Solve the linear equation 7x − (3/2) * (4x − 9) = 6.5 for the real variable x, and give the exact value of x.

Introduction / Context:
This question is a straightforward linear equation involving fractions and a decimal on the right side. It tests your ability to distribute multiplication across brackets, combine like terms, and handle different numerical formats such as fractions and decimals within a single equation. These operations are fundamental skills in algebra and aptitude tests.

Given Data / Assumptions:

• The equation to solve is 7x − (3/2) * (4x − 9) = 6.5.
• x is a real variable.
• The decimal 6.5 can be treated as the fraction 13/2 for exact work.
• Standard properties of equality and linear equations apply.

Concept / Approach:
The plan is to first expand the term (3/2) * (4x − 9) using distribution, taking care with signs. After expansion, we combine the x terms and constant terms on the left side. Converting 6.5 to the fraction 13/2 makes the equation easier to manipulate exactly. Finally, we isolate x by simple algebraic steps. Throughout, we avoid unnecessary decimal approximations to keep the result exact.

Step-by-Step Solution:
1) Rewrite the equation with a fraction on the right side: 7x − (3/2) * (4x − 9) = 13/2. 2) Expand the bracket: (3/2) * (4x − 9) = (3/2) * 4x − (3/2) * 9. 3) Compute each term: (3/2) * 4x = 6x and (3/2) * 9 = 27/2. 4) So the equation becomes 7x − 6x + 27/2 = 13/2. 5) Combine the x terms on the left: 7x − 6x = x. 6) This simplifies the equation to x + 27/2 = 13/2. 7) Subtract 27/2 from both sides: x = 13/2 − 27/2. 8) Compute the difference: 13/2 − 27/2 = (13 − 27) / 2 = −14 / 2 = −7.

Verification / Alternative check:
Substitute x = −7 back into the original equation. Compute 7x = 7 * (−7) = −49. Next compute 4x − 9 = 4 * (−7) − 9 = −28 − 9 = −37. Then (3/2) * (4x − 9) = (3/2) * (−37) = −111/2. The left side becomes 7x − (3/2) * (4x − 9) = −49 − (−111/2) = −49 + 111/2. Rewrite −49 as −98/2, so the sum is (−98/2 + 111/2) = 13/2 = 6.5, which matches the right side. This confirms that x = −7 is the correct solution.

Why Other Options Are Wrong:
Option b (7) would make 7x positive and change the sign of the entire left side, leading to a different result. Options c (20), d (−20), and e (0) similarly fail to satisfy the equation when substituted. Only option a, x = −7, reproduces the right side value 6.5 exactly and therefore solves the equation.

Common Pitfalls:
Common mistakes include incorrect distribution of (3/2) across the bracket, especially forgetting to multiply by both 4x and −9, or mis handling the negative sign in front of the bracket. Some learners also mix up the decimal 6.5 with 6 1/2 and do not convert it correctly to 13/2. Working consistently with fractions instead of switching between decimals and fractions helps maintain accuracy.

Final Answer:
Solving the linear equation gives the unique real solution x = -7.