Problem 75 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{24}{-6} $$

Step-by-Step Solution

Answer

-4

Step 1: Understand the Problem

We need to perform the division operation for the given expression: \( \frac{24}{-6} \). This means we need to divide 24 by -6.

Step 2: Divide the Numerator by the Denominator

Start by dividing the absolute values of the numerator (24) and the denominator (6). The division \( 24 \div 6 \) equals 4.

Step 3: Apply the Sign Rule

Now, we apply the sign rule for division. The division of a positive number by a negative number results in a negative number. Therefore, the answer is \(-4\).

Key Concepts

Absolute Value

The absolute value of a number is like the number's positive shell. It represents the distance from zero on a number line, effectively stripping away any negative sign. For instance, the absolute value of both 6 and -6 is 6. To find the absolute value, simply disregard the sign in front of the number.
In the exercise, we are asked to divide 24 by -6. When calculating the absolute value, we only consider the magnitude of -6, which is 6. This helps in simplifying division without worrying about the negative sign. So for calculation purposes, \( |24| = 24 \ \text{and} \ \ |{-6}| = 6. \) This lets us focus solely on the division 24 by 6.

Negative Numbers

Negative numbers are those which lie below zero on the number line and are always represented with a negative (-) sign. They essentially represent a direction opposite to positive numbers, frequently thought of as moving leftward from zero. This can often be applied in various real-life contexts, such as temperatures below freezing, elevations below sea level, or bank overdrafts.
In division problems, such as dividing 24 by -6, the negative sign indicates a direction or property we must consider. This means that while the absolute values of the numbers determine the basic division operation, the presence of a negative number alters the result's sign. Recognizing this indicator is vital for correctly solving problems involving negative numbers.

Sign Rule in Division

The sign rule in division plays a crucial role in determining the sign of your result. It dictates how positive and negative numbers interact when divided.
This rule can be summarized as follows:

• Positive number divided by a positive number results in a positive number.
• Positive number divided by a negative number results in a negative number.
• Negative number divided by a positive number results in a negative number.
• Negative number divided by a negative number results in a positive number.

In the case of \( rac{24}{-6} \), since 24 is positive and -6 is negative, the result of the division is negative, as per the rule. Thus, after calculating the division using absolute values (which is 4), we then apply the sign rule to make the answer \( -4. \) Understanding this rule ensures the correct sign is applied to your final answer.