest assured that mastering the sign rules will simplify the problem-solving process considerably.

Integer Multiplication

Multiplying integers is a core operation in mathematics and understanding it is crucial for doing more complex math work. Integer multiplication follows specific rules that include accounting for absolute values and the sign of numbers.

• Always start by multiplying the absolute values of the integers. This gives you the numerical value before considering any sign changes.


• Determine the final sign of the product by counting the negatives: one negative makes the product negative, while two negatives make it positive.


• Integer multiplication problems like \(5(-4)\) reinforce the importance of these rules. First, calculate the absolute values, then apply the sign rules to get the answer \(-20\).

These steps are fundamental, as they help solidify your understanding, making it easier to tackle even more complex mathematical tasks.