Question: Which of the following best describes the solution to the equation below?
-6(6x + 11) = -3(12x + 22)
A. no real solutions
B. exactly one real solution, x = 2/3
C. exactly one real solution, x = -2/3
D. infinite real solutions
1. **State the problem:** Solve the equation $$-6(6x + 11) = -3(12x + 22)$$ and determine which description fits the solution.
2. **Expand both sides:**
$$-6 \times 6x = -36x$$
$$-6 \times 11 = -66$$
$$-3 \times 12x = -36x$$
$$-3 \times 22 = -66$$
So the equation becomes:
$$-36x - 66 = -36x - 66$$
3. **Simplify the equation:**
Subtract $$-36x$$ from both sides:
$$-36x - 66 - (-36x) = -36x - 66 - (-36x)$$
$$\cancel{-36x} - 66 - \cancel{-36x} = \cancel{-36x} - 66 - \cancel{-36x}$$
$$-66 = -66$$
4. **Interpret the result:**
Since the equation simplifies to $$-66 = -66$$, which is always true, the equation holds for all values of $$x$$.
5. **Conclusion:**
The equation has infinitely many real solutions.
**Answer:** D. infinite real solutions