Problem 29 - Beginning and Intermediate Algebra

Question

Multiply. $$ -11(11) $$

Step-by-Step Solution

Answer

The product is -121.

Step 1: Identify the Operation

The operation we need to perform is multiplication between a negative number and a positive number. Specifically, we have \[-11 \times 11\].

Step 2: Apply the Rule of Signs

Remember, when multiplying a negative number by a positive number, the result is negative. Therefore, we should expect a negative result for this multiplication.

Step 3: Multiply the Absolute Values

Ignore the signs for a moment and multiply the absolute values of the numbers: 11 multiplied by 11 is \[121\].

Step 4: Attach the Correct Sign

Based on the rule of signs, attach a negative sign to the result from Step 3, making it \[-121\].

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are usually written with a minus sign in front, like -1, -5, or -11. Imagine them as steps below zero on a number line. The further left you move from zero, the more negative the number becomes. When we multiply two numbers, the signs of the numbers matter. Here’s a simple way to remember:

• Negative times Positive = Negative
• Negative times Negative = Positive
• Positive times Positive = Positive

In our exercise, since we have one negative (-11) and one positive (11) number, the result is negative.

Positive Numbers

Positive numbers are greater than zero and don't need any sign, but for clarity, you can also write a plus before them, like +5 or +11. On the number line, these are all the numbers to the right of zero. When multiplying positive numbers by any other numbers, they tend to 'keep the peace' with their non-negative nature, affecting the result based on the rules of multiplication:

• Positive times Positive = Positive
• Positive times Negative = Negative

In our case, the positive 11 influences the final result's nature when combined with a negative number, giving us a negative result.

Absolute Value

The absolute value of a number is simply the number without its sign. It shows how far the number is from zero, regardless of the direction. For example, both \(-5\) and \(5\) have an absolute value of 5 because they are five steps away from zero on a number line. In math, we write the absolute value using vertical lines, such as \(|-11| = 11\).

When multiplying integers, first find the absolute values of the numbers, multiply them, and consider the signs afterward. For \(-11 \times 11\), the absolute values are \(11\) and \(11\), resulting in \(121\). Then attach the negative sign as per the rule of multiplying a negative by a positive, thus giving \(-121\).