To determine the correct statement about the given equation
[tex]\[
-9(x+3) + 12 = -3(2x+5) - 3x,
\][/tex]
we can follow a step-by-step process to simplify and analyze the equation. Here are the steps:
1. Expand both sides of the equation:
- Left side:
[tex]\[
-9(x + 3) + 12 \implies -9x - 27 + 12 \implies -9x - 15
\][/tex]
- Right side:
[tex]\[
-3(2x + 5) - 3x \implies -6x - 15 - 3x \implies -9x - 15
\][/tex]
2. Set the simplified expressions equal to each other:
[tex]\[
-9x - 15 = -9x - 15
\][/tex]
3. Analyze the resulting equation:
Since both sides of the equation are exactly the same, this indicates that:
[tex]\[
-9x - 15 = -9x - 15
\][/tex]
This equation is true for all values of [tex]\( x \)[/tex]. It doesn't matter what [tex]\( x \)[/tex] is, the equation will always hold true because both sides are identical.
4. Conclusion:
Because the simplified forms are the same, the original equation has no constraint on [tex]\( x \)[/tex]; hence, it has infinitely many solutions.
Therefore, the correct answer is:
D. The equation has infinitely many solutions.
