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Rational Expression: Evaluation
Rational expressions can be evaluated if a value is given for the variable. For example, if the rational expression (3 / x + 2) was given with x = 3, the expression could be written (3 / 3 + 2) and solved as (3 / 5). Note that without this given value, nothing could have been done to the expression as it was already in its simplest form.
Rational Expression: Simplifying
Complex rational expressions that can't be evaluated can often be simplified. This is done similarly to simplifying nonrational fractions by finding common factors of the numerator and denominator and canceling them out. For example, simplify the rational expression (x^2 + 7x + 12) / (x^2 + 5x + 6). Begin by factoring the numerator: (x + 3)(x + 4). Factor the denominator: (x + 3)(x + 2). Place back in the fraction: (x + 3)(x + 4) / (x + 3)(x + 2). Cancel out like terms, which here would be the (x + 3), for a final answer of (x + 4) / (x + 2).
Rational Equation: Domains
When solving a rational equation, it is important to establish the domain. The domain is the answers that would cause the denominator to equal 0, which is an invalid answer since a 0 denominator is undefined. The easiest way to find the domain is to isolate the denominator, set it equal to 0 and then solve for the variable. For example, if the rational term in the equation was 3x^2 / 2x + 4. Set the denominator equal to 0: 2x + 4 = 0. Solve for the variable: 2x = -4 becomes x = -2. If the solution of the equation ended up equaling -2, then the equation would in fact have no solution, as this is not a valid answer.
Rational Equations: Solving
Solve a rational equation by using algebra to shift terms away from the variable until it is isolated on one side of the equation. Find the answer then establish the domain to make certain the answer is valid. For example, solve the rational equation (3 / (x (x - 2))) + (5/x) = (3 / (x - 2)). Begin by establishing a common denominator. Since the first denominator shares common terms with the others, it will be the common denominator. Convert the fractions accordingly: (3 / (x(x - 2))) + ((5 * (x - 2)) / (x(x - 2)) = (3x / x(x - 2)). Distribute the 5 in the second numerator: (5x - 10). Ignore the denominators since they're identical and write the equation in terms of numerators: 3 + 5x - 10 = 3x. Combine like terms: 5x - 7 = 3x. Subtract 5x from both sides: -7 = -2x. Divide -2 from both sides: 3.5 = x. Check whether this answer will make any of the denominators equal 0; since it doesn't, this answer is valid.