Certainly! Let's go through the process of solving the equation step by step:
### Given Equation:
[tex]\[ 10(2x - 4) = -12(x + 6) \][/tex]
### Step 1: Distribute the Constants
First, distribute the constants within the parentheses on both sides of the equation.
For the left side:
[tex]\[ 10(2x - 4) = 10 \cdot 2x - 10 \cdot 4 \][/tex]
[tex]\[ 10(2x - 4) = 20x - 40 \][/tex]
For the right side:
[tex]\[ -12(x + 6) = -12 \cdot x - 12 \cdot 6 \][/tex]
[tex]\[ -12(x + 6) = -12x - 72 \][/tex]
Now the equation looks like this:
[tex]\[ 20x - 40 = -12x - 72 \][/tex]
### Step 2: Combine Like Terms
Next, let's move all the terms containing [tex]\( x \)[/tex] to one side and the constant terms to the other side. To do this, add [tex]\( 12x \)[/tex] to both sides of the equation:
[tex]\[ 20x - 40 + 12x = -12x - 72 + 12x \][/tex]
[tex]\[ 32x - 40 = -72 \][/tex]
Then, add 40 to both sides to isolate the term [tex]\( 32x \)[/tex]:
[tex]\[ 32x - 40 + 40 = -72 + 40 \][/tex]
[tex]\[ 32x = -32 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Now, divide both sides by 32 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-32}{32} \][/tex]
[tex]\[ x = -1 \][/tex]
### Step 4: Check the Solution
Substitute [tex]\( x = -1 \)[/tex] back into the original equation to ensure it satisfies the equation:
[tex]\[ 10(2(-1) - 4) = -12((-1) + 6) \][/tex]
[tex]\[ 10(-2 - 4) = -12(5) \][/tex]
[tex]\[ 10(-6) = -60 \][/tex]
[tex]\[ -60 = -60 \][/tex]
Since both sides of the equation are equal, [tex]\( x = -1 \)[/tex] is indeed the correct solution.
### Conclusion
Based on our solution and verification, there is exactly one solution. The correct choice is:
A. There is exactly one solution. The solution set is { -1 }.
