Complex Fractions and Equations with Rational Expressions
First we begin with a complex fraction that contains no variables.
Notice we first multiplied by the total Least Common Denominator, then we simplified.
Complex Fractions Involving Expressions
When we have a complex fraction with rational expressions as the numerator and denominator, we follow similar steps, except, of course factoring plays a key role.
Note: Usually you will not have to do all of the steps.
x + 1 -
x - 6
Recall that if
ad = bc
The same hold true for functions:
fg = hk
3x - 1 x + 2
We cross multiply
9x2 + 12x - 3x - 4 = 5x2 + 10x - 2x - 4
9x2 + 9x - 4 = 5x2 + 8x - 4
4x2 + x = 0
x (4x + 1) = 0
Caution: Always check and see that the solution works by plugging back into the original equation!
To solve equations that involve rational expressions, we following the following steps:
Then multiply by the LCD (x - 6)(x+ 6).
3 4 48
3(x + 6) - 4(x - 6) = 48
3x + 18 - 4x + 24 = 48
-x + 42 = 48
-x = 6
x = -6
Notice that -6 cannot be put back into the original equation, since there would be a zero in the denominator. We can conclude that this equation has no solution.