\frac{2 x-1}{6}=\frac{4 x+1}{12}+ \frac{2 x+2}{2 x+1}
Answer (2)
Multiply both sides of the equation by 12 ( 2 x + 1 ) to eliminate the fractions.
Expand and simplify the equation.
Solve for x .
Check for extraneous solutions. The solution is x = − 10 9 .
Explanation
Problem Analysis We are given the equation 6 2 x − 1 = 12 4 x + 1 + 2 x + 1 2 x + 2 Our goal is to solve for x .
Eliminating Fractions First, let's multiply both sides of the equation by 12 ( 2 x + 1 ) to eliminate the fractions: 12 ( 2 x + 1 ) ⋅ 6 2 x − 1 = 12 ( 2 x + 1 ) ⋅ 12 4 x + 1 + 12 ( 2 x + 1 ) ⋅ 2 x + 1 2 x + 2 This simplifies to: 2 ( 2 x + 1 ) ( 2 x − 1 ) = ( 2 x + 1 ) ( 4 x + 1 ) + 12 ( 2 x + 2 )
Expanding the Terms Now, expand the terms: 2 ( 4 x 2 − 1 ) = ( 8 x 2 + 2 x + 4 x + 1 ) + ( 24 x + 24 ) 8 x 2 − 2 = 8 x 2 + 6 x + 1 + 24 x + 24
Combining Like Terms Combine like terms: 8 x 2 − 2 = 8 x 2 + 30 x + 25 Subtract 8 x 2 from both sides: − 2 = 30 x + 25
Isolating x Subtract 25 from both sides: − 27 = 30 x
Solving for x Divide by 30: x = 30 − 27 Simplify the fraction: x = 10 − 9
Checking for Extraneous Solutions Now, we need to check for extraneous solutions. Substitute x = − 10 9 back into the original equation: 6 2 ( − 10 9 ) − 1 = 12 4 ( − 10 9 ) + 1 + 2 ( − 10 9 ) + 1 2 ( − 10 9 ) + 2 6 − 10 18 − 1 = 12 − 10 36 + 1 + − 10 18 + 1 − 10 18 + 2 6 − 10 28 = 12 − 10 26 + − 10 8 10 2 60 − 28 = 120 − 26 + − 8 2 30 − 14 = 60 − 13 − 4 1 60 − 28 = 60 − 13 − 60 15 60 − 28 = 60 − 28 The solution is valid.
Final Answer Therefore, the solution to the equation is x = − 10 9 .
Examples
When designing a bridge, engineers use equations like this to calculate the forces and stresses acting on different parts of the structure. By solving for variables, they can determine the optimal dimensions and materials needed to ensure the bridge's stability and safety. Similarly, in electrical engineering, such equations help in designing circuits by determining the values of components needed to achieve desired performance characteristics.
To solve the equation 6 2 x − 1 = 12 4 x + 1 + 2 x + 1 2 x + 2 , we eliminate fractions by multiplying by 12 ( 2 x + 1 ) , expand, and isolate x . The solution is x = − 10 9 , which can be verified by substitution. Hence, the correct answer is x = − 10 9 .
;
