Problem 41 - Intermediate Algebra

Question

Perform the operations. See Examples 4 and 5 . $$ \frac{84}{-6} $$

Step-by-Step Solution

Answer

The result of \( \frac{84}{-6} \) is \(-14\).

Step 1: Identify the Numbers

In the expression \( \frac{84}{-6} \), the numerator is 84 and the denominator is -6.

Step 2: Determine the Sign of the Result

Since we are dividing a positive number (84) by a negative number (-6), the result will be negative.

Step 3: Divide the Absolute Values

Ignore the signs for a moment and divide the absolute values: \( \frac{84}{6} = 14 \).

Step 4: Apply the Sign to the Quotient

Based on our conclusion in Step 2, the result of \( \frac{84}{-6} \) is \(-14\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front of them, for example, -1, -2, -3, etc. Negative numbers are used to represent losses, temperatures below zero, or altitudes below sea level, among other things.
When you divide or multiply two negative numbers, the rule says they cancel each other out, resulting in a positive number. However, if you divide a positive number by a negative number, or vice versa, the result will be negative.

Grasping Absolute Value

The absolute value of a number is the non-negative value of the number without regard to its sign. For example, both 6 and -6 have an absolute value of 6 because absolute value measures the distance a number is from zero, not the direction.
When working with operations like division, as with our exercise, finding the absolute value is crucial. This is because it simplifies calculations by allowing you to temporarily ignore the signs to find the magnitude of the result.

Exploring Sign Rules in Division

When dividing numbers, knowing the sign rules can help determine the sign of the result quickly. Here are the basic rules:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

These rules come in handy when performing integer division, allowing you to accurately assign the correct sign to your answer.
In our original exercise with the expression \( \frac{84}{-6} \), we determined the result would be negative because a positive number divided by a negative number gives a negative result.