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When we talk about ‘quotient’ in division, we mean the result you get when you divide one integer by another. In our example, \frac{−28}{7}, we're trying to find the quotient of \(-28\) divided by \(7\).

A quotient can be positive or negative, depending on the signs of the numbers involved. Here’s the basic idea:

- If both numbers are positive, the quotient is positive. - If both numbers are negative, the quotient is positive. - If one number is positive and the other is negative, the quotient is negative.

Knowing the quotient helps us understand how division breaks up a certain number into equal parts, taking into account the direction (positive or negative) of the result.

Absolute value is a concept that expresses the magnitude of a number without regard to its sign. It is always a positive value or zero.

In mathematical terms, the absolute value of a number x, written as \|x\, is: \[|x| = x \textrm { if x ≥ 0} \ \textrm { or } |x| = -x \textrm { if x < 0} \]

For example, in the problem \(\frac{-28}{7}\), you first convert \|-28| = 28\. This step helps by making the division process simpler because you’re only dealing with positive numbers. After you find the quotient of the absolute values, you can then apply the sign rules to get the final result.

When dividing integers, it’s crucial to apply the correct sign rules to determine whether the quotient will be positive or negative. Here are the rules:

- A positive number divided by a positive number gives a positive quotient. - A negative number divided by a negative number gives a positive quotient. - A positive number divided by a negative number gives a negative quotient. - A negative number divided by a positive number gives a negative quotient.

In our example \(\frac{-28}{7}\), the numerator (\(-28\)) is negative, and the denominator (\(7\)) is positive. Applying the sign rules, a negative divided by a positive gives us a negative quotient. This explains why the final result is \(-4\).

Remembering these rules will make it easier to solve similar problems and get the correct sign for your quotient every time.