Solving Rational Equations

We want to solve the given rational equation.

\frac{x}{x + 1} + \frac{x}{x - 2} = 2

To do so, we will first determine the least common denominator (LCD) so that we can clear the denominators. In this case, the LCD is the product of the denominators.

\frac{x}{x + 1} + \frac{x}{x - 2} = 2

MultEqn

LHS \cdot (x + 1) (x - 2) = RHS \cdot (x + 1) (x - 2)

(\frac{x}{x + 1} + \frac{x}{x - 2}) (x + 1) (x - 2) = 2 (x + 1) (x - 2)

DistrDistribute

(x + 1) (x - 2)

\frac{x}{x + 1} (x + 1) (x - 2) + \frac{x}{x - 2} (x + 1) (x - 2) = 2 (x + 1) (x - 2)

CancelCommonFacCancel out common factors

\frac{x}{x + 1} (x + 1) (x - 2) + \frac{x}{x - 2} (x + 1) (x - 2) = 2 (x + 1) (x - 2)

SimpQuotSimplify quotient

x (x - 2) + x (x + 1) = 2 (x + 1) (x - 2)

Solve for

x

DistrDistribute

x

x^{2} - 2 x + x^{2} + x = 2 (x + 1) (x - 2)

DistrDistribute

2

x^{2} - 2 x + x^{2} + x = (2 x + 2) (x - 2)

DistrDistribute

(k - 2)

x^{2} - 2 x + x^{2} + x = 2 x (x - 2) + 2 (x - 2)

DistrDistribute

2 k

x^{2} - 2 x + x^{2} + x = 2 x^{2} - 4 x + 2 (x - 2)

DistrDistribute

2

x^{2} - 2 x + x^{2} + x = 2 x^{2} - 4 x + 2 x - 4

AddSubTermsAdd and subtract terms

2 x^{2} - x = 2 x^{2} - 2 x - 4

SubEqn

LHS - 2 x^{2} = RHS - 2 x^{2}

- x = - 2 x - 4

AddEqn

LHS + 2 x = RHS + 2 x

x = - 4

Finally, we will check our result to make sure that it is not an extraneous solution.

\frac{x}{x + 1} + \frac{x}{x - 2} = 2

Substitute

x = - 4

\frac{- 4}{- 4 + 1} + \frac{- 4}{- 4 - 2} = ? 2

Simplify left-hand side

AddSubTermsAdd and subtract terms

\frac{- 4}{- 3} + \frac{- 4}{- 6} = ? 2

DivNegNeg

\frac{- a}{- b} = \frac{a}{b}

\frac{4}{3} + \frac{4}{6} = ? 2

ReduceFrac

\frac{a}{b} = \frac{a / 2}{b / 2}

\frac{4}{3} + \frac{2}{3} = ? 2

AddFracAdd fractions

\frac{6}{3} = ? 2

CalcQuotCalculate quotient

2 = 2 ✓

Solving Rational Equations 1.11 - Solution