Solve the linear equation with a negative fractional coefficient, (-1/2)(x - 5) + 3 = -5/2, by simplifying step by step and find the exact value of x.

Introduction / Context:
This question involves solving a simple linear equation that contains a bracket and a negative fractional coefficient. Such problems are routine in aptitude tests, and they are designed to check your ability to expand brackets correctly, handle negative signs, and work with fractions without making algebraic mistakes. The final goal is to isolate x and identify its value from the options.

Given Data / Assumptions:

• The equation is (-1/2)(x - 5) + 3 = -5/2.
• x is a real number.
• All fractions should be treated exactly.
• We can clear fractions by multiplying through by a suitable common multiple if convenient.

Concept / Approach:
The key idea is to first expand the bracket (-1/2)(x - 5) and then combine like terms. Because the coefficient is negative, special care is needed with signs. Once the expression on the left hand side is simplified, we can move constants to one side and x terms to the other, or we can clear the denominator by multiplying the entire equation by 2. Either path yields the same solution for x when done carefully.

Step-by-Step Solution:
Start with (-1/2)(x - 5) + 3 = -5/2. Expand the bracket: (-1/2)(x - 5) = -(1/2)x + (5/2). Substitute this expansion: -(1/2)x + 5/2 + 3 = -5/2. Convert 3 to a fraction with denominator 2: 3 = 6/2, so 5/2 + 3 = 5/2 + 6/2 = 11/2. The equation becomes -(1/2)x + 11/2 = -5/2. Multiply the entire equation by 2 to clear denominators: -x + 11 = -5. Rearrange: -x = -5 - 11 = -16, therefore x = 16.
Verification / Alternative check:
Substitute x = 16 back into the original equation. Evaluate x - 5 = 11, so (-1/2)(x - 5) = (-1/2)*11 = -11/2. Then (-11/2) + 3 = -11/2 + 6/2 = -5/2, which matches the right hand side. This confirms that x = 16 satisfies the equation exactly.

Why Other Options Are Wrong:
Option b (4), option c (-4), and option d (-6) do not satisfy the equation when substituted back and result in unequal left and right sides. Option e (-16) comes from a common sign error when moving terms across the equals sign or misapplying the multiplication by 2 step, where -x + 11 = -5 is incorrectly converted to x + 11 = -5.

Common Pitfalls:
Typical mistakes include dropping the negative sign in front of 1/2, incorrectly expanding (-1/2)(x - 5) as -x/2 - 5/2, or failing to convert the integer 3 into a fraction with matching denominator when combining terms. Another error is to multiply only some terms by 2 when clearing denominators instead of multiplying both sides of the equation completely. Writing out each step carefully helps avoid these issues.

Final Answer:
The exact solution of the linear equation (-1/2)(x - 5) + 3 = -5/2 is x = 16.