$\frac{x}{4}-\frac{5}{2 x}=\frac{1}{4 x}$
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 .
Explanation
Analyze the problem We are given the equation 4 x − 2 x 5 = 4 x 1 Our goal is to solve for x . First, we note that x cannot be zero because it appears in the denominator of the fractions.
Eliminate 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 = 4 x ⋅ 4 x 1 x 2 − 10 = 1
Isolate x squared Next, we add 10 to both sides of the equation: x 2 − 10 + 10 = 1 + 10 x 2 = 11
Solve for x Now, we take the square root of both sides: x 2 = 11 x = ± 11
Final Answer Since x cannot be zero, both solutions x = 11 and x = − 11 are valid. Therefore, the solutions 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 4 w − 2 w 5 = 4 w 1 , where w is the width. Solving this equation helps you find the possible values for the width of the garden that satisfy the given condition. This type of problem arises in various engineering and design contexts where certain relationships between dimensions must be satisfied.
