To determine how many solutions the equation [tex]\( 12(x - 3) = -3(x + 12) \)[/tex] has, let's solve it step by step.
1. Expand both sides of the equation:
[tex]\[
12(x - 3) = -3(x + 12)
\][/tex]
Expand the left side:
[tex]\[
12x - 36
\][/tex]
Expand the right side:
[tex]\[
-3x - 36
\][/tex]
So the equation becomes:
[tex]\[
12x - 36 = -3x - 36
\][/tex]
2. Collect like terms:
To isolate [tex]\( x \)[/tex], add [tex]\( 3x \)[/tex] to both sides of the equation:
[tex]\[
12x - 36 + 3x = -3x - 36 + 3x
\][/tex]
Simplify:
[tex]\[
15x - 36 = -36
\][/tex]
3. Isolate the variable [tex]\( x \)[/tex]:
Add 36 to both sides to get:
[tex]\[
15x - 36 + 36 = -36 + 36
\][/tex]
Simplify:
[tex]\[
15x = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 15:
[tex]\[
x = 0
\][/tex]
So, the equation [tex]\( 12(x - 3) = -3(x + 12) \)[/tex] has exactly one solution. Therefore, the correct answer is:
[tex]\[ (\text{A}) \text{ Exactly one} \][/tex]
