Under what situation would one or more solutions of a rational equation be unacceptable? Give an example.
Well, if the solution involves going on a date with a clown, that might be considered unacceptable for some people. For example, let's say you have the rational equation "x/2 = 4". The solution to this equation is x = 8. But if your solution involves going out with a clown named Bozo, well, that might be a deal-breaker for some folks! So, in this case, the solution involving going on a date with a clown would be unacceptable.
One or more solutions of a rational equation would be considered unacceptable if they result in a denominator equal to zero. This is because division by zero is undefined in mathematics.
For example, let's consider the rational equation: (x + 2)/(x - 3) = 5. Solving this equation, we would get:
(x + 2)/(x - 3) = 5
Cross multiplying:
5(x - 3) = x + 2
Expanding:
5x - 15 = x + 2
Combining like terms:
5x - x = 2 + 15
4x = 17
x = 17/4
In this case, the solution x = 17/4 is acceptable because it does not cause any denominator to be zero. However, if the equation were (x + 2)/(x - 3) = 0, solving for x would give x = 3 as a solution. In this case, x = 3 would be an unacceptable solution because it would make the denominator (x - 3) equal to zero, resulting in division by zero.
One or more solutions of a rational equation can be considered unacceptable when they result in a division by zero. This is because division by zero is undefined in mathematics.
To illustrate this, let's consider the rational equation:
(x + 1)/(x - 2) = 3
To solve this equation, we start by setting the numerator equal to the denominator times 3:
x + 1 = 3(x - 2)
Expanding and simplifying both sides of the equation:
x + 1 = 3x - 6
Now, we can isolate the variable x by subtracting x from both sides:
1 = 2x - 6
Adding 6 to both sides:
7 = 2x
Finally, dividing both sides by 2:
x = 3.5
In this example, the solution to the equation is x = 3.5. However, if we substitute this solution back into the original equation:
(3.5 + 1)/(3.5 - 2) = 3
We would end up with:
4.5/1.5 = 3
Which simplifies to:
3 = 3
This means that x = 3.5 is an acceptable solution. However, if we had obtained x = 2 as a solution, substituting it back into the original equation would lead to a division by zero:
(2 + 1)/(2 - 2) = 3/0
This expression is undefined since division by zero is not allowed in mathematics. Therefore, the solution x = 2 would be considered unacceptable for this rational equation.