What is the solution set to the rational equation $\frac{3}{x}+\frac{5}{x+2}=2$?

To solve the equation x 3 ​ + x + 2 5 ​ = 2 , we eliminate the fractions by multiplying through by x ( x + 2 ) . This leads to the quadratic equation x 2 − 2 x − 3 = 0 , which factors to give solutions x = − 1 and x = 3 . Thus, the solution set is { − 1 , 3 } .
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Answered by Anonymous | 2025-07-04

Multiply both sides of the equation by x ( x + 2 ) to eliminate the fractions: 3 ( x + 2 ) + 5 x = 2 x ( x + 2 ) .
Expand and simplify the equation to obtain a quadratic equation: x 2 − 2 x − 3 = 0 .
Solve the quadratic equation by factoring: ( x − 3 ) ( x + 1 ) = 0 , which gives x = 3 and x = − 1 .
Check for extraneous solutions and confirm that both x = 3 and x = − 1 are valid. The solution set is { − 1 , 3 } ​ .

Explanation

Understanding the Problem We are given the rational equation x 3 ​ + x + 2 5 ​ = 2 Our goal is to find the solution set for x .

Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by x ( x + 2 ) :
x ( x + 2 ) ( x 3 ​ + x + 2 5 ​ ) = 2 x ( x + 2 ) This simplifies to: 3 ( x + 2 ) + 5 x = 2 x ( x + 2 )

Expanding the Equation Expanding both sides, we get: 3 x + 6 + 5 x = 2 x 2 + 4 x Combining like terms, we have: 8 x + 6 = 2 x 2 + 4 x

Forming a Quadratic Equation Rearranging the terms to form a quadratic equation, we get: 2 x 2 + 4 x − 8 x − 6 = 0 2 x 2 − 4 x − 6 = 0 Dividing the entire equation by 2, we simplify it to: x 2 − 2 x − 3 = 0

Solving the Quadratic Equation Now, we can factor the quadratic equation: ( x − 3 ) ( x + 1 ) = 0 This gives us two possible solutions for x :
x − 3 = 0 ⇒ x = 3 x + 1 = 0 ⇒ x = − 1

Checking for Extraneous Solutions We need to check for extraneous solutions by substituting the solutions back into the original equation. We must ensure that the solutions do not make the denominator zero in the original equation.


For x = 3 :
3 3 ​ + 3 + 2 5 ​ = 1 + 5 5 ​ = 1 + 1 = 2 So, x = 3 is a valid solution.
For x = − 1 :
− 1 3 ​ + − 1 + 2 5 ​ = − 3 + 1 5 ​ = − 3 + 5 = 2 So, x = − 1 is also a valid solution.

Final Answer Therefore, the solution set to the rational equation is { − 1 , 3 } .

Examples
Rational equations are useful in various real-world scenarios, such as calculating work rates. For example, if two people are working together to complete a task, and one person can complete the task in x hours while the other can complete it in x + 2 hours, you can use a rational equation to determine how long it will take them to complete the task together. Understanding how to solve these equations helps in optimizing time and resources in collaborative projects.

Answered by GinnyAnswer | 2025-07-04