If 3x - 10 = -(2x + 5), then what is the numerical value of (x - 4)^3?

Introduction / Context: This problem combines solving a simple linear equation with evaluating a cubic expression based on the solution. It is a straightforward but classic aptitude style question, testing algebraic manipulation and substitution skills.

Given Data / Assumptions:

• The equation is 3x - 10 = -(2x + 5).
• We must first find the value of x.
• Then we must compute (x - 4)^3 using that value.

Concept / Approach: We start by solving the linear equation in one variable. Once x is known, substitution into the expression (x - 4)^3 is straightforward. The cube can be computed directly because the resulting base is a simple integer. Careful handling of the negative sign on the right side of the given equation is important to avoid mistakes.

Step-by-Step Solution: Step 1: Start with 3x - 10 = -(2x + 5). Step 2: Distribute the minus sign on the right: -(2x + 5) = -2x - 5. Step 3: The equation becomes 3x - 10 = -2x - 5. Step 4: Add 2x to both sides: 5x - 10 = -5. Step 5: Add 10 to both sides: 5x = 5. Step 6: Divide by 5: x = 1. Step 7: Now compute (x - 4)^3 = (1 - 4)^3 = (-3)^3. Step 8: Evaluate the cube: (-3)^3 = -27.

Verification / Alternative check: Substitute x = 1 back into the original equation to confirm the solution. The left side becomes 3(1) - 10 = 3 - 10 = -7. The right side becomes -(2(1) + 5) = -(2 + 5) = -7. Both sides are equal, confirming that x = 1 is correct and therefore (x - 4)^3 = -27 is also correct.

Why Other Options Are Wrong:

• Option A: 27 would be the cube of 3, which would require x - 4 = 3, not -3.
• Option B: 125 corresponds to a base of 5, which does not arise here.
• Option C: -125 would be (-5)^3, unrelated to the solved value of x.
• Option E: 0 would occur only if x - 4 = 0, that is x = 4, which does not satisfy the given equation.

Common Pitfalls: The most common mistake is mishandling the negative sign when expanding -(2x + 5), which can lead to incorrect coefficients and a wrong value for x. Another error is miscalculating the cube of a negative number and ignoring the negative sign. Working step by step and checking signs at each stage helps avoid these problems.

Final Answer: Thus, the required value of (x - 4)^3 is -27.