Answer:
- x = 7 is a solution
- x = -2 is an extraneous solution
Step-by-step explanation:
The recommended method of solving this sort of equation is to ...
a) recognize that the denominators (x+2) and (x+3) are factors of the denominator (x^2 +5x +6).
b) multiply the whole equation by (x^2 +5x +6)
Doing this will give the equation ...
3 + (x -1)(x +3) = 7(x +2)
In standard form, this is ...
x^2 -5x -14 = 0
(x +2)(x -7) = 0 . . . . factored form
This has solutions x = -2 and x = 7.
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The original equation is undefined for x = -2 (and x = -3), so the solution x = -2 is extraneous.
The only solution is x = 7.
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Comment on avoiding extraneous solutions
If you don't try to eliminate fractions, but work with the equation as a rational equation, you get ...
(x +2)(x -7) / ((x +2)(x +3)) = 0
This reduces to
(x -7)/(x +3) = 0
which has one solution at x=7. The factor that results in the extraneous solution cancels itself from numerator and denominator.
