Solve the fractional linear equation (5x/2) − [7(6x − 3/2)]/4 = 5/8 and determine the exact value of the variable x.

Introduction / Context: This equation involves fractional coefficients and a bracketed term, which is common in aptitude questions on algebraic simplification. The focus is on careful expansion, collection of like terms and solving a resulting linear equation in x.

Given Data / Assumptions:

• Equation: (5x/2) − [7(6x − 3/2)]/4 = 5/8
• x is a real number
• Goal is to find the exact value of x

Concept / Approach: We first simplify the numerator inside the bracket, then divide by 4, and then combine the term with 5x/2. The easiest method is to bring all terms to one side, express them with a common denominator and solve. Alternately, we can clear denominators at an early stage by multiplying through by the least common multiple.

Step-by-Step Solution: Start with (5x/2) − [7(6x − 3/2)]/4 = 5/8 Simplify the bracket: 6x − 3/2 Multiply by 7: 7(6x − 3/2) = 42x − 21/2 Divide by 4: [7(6x − 3/2)]/4 = (42x − 21/2)/4 This equals (42x)/4 − (21/2)/4 = (21x/2) − 21/8 So the equation becomes (5x/2) − (21x/2) + 21/8 = 5/8 Combine x terms: (5x/2 − 21x/2) = −16x/2 = −8x So −8x + 21/8 = 5/8 Subtract 21/8 from both sides: −8x = 5/8 − 21/8 = −16/8 = −2 Thus x = (−2)/ (−8) = 1/4

Verification / Alternative check: Substitute x = 1/4 into the original equation. Evaluate 5x/2 = 5*(1/4)/2 = 5/8. Inside the bracket 6x − 3/2 = 6*(1/4) − 3/2 = 3/2 − 3/2 = 0, so the bracketed term contributes zero. The left side is therefore 5/8, matching the right side 5/8 exactly.

Why Other Options Are Wrong: Using −1/4 changes the signs and gives a different left side. Values 4 and −4 give very large magnitudes which clearly do not equal 5/8. Zero would also cause the bracket term to be non zero and the equality fails. Only x = 1/4 satisfies the equation perfectly.

Common Pitfalls: Common mistakes include incorrect distribution of 7 across the bracket, mishandling halves and eighths, and forgetting that subtracting a negative term changes the sign. Clearing denominators too early without tracking each step properly can also lead to sign errors.

Final Answer: Therefore, the exact solution is x = 1/4.