$\frac{x+4}{x-2}-\frac{x-2}{x}+4=0$
Answer (2)
To solve the equation x − 2 x + 4 − x x − 2 + 4 = 0 , we cleared the fractions and simplified to find the solutions are x = 1 and x = − 1 . Both solutions are valid since they do not violate any restrictions from the original equation's denominators. Hence, the final answer consists of both solutions.
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Multiply both sides by x ( x − 2 ) to eliminate fractions.
Expand and simplify the equation to get 4 x 2 − 4 = 0 .
Solve for x , obtaining x = ± 1 .
Check for restrictions; since x = 0 and x = 2 , both solutions are valid: − 1 , 1 .
Explanation
Understanding the Problem We are given the equation x − 2 x + 4 − x x − 2 + 4 = 0 . Our goal is to find the value(s) of x that satisfy this equation. We must also consider the restrictions on x : x = 2 and x = 0 , since these values would make the denominators of the fractions equal to zero, which is undefined.
Eliminating Fractions To solve the equation, we first eliminate the fractions by multiplying both sides of the equation by x ( x − 2 ) . This gives us:
x ( x − 2 ) ( x − 2 x + 4 − x x − 2 + 4 ) = 0 × x ( x − 2 )
x ( x + 4 ) − ( x − 2 ) ( x − 2 ) + 4 x ( x − 2 ) = 0
Expanding and Simplifying Next, we expand and simplify the equation:
x 2 + 4 x − ( x 2 − 4 x + 4 ) + 4 x 2 − 8 x = 0
x 2 + 4 x − x 2 + 4 x − 4 + 4 x 2 − 8 x = 0
4 x 2 + 0 x − 4 = 0
4 x 2 − 4 = 0
Solving for x Now, we solve the simplified equation for x :
4 x 2 − 4 = 0
4 x 2 = 4
x 2 = 1
x = ± 1
x = ± 1
Checking Restrictions We have two potential solutions: x = 1 and x = − 1 . We need to check if these solutions satisfy the initial restrictions x = 2 and x = 0 . Since neither of these solutions violates the restrictions, both are valid.
Final Answer Therefore, the solutions to the equation are x = 1 and x = − 1 .
Examples
Understanding how to solve rational equations is crucial in many real-world applications, such as calculating the flow rate in pipes or determining the optimal concentration of a mixture in chemical processes. For instance, if you're designing a water distribution system, you might use rational equations to model the relationship between pipe diameter, water pressure, and flow rate to ensure efficient water delivery. Similarly, in chemical engineering, these equations can help you optimize reaction rates and yields by carefully controlling the concentrations of reactants.
