Adding and subtracting rational expressions

Rational expressions are algebraic fractions whose numerators and denominators contain numbers, variables, or polynomials. Examples include (\frac{x+2}{3}), (\frac{1}{x}), and (\frac{x^2-4}{x+1}). Unlike an ordinary numerical fraction, a rational expression may have a variable in its denominator, so some values of that variable cannot be used.

When two rational expressions are combined, their denominators may differ. The central rule is still familiar: rewrite each fraction so that the denominators match, then combine the numerators. A dependable method also records domain restrictions, factors where possible, and checks that the final answer is equivalent to the original expression.

Understanding rational expressions

A rational expression is any fraction made from polynomial expressions. Its numerator and denominator can include addition, subtraction, multiplication, division, and integer powers of variables. Terms such as (\frac{3}{x-5}) and (\frac{x+1}{x^2+4}) therefore belong to the same family as numerical fractions.

Every rational expression has a domain restriction wherever its denominator equals zero. For (\frac{1}{x+3}), the restriction is (x\ne-3). For (\frac{5}{x^2-4}), factorising the denominator gives ((x-2)(x+2)), so both (x=2) and (x=-2) are excluded. Recording these exclusions before starting prevents an answer from appearing valid at a value where the original expression is undefined.

Finding a common denominator

Fractions can be added directly when their denominators are identical. With unlike denominators, the first task is to find their least common denominator, often called the LCD. For algebraic fractions, factorising each denominator makes this process clearer.

For example, consider (\frac{1}{x}+\frac{1}{x+2}). The factors of (x) and (x+2) have no common factor, so the LCD is (x(x+2)). The fractions become (\frac{x+2}{x(x+2)}) and (\frac{x}{x(x+2)}). Both now have the same denominator and can be combined.

The LCD is not always the product of the denominators. In (\frac{1}{x}+\frac{1}{x^2}), for instance, the denominators already share a factor. Since (x^2=x\cdot x), the LCD is (x^2), not (x^3). Factoring first avoids unnecessary algebra later.

Adding expressions with unlike denominators

Once each numerator has been rewritten over the common denominator, add the numerators and keep the denominator unchanged. Using the earlier example, (\frac{x+2}{x(x+2)}+\frac{x}{x(x+2)}) becomes (\frac{2x+2}{x(x+2)}), which simplifies to (\frac{2(x+1)}{x(x+2)}).

It is important not to add the denominators together. The rule (\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}) is incorrect. The denominators must first be made identical; only then do the numerators combine. Any restriction from the original denominators remains in force, even if the final fraction appears simpler.

Subtracting expressions with unlike denominators

Subtraction follows the same preparation, but the subtraction sign applies to the entire second numerator. This matters because (\frac{a}{b}-\frac{c}{d}) is not the same as (\frac{a-c}{b-d}). After converting both fractions to a common denominator, keep the minus sign in front of the complete second numerator.

For (\frac{1}{x}-\frac{1}{x+2}), the rewritten expression is (\frac{x+2}{x(x+2)}-\frac{x}{x(x+2)}). Combining gives (\frac{(x+2)-x}{x(x+2)}=\frac{2}{x(x+2)}), with (x\ne0,-2).

Polynomial numerators require the same care. In (\frac{5}{x-4}-\frac{3}{x}), the common denominator is (x(x-4)), so the result is (\frac{5x-3(x-4)}{x(x-4)}=\frac{2(x+6)}{x(x-4)}). Distributing the minus sign changes the sign of every term inside the bracket.

Factorising before simplifying

Factorising can reveal common factors that make simplification much easier. Consider (\frac{x^2-9}{x+3}+\frac{4}{x-3}). The first numerator factors as ((x-3)(x+3)), allowing ((x+3)) to cancel with its denominator, provided the original restriction (x\ne-3) is retained.

The expression then becomes (\frac{x-3}{x-3}+\frac{4}{x-3}). Since the first fraction equals (1) for allowed values, the sum is (\frac{x-3+4}{x-3}=\frac{x+1}{x-3}). The original domain still excludes both (x=-3) and (x=3).

Cancellation is valid only when the numerator and denominator contain exactly the same factor. It does not mean that any similar-looking terms can be removed. If no common factor remains, the combined fraction is already in its simplest form, although a different algebraic form may still exist.

Domain restrictions and cancellations

A simplified expression and the original rational expression do not always have identical domains. For example, (\frac{x^2-4}{x-2}) cancels to (x+2), but the value (x=2) must remain excluded. At that point, the original has a zero denominator and is undefined, whereas the simplified expression would otherwise suggest a value.

These restrictions are particularly important when graphing or interpreting rational functions. A cancelled factor can create a hole in a graph, while a factor left in the denominator can create a vertical asymptote. When studying the related idea of reversing a function’s direction, inverse functions offer useful background, but an inverse is not created merely by swapping a rational expression’s numerator and denominator.

Applying the method in Australia

Rational expressions appear in practical comparisons involving distances, prices, rates, and travel. A passenger in Sydney using an Opal card or in Melbourne using a Myki card might compare the cost per journey for different ticket options. If one option costs ($12) for three trips and another costs ($18) for four trips, the relevant unit prices are ($4) per trip and ($4.50) per trip. Choosing a common unit makes the comparison meaningful.

Australian supermarkets such as Woolworths and Coles also provide familiar examples. Shoppers often compare prices per litre, kilogram, or 100 grams rather than focusing only on the amount paid at the till. Because Australia uses dollars and cents and measures fuel, milk, and distances in metric units, unit-price comparisons naturally lead to numerical fractions. Under the Australian Consumer Law, traders must not mislead consumers about pricing, which makes clear unit prices and other pricing terms important to check.

Whether the fractions contain plain numbers, variables, or polynomials, the same sequence remains: identify excluded values, find the least common denominator, rewrite each fraction, combine the numerators with the correct signs, and factor the result where possible. The essential habit to remember is that matching denominators comes first, and every original restriction must survive the simplification.