Sure, let's solve the given equation step-by-step.
### Step 1: Distribute the 2 on both sides of the equation.
Given:
[tex]\[ 2(5 - 3x) = 2(5x + 1) \][/tex]
Distribute the 2:
[tex]\[ 2 \cdot 5 - 2 \cdot 3x = 2 \cdot 5x + 2 \cdot 1 \][/tex]
This simplifies to:
[tex]\[ 10 - 6x = 10x + 2 \][/tex]
### Step 2: Combine like terms to solve for [tex]\( x \)[/tex].
Move all terms involving [tex]\( x \)[/tex] to one side and constants to the other side:
First, we subtract [tex]\( 10x \)[/tex] from both sides:
[tex]\[ 10 - 6x - 10x = 2 \][/tex]
[tex]\[ 10 - 16x = 2 \][/tex]
Next, we subtract 2 from both sides:
[tex]\[ 10 - 16x - 2 = 0 \][/tex]
[tex]\[ 8 - 16x = 0 \][/tex]
### Step 3: Solve the equation for [tex]\( x \)[/tex].
Subtract 8 from both sides:
[tex]\[ -16x = -8 \][/tex]
Now divide both sides by -16:
[tex]\[ x = \frac{-8}{-16} \][/tex]
[tex]\[ x = \frac{8}{16} \][/tex]
[tex]\[ x = 0.5 \][/tex]
So, the unique solution is [tex]\( x = 0.5 \)[/tex].
### Step 4: Check the solution.
Substitute [tex]\( x = 0.5 \)[/tex] back into the original equation to verify:
Original equation:
[tex]\[ 2(5 - 3x) = 2(5x + 1) \][/tex]
Substitute [tex]\( x = 0.5 \)[/tex]:
[tex]\[ 2(5 - 3 \cdot 0.5) = 2(5 \cdot 0.5 + 1) \][/tex]
Simplify:
[tex]\[ 2(5 - 1.5) = 2(2.5 + 1) \][/tex]
[tex]\[ 2 \cdot 3.5 = 2 \cdot 3.5 \][/tex]
[tex]\[ 7 = 7 \][/tex]
Both sides are equal, so [tex]\( x = 0.5 \)[/tex] is indeed a solution.
### Conclusion:
The correct choice is:
A. The solution set is [tex]\(\{0.5\}\)[/tex].
