Solving Rational Equations

Arational equationis an equation withrational expressionson either side of the equals sign.

ONE TECHNIQUEfor solving rational equations iscross-multiplication— what some textbooks call themeans/extremes property.

This method works only if on each side of the equation there is only one rational expression.

Example 1:

Solve:

$\frac{7}{x + 2} = \frac{x}{x + 2}$

Cross multiplying, we get:

$x^{2} + 2 x = 7 x + 14$

Thisquadratic equationcan be solved byfactoring.

$x^{2} - 5 x - 14 = 0$

$(x - 7) (x + 2) = 0$

Rememberto check in the original equation for validity of solutions. In this case, $x = 7$ is valid but $x = - 2$ isn't, since it means division by zero in the original equation.

ANOTHER METHODis to multiply through by theleast common denominatorof all of the fractions on either side of the equation.

Example 2:

Solve:

$\frac{x}{16} - \frac{3}{8 x} = \frac{5}{16}$

The least common denominator (LCD) in this case is $16 x$ . So, multiply both sides of the equation by $16 x$ .

$\frac{x (16 x)}{16} - \frac{3 (16 x)}{8 x} = \frac{5 (16 x)}{16}$

$x^{2} - 6 = 5 x$

Solve the quadratic equation by factoring.

$x^{2} - 5 x - 6 = 0$

$(x - 6) (x + 1) = 0$

$x = 6 or x = - 1$

Rememberto check back to make sure these solutions are valid – that is, that they don't result in division by zero when substituted in the original equation. In this case, both solutions are valid.

Solving Rational Equations

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