Solve the following equation. [tex]$\frac{3}{x-9}-\frac{4 x}{x^2-81}=\frac{2}{x+9}$[/tex]
Answer (2)
Factor the denominator and rewrite the equation: x − 9 3 − ( x − 9 ) ( x + 9 ) 4 x = x + 9 2 .
Multiply both sides by ( x − 9 ) ( x + 9 ) to eliminate denominators: 3 ( x + 9 ) − 4 x = 2 ( x − 9 ) .
Simplify and solve for x : 3 x + 27 − 4 x = 2 x − 18 ⇒ − x + 27 = 2 x − 18 ⇒ 3 x = 45 ⇒ x = 15 .
Check the solution against restrictions: x = 15 is a valid solution.
The final answer is 15 .
Explanation
Rewrite the equation We are given the equation x − 9 3 − x 2 − 81 4 x = x + 9 2 . Our goal is to solve for x . First, we notice that x 2 − 81 can be factored as ( x − 9 ) ( x + 9 ) . This means we can rewrite the equation with a common denominator.
Eliminate the denominators We can rewrite the equation as x − 9 3 − ( x − 9 ) ( x + 9 ) 4 x = x + 9 2 . To eliminate the denominators, we multiply both sides of the equation by ( x − 9 ) ( x + 9 ) . This gives us: 3 ( x + 9 ) − 4 x = 2 ( x − 9 )
Simplify the equation Now, we simplify the equation by distributing and combining like terms: 3 x + 27 − 4 x = 2 x − 18 − x + 27 = 2 x − 18
Isolate x Next, we isolate x by adding x to both sides and adding 18 to both sides: 27 + 18 = 2 x + x 45 = 3 x
Solve for x Finally, we solve for x by dividing both sides by 3 : x = 3 45 = 15
Check the solution We need to check if our solution x = 15 satisfies the restrictions x = 9 and x = − 9 . Since 15 is not equal to 9 or − 9 , it is a valid solution.
Examples
Solving rational equations like this one is useful in many real-world scenarios, such as calculating the flow rate of fluids in pipes or determining the optimal amount of resources to allocate in a project. For example, if you are designing a water distribution system, you might use rational equations to model the flow of water through different pipes and ensure that the system is balanced and efficient. Understanding how to solve these equations allows engineers to optimize designs and avoid potential problems.
The solution to the equation x − 9 3 − x 2 − 81 4 x = x + 9 2 is x = 15 . This was found by eliminating the denominators and solving for x . The solution is valid as it does not violate any restrictions from the equation.
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