In basic algebraic simplification, solve the linear equation with brackets and find the exact value of x: (4x - 3) - (2x + 1) = 4.

Introduction / Context:
This is a straightforward linear equation in one variable that involves subtracting a bracketed expression. Such questions appear frequently in aptitude and school exams and assess your ability to correctly expand brackets, combine like terms, and isolate the variable. Mastering these basics is essential before moving on to more advanced algebraic techniques.

Given Data / Assumptions:

• The equation is (4x - 3) - (2x + 1) = 4.
• x is a real number.
• We must find the unique value of x that satisfies the equation.
• Standard algebraic operations apply (addition, subtraction, etc.).

Concept / Approach:
The key idea is to remove the parentheses carefully. When subtracting a bracket, you must change the sign of each term inside the bracket. After expansion, collect all x terms together and all constant terms together. Then solve the resulting simple linear equation by isolating x. This method is efficient and avoids errors that come from skipping steps or mis-handling signs.

Step-by-Step Solution:
1) Start with the equation: (4x - 3) - (2x + 1) = 4. 2) Expand the first bracket: (4x - 3) stays as 4x - 3. 3) Subtract the second bracket: −(2x + 1) becomes −2x − 1. 4) Combine like terms on the left side: 4x − 3 − 2x − 1 = (4x − 2x) + (−3 − 1) = 2x − 4. 5) The equation is now 2x − 4 = 4. 6) Add 4 to both sides to isolate the x term: 2x = 8. 7) Divide both sides by 2: x = 8 / 2 = 4.

Verification / Alternative check:
Substitute x = 4 back into the original equation to ensure the solution is correct. Compute 4x − 3 = 4 * 4 − 3 = 16 − 3 = 13. Then compute 2x + 1 = 2 * 4 + 1 = 8 + 1 = 9. The left-hand side becomes 13 − 9 = 4, which matches the right-hand side exactly. This confirms that x = 4 is the correct and unique solution of the given linear equation.

Why Other Options Are Wrong:
If x = 2, then (4x − 3) − (2x + 1) becomes (8 − 3) − (4 + 1) = 5 − 5 = 0, not 4. If x = 1, we get (4 − 3) − (2 + 1) = 1 − 3 = −2. For x = 0, we get (0 − 3) − (0 + 1) = −3 − 1 = −4. For x = 3, we get (12 − 3) − (6 + 1) = 9 − 7 = 2. None of these equal 4, so those options are incorrect.

Common Pitfalls:
A common error is forgetting to distribute the negative sign when subtracting the second bracket, which leads to incorrect expressions like 4x − 3 − 2x + 1. Another mistake is performing incorrect arithmetic when combining the constant terms. Writing each step clearly and carefully handling signs will help you avoid these mistakes in similar problems.

Final Answer:
The exact value of x that satisfies the equation is 4.