Solve the linear equation (−3/2) + (2/3)(3x + 9) = x/2 and find the exact value of x.

Introduction / Context:
This is a standard linear equation with fractional coefficients. It tests your ability to expand brackets, distribute fractions correctly, and then collect like terms to isolate the variable x. Such questions are very common in aptitude tests under the simplification or algebra section.

Given Data / Assumptions:

• Equation: (−3/2) + (2/3)(3x + 9) = x/2.
• x is a real number.
• All algebraic operations follow the usual rules of arithmetic.
• No denominator is zero in the process.

Concept / Approach:
The equation can be simplified step by step. First expand the product (2/3)(3x + 9) using distribution. This will give a linear expression in x plus a constant term. Then move all x terms to one side and constants to the other. Because the equation is linear, you will end up with a single solution for x. It is often easier to clear fractions early, but even without that, systematic manipulation works well.

Step-by-Step Solution:
Start from (−3/2) + (2/3)(3x + 9) = x/2. Expand (2/3)(3x + 9): it equals (2/3)·3x + (2/3)·9 = 2x + 6. So the equation becomes (−3/2) + 2x + 6 = x/2. Combine the constant terms on the left: 6 − 3/2 = (12/2 − 3/2) = 9/2. Thus 2x + 9/2 = x/2. Multiply every term by 2 to clear the denominator: 4x + 9 = x. Subtract x from both sides: 3x + 9 = 0. So 3x = −9 and x = −9/3 = −3.

Verification / Alternative check:
Substitute x = −3 back into the original equation. Then 3x + 9 = 3(−3) + 9 = −9 + 9 = 0, so (2/3)(3x + 9) becomes (2/3)·0 = 0. The left side is then (−3/2) + 0 = −3/2. The right side is x/2 = (−3)/2 = −3/2. Both sides match, confirming that x = −3 satisfies the equation exactly.

Why Other Options Are Wrong:
If x = −9, then x/2 = −9/2, but the left side becomes much more negative, so it does not balance. Positive values like 11 or 9 make the right side positive, while the left side evaluates to a different number completely. Zero would give −3/2 on the left and 0 on the right. Only x = −3 satisfies the equality.

Common Pitfalls:
A typical error is distributing (2/3) incorrectly or mishandling the sign of the −3/2 term. Some learners forget to multiply all terms when clearing denominators, causing inconsistencies. Others accidentally move terms across the equality sign without changing their signs. Working slowly and checking each arithmetic step helps avoid such mistakes.

Final Answer:
Therefore, the solution of the equation is x = −3.