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Algebraic fractions are similar to the regular fractions you see with numbers, but they include variables (like x or y). The key to working with algebraic fractions is the same as with numerical fractions: you need to understand the numerator (the top part), the denominator (the bottom part), and the operation involved, whether it's addition, subtraction, multiplication, or division.

In dividing algebraic fractions, you essentially invert the second fraction and then perform multiplication. For example, \( \frac{a}{b} \div \( \frac{c}{d} \) = \frac{a}{b} \times \( \frac{d}{c} \) \). The rules for dividing negative numbers apply here just as they would with integers, ensuring that you arrive at the correct sign for your answer.

When dividing integers, the process is straightforward: divide the absolute values of the numbers and then determine the sign of the result. If the integers have the same sign, whether both negative or positive, the result of the division is positive. Conversely, if they have different signs, the result will be negative.

Let's say you're working with \( -12 \div 4 \). Since one number is negative and the other positive, your answer will be negative. The division of their absolute values, 12 divided by 4, is 3, so with the different signs factored in, the result is \( -3 \). This rule keeps division of integers consistent and predictable.

In mathematics, some expressions are considered undefined. The most common type of these is division by zero. Dividing by zero is undefined because there's no number that you can multiply by zero to get any other number.

For instance, \( \frac{5}{0} \) is undefined because whatever number you might think of multiplying by zero to get 5 doesn't exist – zero times anything will always be zero. Always check the denominator in a division problem; if it is zero, you can immediately state that the expression is undefined.

When dividing negative numbers, use the following rules: if both numbers are negative, the result is positive; if one number is negative, the result is negative. This might seem counterintuitive at first, but it ensures consistency across mathematical operations.

The division problem from our textbook \( \frac{-90}{-3} \) is an excellent example. Both numbers are negative, so the result is a positive 30, as calculating their absolute values gives 90 divided by 3, which equals 30. Remembering this rule helps prevent mistakes when dealing with negative numbers in division.