If 8x − 4 = 3x + 6 for a real number x, what is the numerical value of (x + 1)^3 after solving the linear equation?

Introduction / Context:
This question combines solving a simple linear equation with evaluating a power expression. First you must determine the value of x that satisfies the equation 8x − 4 = 3x + 6. Once x is obtained, you substitute it into the expression (x + 1)^3. This tests basic algebra skills and familiarity with cubes of small integers.

Given Data / Assumptions:

• The linear equation is 8x − 4 = 3x + 6.
• x is a real number.
• We then evaluate the expression (x + 1)^3.
• No approximations are needed; all values are exact integers.

Concept / Approach:
The goal is to isolate x in the linear equation using standard algebraic operations: collect like terms, move constants to one side, and divide by the coefficient of x. After finding x, we compute (x + 1)^3. Recognising perfect cubes such as 2^3 = 8 and 3^3 = 27 helps to answer quickly once x + 1 is known.

Step-by-Step Solution:
Start from the given equation: 8x − 4 = 3x + 6.Move all x terms to one side by subtracting 3x from both sides: 5x − 4 = 6.Move constants to the other side by adding 4 to both sides: 5x = 10.Divide by 5 to isolate x: x = 10 / 5 = 2.Now compute x + 1: x + 1 = 2 + 1 = 3.Finally evaluate (x + 1)^3: (3)^3 = 3 * 3 * 3 = 27.

Verification / Alternative check:
Substitute x = 2 back into the original equation to verify. The left-hand side is 8x − 4 = 8 * 2 − 4 = 16 − 4 = 12. The right-hand side is 3x + 6 = 3 * 2 + 6 = 6 + 6 = 12. Since both sides equal 12, x = 2 is correct. With x confirmed, (x + 1)^3 = 3^3 = 27 is undoubtedly the correct value.

Why Other Options Are Wrong:

• 3 and 9 correspond to (x + 1) or (x + 1)^2, not to (x + 1)^3 for x = 2.
• 81 would be the cube of 4, which would occur only if x + 1 = 4, that is, x = 3, which does not satisfy the equation.
• 64 is 4^3 or 2^6 and does not result from the correct value of x in this problem.

Common Pitfalls:

• Making sign mistakes when moving terms across the equals sign.
• Forgetting to add 1 to x before cubing, and instead cubing x itself.
• Miscomputing 3^3 and writing 9 instead of 27.

Final Answer:
27