Rational Equations

Rational equations contain variable denominators and are solved by recording restrictions, clearing denominators, and checking candidates.

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Excluded inputs appear as breaks or vertical asymptotes. Two branches approach dashed horizontal and vertical asymptotes without touching them. The separation emphasizes that denominator restrictions belong to both the algebra and the graph.

Clearing denominators simplifies the algebra, but it never makes forbidden inputs legal.

Identify restrictions first

A rational equation contains one or more rational expressions. Before manipulating it, find values that make any original denominator zero. These are excluded even if a factor later cancels.

For

\frac2{x-1}+1=\frac5{x-1},

the restriction is $x\ne1$.

Worked example: a linear result

Worked example: a quadratic result

Solve

\frac1x+\frac1{x-2}=1,qquad x\ne0,2.

Multiply by $x(x-2)$:

$$ (x-2)+x=x(x-2). $$

Rearrange:

$$ x^2-4x+2=0. $$

The quadratic formula gives

x=2\pm\sqrt2.

Neither value is excluded, so both are valid.

Applications and units

Rational equations model combined work rates, travel time, lens formulas, electrical resistance, and inverse variation. Define variables and units before solving.

Context may add restrictions beyond denominators: time, speed, length, and count are often positive. Algebraic candidates must satisfy both mathematical and contextual domains.

Graphical interpretation

An equation $f(x)=g(x)$ asks where two graphs intersect. Vertical asymptotes and holes can explain excluded candidates or a lack of solutions.

Graphing is a useful check, but exact solutions require algebra when the question asks for exact form.

Common mistakes

Writing restrictions after cancelling. Use original denominators.

Multiplying only the fractions by the LCD. Every term must be multiplied.

Cross multiplying an equation with more than one fraction per side without combining first. Use the full LCD method.

Keeping an excluded candidate. A zero denominator is never allowed.

Checking only the cleared equation. Verify in the original.