Chapter Test
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When is a solution of an equation extraneous?
Exactly One Solution: All real numbers except 3
No Solutions: a=3
LHS * (x-a)=RHS* (x-a)
a/(x-a)* (x-a) = a
Rearrange equation
We have that x is equal to 3. Therefore, x=3 is the only potential solution of this equation. However, this value is not always a solution of the original equation! When dealing with rational equations we have to remember to consider the possibility of extraneous solutions.
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An extraneous solution is a solution that is derived from the original equation, but it is not actually a solution. |
In our equation, a solution is extraneous when it makes the denominator in the original equation equal to 0. x- a=0 ⇔ x= a Let's summarize what we already know.
We will use this information to determine the values of a for which the equation has exactly one solution and for which it has no solutions.
For this equation, when x=3 is not an extraneous solution, it is the only solution of the original equation. Also, in this case, a solution is extraneous if and only if it is equal to a. Therefore, a solution is not extraneous if and only if it is not equal to a. x= 3 isnotan extraneous solution ⇕ a≠3 We have that our equation has exactly one solution for any value of a that is different from 3.
Conversely to the above explanation, when x=3 is an extraneous solution, the equation has no solutions. Again, a solution is extraneous if and only if it is equal to a. x= 3 is an extraneous solution ⇕ a= 3 We have that our equation has no solutions for a=3.