$\frac{x}{x-1}-\frac{2}{x+2}=\frac{4 x-1}{x^2+x-2}$
What are all the solutions for the given equation?
1
3
1 and 3
-1 and -3
Answer (1)
Factor the denominator and identify restrictions: x 2 + x − 2 = ( x − 1 ) ( x + 2 ) , x = 1 , − 2 .
Multiply by the common denominator to clear fractions: ( x − 1 ) ( x + 2 ) .
Simplify to a quadratic equation: x 2 − 4 x + 3 = 0 .
Solve the quadratic equation and check for extraneous solutions: x = 1 (extraneous), x = 3 . The solution is 3 .
Explanation
Problem Analysis We are given the equation x − 1 x − x + 2 2 = x 2 + x − 2 4 x − 1 We want to find all solutions for x .
Factoring and Restrictions First, notice that the denominator on the right side can be factored as x 2 + x − 2 = ( x − 1 ) ( x + 2 ) Thus, the equation can be rewritten as x − 1 x − x + 2 2 = ( x − 1 ) ( x + 2 ) 4 x − 1 We must have x = 1 and x = − 2 to avoid division by zero.
Eliminating Denominators and Simplifying Multiply both sides of the equation by ( x − 1 ) ( x + 2 ) to eliminate the denominators: x ( x + 2 ) − 2 ( x − 1 ) = 4 x − 1 Expand and simplify the equation: x 2 + 2 x − 2 x + 2 = 4 x − 1 Combine like terms: x 2 + 2 = 4 x − 1 Rearrange the equation into a quadratic equation: x 2 − 4 x + 3 = 0
Solving the Quadratic Equation Factor the quadratic equation: ( x − 1 ) ( x − 3 ) = 0 Solve for x :
x = 1 or x = 3
Checking for Extraneous Solutions Check for extraneous solutions. Since x = 1 and x = − 2 , x = 1 is an extraneous solution because it makes the denominators in the original equation equal to zero. Therefore, the only solution is x = 3 .
Final Answer Thus, the only solution to the equation is x = 3 .
Examples
When designing a bridge, engineers often use rational equations to model the distribution of forces and stresses. By solving these equations, they can determine the optimal placement of supports and the necessary strength of materials to ensure the bridge's stability. Similarly, in electrical engineering, rational equations are used to analyze circuits and determine the values of components needed to achieve desired performance characteristics. These applications highlight the importance of understanding and solving rational equations in real-world engineering scenarios.
