$\frac{x}{4}-\frac{5}{2 x}=\frac{1}{4 x}$
Answer (1)
Multiply both sides of the equation by 4 x to eliminate the fractions: x 2 − 10 = 1 .
Add 10 to both sides: x 2 = 11 .
Take the square root of both sides: x = ± 11 .
The solutions are x = 11 and x = − 11 , so the final answer is ± 11 .
Explanation
Problem Analysis We are given the equation 4 x − 2 x 5 = 4 x 1 and we want to solve for x .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by 4 x :
4 x ⋅ ( 4 x − 2 x 5 ) = 4 x ⋅ 4 x 1 4 x ⋅ 4 x − 4 x ⋅ 2 x 5 = 1 x 2 − 10 = 1
Isolating x 2 Now, we add 10 to both sides of the equation: x 2 − 10 + 10 = 1 + 10 x 2 = 11
Solving for x Taking the square root of both sides, we get: x = ± 11
Checking Solutions We need to check if these solutions are valid. Since x appears in the denominator in the original equation, x cannot be 0. Since 11 = 0 and − 11 = 0 , both solutions are valid. Thus, the solutions are x = 11 and x = − 11 .
Final Answer The solutions to the equation are x = 11 and x = − 11 .
Examples
Imagine you are designing a rectangular garden where the length is related to the width by the equation given. Solving this equation helps you determine the possible dimensions of the garden, ensuring it meets specific area requirements. Understanding how to manipulate and solve such equations is crucial in various fields like engineering, physics, and economics, where relationships between variables are often expressed in similar mathematical forms.
