Rationale
x ≤ -40
To solve the inequality (1/8)x ≤ (1/2)x + 15, we first isolate x by simplifying the expression and performing algebraic operations. This leads us to the conclusion that the solution for x is less than or equal to -40.
A) The choice x ≤ -24 is incorrect because, when we solve the inequality, we find that the critical value of x occurs at -40, not -24. This value results from properly isolating x and does not satisfy the original inequality when substituted back in.
B) Similar to choice A, x ≤ -40 is incorrectly listed here. The inequality does not yield -40 as a solution due to a miscalculation or misunderstanding of the simplification process. The correct inequality indicates that x must be less than or equal to -40, which is the intended interpretation of the solution.
C) This choice correctly states that x ≤ -40, as derived from solving the inequality (1/8)x ≤ (1/2)x + 15. By isolating x and simplifying step-by-step, we confirm that -40 is the boundary point where the inequality holds true.
D) The choice x ≤ -24 is also incorrect for the same reasons as option A. The solution derived from the inequality indicates a different threshold for x, which is clearly above -40, making -24 an invalid conclusion in the context of the given problem.
Conclusion
In solving the inequality (1/8)x ≤ (1/2)x + 15, we found that x must be less than or equal to -40. This highlights an essential part of algebraic manipulation and inequality solving, where identifying boundaries accurately is crucial. Options A, B, and D fail to represent the correct solution, while C appropriately captures the necessary condition for x.