Solve the linear equation with fractional coefficients 5/2 - (6/5)(x - 15/2) = -x/5 by clearing denominators carefully and find the exact value of x from the given options.

Introduction / Context:
This simplification question focuses on solving a linear equation that contains fractions and a bracketed term. Such equations are very common in aptitude tests and require careful handling of brackets and denominators. The goal is to isolate x by performing valid algebraic operations and then matching the result with one of the given answer choices.

Given Data / Assumptions:

• The equation is 5/2 - (6/5)(x - 15/2) = -x/5.
• x is a real number.
• All fractions are to be treated exactly, without decimal approximation.
• We can clear denominators by multiplying through by the least common multiple of the denominators.

Concept / Approach:
The main idea is to first expand the bracket (x - 15/2) by multiplying it with 6/5, then simplify the left hand side. After that, it is convenient to clear all fractional denominators by multiplying the entire equation by a common multiple, such as 10. Finally, we collect like terms and solve the resulting simple linear equation for x. This step by step approach greatly reduces the chance of sign or arithmetic errors.

Step-by-Step Solution:
Start with 5/2 - (6/5)(x - 15/2) = -x/5. Expand (6/5)(x - 15/2) = (6/5)x - (6/5)*(15/2) = (6/5)x - 9. Substitute back: 5/2 - [(6/5)x - 9] = -x/5. Distribute the minus sign: 5/2 - (6/5)x + 9 = -x/5. Combine constants on the left: 5/2 + 9 = 5/2 + 18/2 = 23/2, so 23/2 - (6/5)x = -x/5. Multiply the entire equation by 10 to clear denominators: 10*(23/2) - 10*(6/5)x = 10*(-x/5). Compute: 10*(23/2) = 115, and 10*(6/5)x = 12x, and 10*(-x/5) = -2x, giving 115 - 12x = -2x. Bring x terms together: 115 - 12x + 2x = 0, so 115 - 10x = 0, hence -10x = -115 and x = 23/2.
Verification / Alternative check:
You can substitute x = 23/2 back into the original equation to verify. Compute x - 15/2 = 23/2 - 15/2 = 8/2 = 4, so (6/5)(x - 15/2) = (6/5)*4 = 24/5. Then 5/2 - 24/5 = (25/10 - 48/10) = -23/10. On the right side, -x/5 = -(23/2)/5 = -23/10. Both sides match, confirming the solution is correct.

Why Other Options Are Wrong:
Option b ( -23/2 ) results from losing a negative sign while rearranging. Options c and d (13/2 and -13/2) often appear if someone miscomputes constants when combining 5/2 and 9 or miscalculates 10*(23/2). Option e ( -11 ) suggests that denominators were not cleared properly or that the bracket was expanded incorrectly.

Common Pitfalls:
Common mistakes include forgetting to distribute the minus sign across both terms inside the bracket, incorrectly multiplying fractions like (6/5)*(15/2), and making sign errors when moving terms across the equals sign. Always expand carefully, convert all terms to a common denominator, and then simplify step by step to avoid these issues.

Final Answer:
The correct solution of the equation 5/2 - (6/5)(x - 15/2) = -x/5 is x = 23/2.