Problem 76 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{-78}{6} $$

Step-by-Step Solution

Answer

The result is -13.

Step 1: Recognize the Operation

This problem requires us to perform a division operation between -78 and 6. We are dividing a negative number by a positive number.

Step 2: Calculate the Absolute Division

First, divide the absolute values of the numbers: \[ |78| \div |6| = 13 \] So, 78 divided by 6 equals 13.

Step 3: Determine the Sign

Since the original number (-78) is negative and we are dividing by a positive number (6), the result will be negative according to the rules of division. Therefore, the result of the operation will be -13.

Key Concepts

Understanding Negative Numbers

Negative numbers are those that are less than zero. They are typically accompanied by a minus sign. Negative numbers are part of the integers family, which includes positive numbers, negative numbers, and zero.

These numbers signal a value that is opposite in direction from a positive number. For example,

• The number \( -2 \) is two steps left of zero on the number line.
• Meanwhile, the number \( 2 \) is two steps to the right.

When it comes to operations involving negative numbers, especially in division, remembering the direction shift is crucial. A negative number divided by a positive number or vice versa will result in a negative outcome.

Remember that the sign change is a critical part of handling negative values in any mathematical operation.

The Role of Absolute Value

Absolute value takes into account only the size, not the direction, of a number. It's like asking, "How far is it from zero?" without worrying whether it's to the left or right.

For any number \( x \), the absolute value is represented as \( |x| \). This expression always yields a non-negative number. For instance:

• The absolute value of \( -5 \) is \( 5 \).
• The absolute value of \( 5 \) is also \( 5 \).

This characteristic is valuable in division because it simplifies the computation to a one-dimensional scale, allowing us to process calculations ignoring the negative sign initially. Once the division is computed, the sign of the result is determined separately based on the original numbers.

Mastering Division Rules

When dividing numbers, it's crucial to follow division rules which dictate the sign of the result as well as how to perform the calculations.

Here’s a simple guide to division rules:

• Dividing two positive numbers always gives a positive result, \( e.g., \; rac{8}{2} = 4 \).
• Dividing two negative numbers also results in a positive outcome, \( e.g., \; rac{-8}{-2} = 4 \).
• Dividing a positive number by a negative one yields a negative result, \( e.g., \; rac{8}{-2} = -4 \).
• Similarly, dividing a negative number by a positive one also gives a negative result, \( e.g., \; rac{-8}{2} = -4 \).

These rules are grounded in the logic of the number line and the concept of symmetry around zero. Remember, when performing operations dealing with division, always pay close attention to the signs of the numbers involved to determine the final sign of the result.