What is an example of a rational equation where the solution makes the denominator equal to zero and are extraneous solutions?
An example of a rational equation where the solution makes the denominator equal to zero and results in extraneous solutions is:
(3x + 2)/(x - 1) = 4
In this equation, if we solve for x, we will find that when x = 1, the denominator becomes zero. However, this value of x would make the equation undefined, as division by zero is not allowed in mathematics. So, x = 1 is a solution that results in a denominator equal to zero and is considered an extraneous solution.
To solve this rational equation and identify the extraneous solution, we can follow these steps:
Step 1: Eliminate the denominator by multiplying both sides of the equation by (x - 1):
(x - 1) * (3x + 2)/(x - 1) = 4 * (x - 1)
Simplifying, we get:
3x + 2 = 4x - 4
Step 2: Bring all the terms to one side of the equation by subtracting 3x and adding 4 to both sides:
3x - 4x = -4 - 2
Simplifying further:
-x = -6
Step 3: Multiply both sides by -1 to isolate x:
x = 6
Step 4: Check for extraneous solutions by substituting the value of x back into the original equation:
(3(6) + 2)/(6 - 1) = 4
Simplifying:
20/5 = 4
The result of this equation is true, so x = 6 is a valid solution. There are no extraneous solutions in this case.
Remember, extraneous solutions can occur when we perform operations that are not mathematically allowed, such as dividing by zero or taking the square root of a negative number. To identify extraneous solutions, it's essential to check the obtained solutions against the original equation and ensure they satisfy all necessary conditions.