Final answer:
The expression 'the quotient of 4 and the difference of 7 and a number' refers to dividing 4 by the result of subtracting a number from 7. The rules of signs must be applied to determine the sign of the result once the difference has been calculated and the division has been performed.
Step-by-step explanation:
Understanding the Quotient and Difference in Mathematics
The question is asking for the result when 4 is divided by the difference between 7 and a number, say x. This operation is expressed as 4 / (7 - x). Let's break down the process step-by-step:
Identify the difference in the denominator: 7 - x.
Divide the number 4 by this difference to find the quotient.
This mathematical operation involves rules of signs for division and subtraction. When we subtract a number from another, if the signs of both numbers are different, we subtract the smaller number from the larger and keep the sign of the larger number (Rule 3). When we divide, the sign of the result will follow the multiplication rules (Rule 8); it will be positive if the signs of both the numbers are the same, negative if the signs are different.
If the value of x is known, you could compute an actual numerical answer. If x is positive and less than 7, then 7 - x will also be positive, and the quotient will be positive. If x is more than 7, then 7 - x would be negative, and the quotient would be negative as per the rules of division by negative numbers.