Solve the following rational equation. If the equation has no solution, so state.
$\frac{6}{x}+2=\frac{14}{x}$
Select the correct choice below and, if necessary, fill in the answer box to complete.
A. The solution set is {$\square$}.
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are infinitely many solutions.
C. There is no solution.
Answer (1)
Subtract x 6 from both sides: 2 = x 14 − x 6 .
Combine the fractions: 2 = x 8 .
Multiply both sides by x : 2 x = 8 .
Divide both sides by 2 to find the solution: x = 4 , so the solution set is { 4 } .
Explanation
Isolating terms with x We are given the rational equation x 6 + 2 = x 14 . Our goal is to solve for x . First, we want to isolate the terms with x on one side of the equation.
Subtracting fractions Subtract x 6 from both sides of the equation: x 6 + 2 − x 6 = x 14 − x 6 2 = x 14 − x 6
Combining fractions Combine the fractions on the right side of the equation: 2 = x 14 − 6 2 = x 8
Multiplying by x Now, we want to solve for x . Multiply both sides of the equation by x : 2 × x = x 8 × x 2 x = 8
Dividing by 2 Divide both sides of the equation by 2: 2 2 x = 2 8 x = 4
Checking for extraneous solutions Now, we need to check if x = 4 is an extraneous solution. An extraneous solution is a value of x that makes the denominator of the original equation equal to zero. In this case, the denominator is x , so x cannot be 0. Since x = 4 , it is not an extraneous solution. Therefore, the solution is x = 4 .
Final Answer The solution set is {4}.
Examples
Rational equations are useful in many real-world scenarios, such as calculating average speeds or determining the time it takes to complete a task. For example, if two people are working together to complete a job, and one person can complete the job in x hours and the other person can complete the job in y hours, then the time it takes for them to complete the job together can be modeled by a rational equation. Understanding how to solve rational equations allows us to solve these types of problems efficiently.
