1.51 Rational Equation Definitions

Rational Equation Definitions establishes the vocabulary for equations containing rational expressions, describing the preparatory technique used to eliminate the denominators before solving, the application of such equations to model real-world rate and work-based situations, and the specific type of false solution that can arise as an artifact of the solving process itself.

Rational Equation

A rational equation is an equation in which one or more terms are rational expressions, containing a variable in at least one denominator, such as 1/x + 1/3 = 1/2. Solving a rational equation ultimately requires eliminating its denominators so that the resulting equation can be handled using the same isolation techniques applied to polynomial equations.

Rational Clearing

Rational clearing is the technique of multiplying every term on both sides of a rational equation by the least common denominator of all the rational expressions present, which cancels every denominator and produces an equivalent polynomial equation free of fractions. Applying rational clearing to the equation above, multiplying every term by the least common denominator 6x, produces the polynomial equation 6 + 2x = 3x.

Rational Model

A rational model is the application of a rational equation to represent a real-world situation involving rates or shared work, such as combined work rates for two people completing a task together, or combined rates for two objects covering a shared distance. Rational models frequently arise whenever a quantity is described as a rate—an amount per unit of time or per unit of another quantity—since combining several such rates typically requires adding fractions with the varying quantity appearing in the denominator.

Extraneous Solution

An extraneous solution is a value obtained during the solving process of a rational equation that satisfies the cleared, denominator-free equation but does not actually satisfy the original rational equation, typically because it makes one of the original denominators equal to zero. After solving a rational equation, every candidate solution must be checked against the original equation's excluded values, and any candidate solution matching an excluded value must be discarded as extraneous rather than accepted as a true solution.

Together, these definitions describe rational equations as equations requiring rational clearing before ordinary solving techniques can be applied, extend naturally into rational models describing rate and work-based real-world scenarios, and caution that every resulting candidate solution must be checked for the possibility of being an extraneous solution before it is accepted as valid.