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When subtracting fractions, the most important initial step is to ensure that the fractions share a common denominator. A common denominator is a number that is a multiple of both denominators in the fractions you are working with. In our exercise, both fractions have 9 as their denominator: \( \frac{-2}{9} \) and \( \frac{5}{9} \). Since they already share this common denominator, we can move on to subtracting the numerators directly.

If the denominators were different, we would first need to find the Least Common Denominator (LCD), which is the smallest number that both denominators can divide evenly into. For example, in the fractions \( \frac{2}{3} \) and \( \frac{5}{4} \), the denominators are 3 and 4. The LCD for these fractions would be 12, because 12 is the smallest number both 3 and 4 can divide into without leaving a remainder.

After confirming a common denominator, we now subtract the numerators of the fractions. The numerators are the top numbers in a fraction.

In the exercise, the numerators are -2 and 5. When subtracting fractions, you subtract the second numerator from the first. Here we calculate: \(-2 - 5 = -7\).

One of the key points to remember is that subtraction of fractions involves only the numerators, while the denominator remains unchanged. This means that the result of this subtraction step gives us: \(\frac{-7}{9} \). The process can be remembered by this simplified formula: \( \frac{a}{d} - \frac{b}{d} = \frac{(a-b)}{d} \).

Fractions are a key component of rational numbers. A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where 'a' and 'b' are integers and 'b' is not zero. Both \( \frac{-2}{9} \) and \( \frac{5}{9} \) are rational numbers. The result of their subtraction, \( \frac{-7}{9} \), is also a rational number.

Understanding rational numbers helps when working with fractions because it tells us that we can operate (add, subtract, multiply, or divide) on these numbers, and the result will still be a rational number, provided we do not end up dividing by zero. This makes calculations with fractions consistent and predictable. Rational numbers include positive fractions, negative fractions, and zero, all of which can be written in the form \( \frac{a}{b} \).