\frac{5}{2 x+6}-2=\frac{1-8 x}{4 x}
Answer (2)
Multiply both sides by the LCD: 4 x ( 2 x + 6 ) .
Simplify and expand the equation.
Solve for x .
Check for extraneous solutions: x = 3 1 is a valid solution.
The solution is 3 1 .
Explanation
Analyze the problem We are given the equation 2 x + 6 5 − 2 = 4 x 1 − 8 x Our goal is to solve for x .
Eliminate fractions First, let's multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions. The LCD is 4 x ( 2 x + 6 ) = 8 x ( x + 3 ) .
4 x ( 2 x + 6 ) ( 2 x + 6 5 − 2 ) = 4 x ( 2 x + 6 ) ( 4 x 1 − 8 x ) 4 x ( 5 ) − 2 ( 4 x ) ( 2 x + 6 ) = ( 2 x + 6 ) ( 1 − 8 x ) 20 x − 8 x ( 2 x + 6 ) = 2 x − 16 x 2 + 6 − 48 x
Simplify the equation Now, let's expand and simplify the equation: 20 x − 16 x 2 − 48 x = 2 x − 16 x 2 + 6 − 48 x − 16 x 2 − 28 x = − 16 x 2 − 46 x + 6 Add 16 x 2 to both sides: − 28 x = − 46 x + 6 Add 46 x to both sides: 18 x = 6
Solve for x Now, divide both sides by 18 to solve for x :
x = 18 6 = 3 1 So, x = 3 1 .
Check for extraneous solutions We need to check for extraneous solutions by substituting x = 3 1 back into the original equation: 2 ( 3 1 ) + 6 5 − 2 = 4 ( 3 1 ) 1 − 8 ( 3 1 ) 3 2 + 6 5 − 2 = 3 4 1 − 3 8 3 20 5 − 2 = 3 4 − 3 5 20 15 − 2 = − 4 5 4 3 − 2 = − 4 5 4 3 − 4 8 = − 4 5 − 4 5 = − 4 5 The solution is valid.
Final Answer Therefore, the solution to the equation is x = 3 1 .
Examples
When solving problems involving rates, such as determining the time it takes for two workers to complete a task together, you often encounter equations with rational expressions. The ability to solve such equations is crucial in optimizing work schedules and resource allocation. For instance, if one worker can complete a job in 2 x + 6 hours and another can do it in 4 x hours, setting up and solving a rational equation helps determine how long they take working together. This skill is also applicable in mixture problems, where you need to find the right proportions of different solutions to achieve a desired concentration, ensuring accuracy in fields like chemistry and pharmaceuticals.
To solve the equation 2 x + 6 5 − 2 = 4 x 1 − 8 x , multiply both sides by the least common denominator to eliminate fractions. This leads to the solution x = 3 1 . The solution is verified by substituting it back into the original equation, confirming it is valid.
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