8.6 Solve Rational
Equations
Cross multiplying
A method of
solving a simple rational equation for which each side of the equation is a
single rational expression. Equal
products are formed by multiplying the numerator of each expression by the
denominator of the other.
Example 1
Solve a rational equation by cross multiplying
|
|
Original equation |
|
20
__(x - 2)__ = 5
__(3x - 5)__ |
Cross multiply. |
|
__20 x - 40__ =
__15x - 25__ |
Distributive property |
|
__5x - 40__ =
__-25__ |
Subtract __15 x__
from each side. |
|
__5 x__
= __15__ |
Add __40__ to each
side |
|
__x__
= __3__ |
Divide each side by __5__. |
Example 2
Solve a rational
equation with one solution
|
|
Original equation |
|
__3x__ .= __3x__ |
Multiply each side by the
LCD, __3x__. |
|
__24 +11x__ =
__- 42__ |
Simplify. |
|
__x__ = __-6__ |
Solve for __x__ . |
The
solution is __-6__. Check
this in the original equation.
Example 3
Solve a rational
equation with two solutions
Solve:
The LCD is __(x
+ 3)(x - 4)__ .
__(x + 3)(x - 4)__ = __(x + 3)(x - 4)__
__(x - 4)(3x - 5)__ = __2(x
+ 3)(x - 4) + 8__
__3x2 - 17x + 20__ = __2x2 - 2x - 16__
__x2 - 15x + 36__
= 0
__(x - 3)(x - 12)__ = 0
x = __3__
or x = __12__
The solutions are __3__ and __12__ . Check these in the original equation.