Solve the linear fractional equation [7(5x/3 − 3/2)]/2 + 3/2 = 1/4 and determine the exact value of the variable x.

Introduction / Context:
This is a linear equation with fractions inside brackets. Such questions examine handling of nested fractions, distribution, and simplification of rational expressions. The main skill is to clear denominators systematically without making arithmetic errors.

Given Data / Assumptions:

• Equation: [7(5x/3 − 3/2)]/2 + 3/2 = 1/4
• x is a real number
• We seek the exact simplified value of x

Concept / Approach:
We first simplify the inner bracket, then multiply by 7 and divide by 2. After that we combine constant terms and isolate x. A good method is to multiply through by the least common multiple of denominators so that the equation becomes an integer linear equation.

Step-by-Step Solution:
Start with [7(5x/3 − 3/2)]/2 + 3/2 = 1/4 Let us simplify the bracket: 5x/3 − 3/2 LCM of 3 and 2 is 6 5x/3 − 3/2 = (10x/6) − (9/6) = (10x − 9)/6 Multiply by 7: 7 * (10x − 9)/6 = (70x − 63)/6 Divide by 2: [(70x − 63)/6] / 2 = (70x − 63)/12 Equation becomes (70x − 63)/12 + 3/2 = 1/4 Rewrite 3/2 with denominator 12: 3/2 = 18/12 So (70x − 63)/12 + 18/12 = 1/4 Combine numerators: (70x − 63 + 18)/12 = (70x − 45)/12 So (70x − 45)/12 = 1/4 Cross multiply: 4(70x − 45) = 12 280x − 180 = 12 280x = 192 x = 192/280 = 24/35

Verification / Alternative check:
Substitute x = 24/35 back into the original expression. The inner term 5x/3 becomes 5*(24/35)/3 and evaluates correctly, and the entire left side simplifies to 1/4. A quick check with approximate decimals also confirms equality of both sides, validating the solution.

Why Other Options Are Wrong:
35/24 and −35/24 are reciprocals or sign changes that do not satisfy the equation when substituted. −24/35 changes the sign of x and reverses the direction of the linear term, giving a different left side. Zero clearly fails because the constant terms alone do not satisfy the equality.

Common Pitfalls:
Sometimes learners forget to divide by 2 after multiplying by 7, or they mishandle the LCM of 3 and 2. Losing track of signs when combining −63 and +18 is also common. Always work carefully with numerators and denominators and only cross multiply once the fractions are correctly arranged.

Final Answer:
Hence, the solution of the equation is x = 24/35.