SOLVING RATIONAL EQUATIONS

Equations of this type occur frequently in applications. It is easiest to solve a rational equation if the fractions are eliminated. This can be done by multiplying each side of the equation by the least common denominator (LCD). Remember that when you multiply each side by the LCD, each term on each side must be multiplied by the LCD.

Example: Solve $\frac{3}{x - 7} + 1 = \frac{8}{x^{2} - 9 x + 14}$ . Verify the solution.

Solution:

\frac{3}{x - 7} + 1 = \frac{8}{x^{2} - 9 x + 14}

Multiply by LCD $(x-2)(x-7)$ :

(x - 2) (x - 7) \frac{3}{x - 7} + (x - 2) (x - 7) = (x - 2) (x - 7) \frac{8}{(x - 2) (x - 7)}

3(x-2) + (x^2 - 9x + 14) = 8

x^2 - 6x + 8 = 8

x^2 - 6x = 0

x(x-6) = 0

$x = 0$ or $x = 6$ . The solutions are 0 and 6.

Solving Rational Equations

Solving Rational Equations

Example: Solve 3x−7+1=8x2−9x+143x−7+1=8x2−9x+14\frac{3}{x-7} + 1 = \frac{8}{x^2 - 9x + 14}x−73​+1=x2−9x+148​x−73​+1=x2−9x+148​. Verify the solution.

Example: Solve $\frac{3}{x - 7} + 1 = \frac{8}{x^{2} - 9 x + 14}$ . Verify the solution.