53 Rational Equations and Rational Models

Rational Equations and Rational Models is the study of solving equations that contain rational expressions by clearing denominators to produce an ordinary polynomial equation, together with checking every resulting candidate solution against the original equation's domain restrictions, and applying this technique to real-world situations involving rates and shared work.

The Scope of Rational Equations

A rational equation is an equation containing one or more rational expressions, meaning at least one term has a variable in its denominator, such as 1/x + 2 = 5/x. Solving a rational equation requires eliminating the denominators to reach a polynomial equation solvable by ordinary algebraic techniques, followed by a mandatory check of each resulting solution against the values that were excluded from the equation's domain from the outset.

Preparing the Equation's Restrictions

Before solving, every denominator present in the equation should be examined to determine which values of the variable would make it zero; these values are recorded as restrictions and must be excluded from consideration as valid solutions, regardless of what the later algebraic solving process produces. Identifying these restrictions at the start, rather than only at the end, ensures they are not overlooked once the equation has been transformed into a different form.

Clearing Denominators

Rational equations are solved by multiplying every term on both sides of the equation by the least common denominator of all the rational expressions present, canceling each denominator and converting the equation into an equation involving only polynomials.

Every term, including any term that does not itself contain a fraction, must be multiplied by the LCD, exactly as in clearing denominators for fractional linear equations generally.

Solving Linear Rational Equations

When clearing denominators reduces a rational equation to a linear equation, that equation is solved using ordinary one-step, multi-step, or two-sided linear equation techniques. Continuing the earlier example, 1 + 2x = 5 simplifies to 2x = 4, giving x = 2, a candidate solution that must still be checked against the restriction x ≠ 0 established at the outset before being accepted as final.

Rational Equations Producing Factorable Quadratic Outcomes

Some rational equations, once denominators are cleared, reduce to a quadratic rather than a linear equation, requiring the quadratic to be solved using factoring or another appropriate quadratic-solving technique, with every resulting solution then checked against the equation's original restrictions.

Factoring this quadratic gives (x - 2)(x + 1) = 0, producing two candidate solutions, x = 2 and x = -1, each of which must be individually checked against the restrictions x ≠ 0 and x ≠ -2 established from the original denominators.

Classifying Rational Equation Solutions

A candidate solution obtained after clearing denominators is classified as valid if it does not match any of the equation's original restrictions, and as extraneous if it does match a restriction — meaning it satisfies the cleared polynomial equation but would make a denominator in the original equation equal to zero, and must therefore be discarded. A rational equation can have zero, one, or several valid solutions after this classification is applied, even if the cleared polynomial equation initially produced more candidate solutions than that.

Elementary Rational Models

Rational equations model real-world situations involving rates, particularly work problems, in which two or more agents complete a task at individual rates and a combined rate is needed, and rate problems more generally, in which a quantity such as distance is expressed as the product of a rate and time, requiring a variable to appear in a denominator when time or rate itself is the unknown being solved for. A work problem is commonly modeled using the relationship that the sum of the fractional amounts of work each agent completes per unit time equals the fractional amount of work completed together per unit time.

Solving a Rational Model

Solving an applied rational model requires the same full procedure as any rational equation: identifying restrictions (typically excluding zero or negative time values, which are also checked against real-world plausibility), clearing denominators, solving the resulting polynomial equation, and checking each candidate solution against both the algebraic restrictions and the contextual requirements of the original scenario, such as rejecting a negative time value even if it is not algebraically restricted.

Verifying Rational Equation Solutions

A rational equation's solution is verified in two stages: algebraically, by substituting it into the original, unsimplified rational equation and confirming both sides are equal, and by restriction, confirming the value does not match any of the values excluded at the outset. Both checks are necessary, since an algebraically satisfying value that violates a restriction must still be discarded as extraneous.

Diagnosing Errors in Rational Equations

Common errors in this area include forgetting to determine the equation's restrictions before solving, failing to check candidate solutions against those restrictions after solving, multiplying only some terms of the equation by the LCD rather than every term, and accepting an extraneous solution as valid because it satisfies the cleared polynomial equation without recognizing that it separately violates the original equation's domain.