Answer :

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]