Solve the linear equation with fractions: (1/3) * (12x / 5 − 1/2) + 6/5 = 7/6 for the real variable x, and find the exact value of x.

Introduction / Context:
This problem involves solving a linear equation that contains several fractional terms. Such equations are standard in aptitude tests because they assess your ability to handle fractions, distribute constants correctly, and simplify expressions systematically. Once these steps are done carefully, the equation reduces to a simple form from which the value of x can be obtained exactly.

Given Data / Assumptions:

• The equation to solve is (1/3) * (12x / 5 − 1/2) + 6/5 = 7/6.
• x is a real variable.
• All denominators are non zero, so standard arithmetic operations on fractions are valid.
• We seek the exact fractional value of x, not a decimal approximation.

Concept / Approach:
The main strategy is to simplify step by step. First distribute the factor 1/3 inside the brackets. Then combine all constant and variable terms on one side. To avoid mistakes, it is convenient to work with a common denominator when adding or subtracting fractions. Finally, once we have a simple linear expression in x equal to another rational number, we can isolate x by basic algebra. Keeping everything in fractional form prevents rounding errors and keeps the solution exact.

Step-by-Step Solution:
1) Start with (1/3) * (12x / 5 − 1/2) + 6/5 = 7/6. 2) Distribute 1/3 inside the bracket: (1/3) * (12x / 5) = 12x / 15 = 4x / 5, and (1/3) * (−1/2) = −1/6. 3) So the left side becomes 4x / 5 − 1/6 + 6/5. 4) Combine the constant terms −1/6 and 6/5 using a common denominator of 30: −1/6 = −5/30 and 6/5 = 36/30. 5) Their sum is (−5 + 36) / 30 = 31 / 30. 6) Thus the equation becomes 4x / 5 + 31 / 30 = 7 / 6. 7) Rewrite 4x / 5 with denominator 30: 4x / 5 = (24x) / 30. 8) Now we have (24x + 31) / 30 = 7 / 6. 9) Cross multiply: 6 * (24x + 31) = 7 * 30. 10) This gives 144x + 186 = 210. 11) Subtract 186 from both sides: 144x = 24. 12) Divide both sides by 144: x = 24 / 144 = 1 / 6.

Verification / Alternative check:
Substitute x = 1/6 back into the original equation. Compute 12x / 5 = 12 * (1/6) / 5 = 2 / 5. Then 2 / 5 − 1/2 has a common denominator 10 and equals (4 − 5) / 10 = −1 / 10. Multiply by 1/3 to get −1 / 30. Next, add 6/5, which is 36/30, giving (−1 + 36) / 30 = 35/30 = 7/6. The right side is 7/6 as well, so the equation is satisfied, confirming that x = 1/6 is correct.

Why Other Options Are Wrong:
Options b (1/5) and e (5/6) place x at different fractional values, which, when substituted, do not produce 7/6 on the right side. Options c (−1/6) and d (−1/5) change the sign of x and again fail to satisfy the equation. Only option a produces an exact match when substituted back into the original equation.

Common Pitfalls:
The most common errors involve sign mistakes when distributing 1/3, or mishandling the combination of fractions −1/6 and 6/5. Some learners may attempt to convert everything to decimals early, which can introduce rounding errors. Others might forget to multiply both sides by the common denominator when clearing fractions. Working carefully with exact fractions and checking the solution by substitution is the best way to avoid these issues.

Final Answer:
Solving the fractional linear equation gives the exact value x = 1/6.