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When simplifying algebraic expressions, understanding how to multiply negative numbers is essential. The rule is simple yet crucial: the product of two negative numbers is a positive number. This might seem counterintuitive at first, as we often associate negative signs with subtraction or loss. However, in multiplication, think of the negative signs as directions. If both numbers are 'going in the same direction' (both negative), their product moves in the 'positive direction'.

To visualize this, consider the fact that if you owe money (a negative value) and you owe it multiple times (-1 for each instance), your debt being eliminated (times -1) would actually be a positive outcome. So, multiplying a negative by a negative leaves you in the positive. In the exercise example, the product of -1/9 and -9 is positive 1, as the negative signs cancel each other out.

Algebraic fractions are fractions that contain variables. They follow the same principles as regular fractions, but instead of numerical denominators and numerators, we often deal with expressions. The key to working with algebraic fractions is to simplify them to make them more manageable.

In our example, we simplify the algebraic fraction \( -\frac{1}{9}(-9 b) \). The process involves multiplying the numerator by the reciprocal of the denominator when possible. Here, though, we are multiplying a fraction by a number. It's like simplifying the fraction -1/9 by multiplying the numerator and the denominator by -9. The -9 in the denominator cancels out with the -9 in the numerator, leaving us with 1. Then simply apply that result to the variable ‘b’. The final simplified form is ‘b’, free from any fractions.

Variables are symbols that represent unknown numbers, and they are fundamental in algebra. Working with them involves understanding that they follow all the rules of arithmetic that apply to numbers. When variables are combined with numbers, as in the exercise \( -\frac{1}{9}(-9 b) \), they are treated the same way you would handle the numbers alone.

In our solution process, after simplifying the negative number multiplication, we are left with multiplying the number 1 by the variable 'b'. Since any number multiplied by 1 is itself, the variable 'b' remains unchanged. Therefore, the expression simplifies to just 'b', demonstrating that the variable acts as a placeholder for any number which, when multiplied by 1, stays the same.