To solve the equation
[tex]\[
\frac{3x - 2}{18} + x = \frac{3x + 2}{3} + 2
\][/tex]
we will follow these steps:
1. Simplify both sides of the equation:
First, let's simplify the left-hand side (LHS):
[tex]\[
\frac{3x - 2}{18} + x
\][/tex]
Rewrite [tex]\( x \)[/tex] as a fraction with a common denominator of 18:
[tex]\[
x = \frac{18x}{18}
\][/tex]
So the LHS becomes:
[tex]\[
\frac{3x - 2}{18} + \frac{18x}{18} = \frac{3x - 2 + 18x}{18} = \frac{21x - 2}{18}
\][/tex]
Next, let's simplify the right-hand side (RHS):
[tex]\[
\frac{3x + 2}{3} + 2
\][/tex]
Rewrite 2 as a fraction with a common denominator of 3:
[tex]\[
2 = \frac{6}{3}
\][/tex]
So the RHS becomes:
[tex]\[
\frac{3x + 2}{3} + \frac{6}{3} = \frac{3x + 2 + 6}{3} = \frac{3x + 8}{3}
\][/tex]
2. Set the simplified left-hand side equal to the simplified right-hand side:
[tex]\[
\frac{21x - 2}{18} = \frac{3x + 8}{3}
\][/tex]
3. Eliminate the denominators by finding a common multiple. Multiply both sides by 18:
[tex]\[
18 \cdot \frac{21x - 2}{18} = 18 \cdot \frac{3x + 8}{3}
\][/tex]
The 18 cancels out the denominators:
[tex]\[
21x - 2 = 6(3x + 8)
\][/tex]
4. Expand and simplify:
[tex]\[
21x - 2 = 18x + 48
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
21x - 18x = 48 + 2
\][/tex]
[tex]\[
3x = 50
\][/tex]
[tex]\[
x = \frac{50}{3}
\][/tex]
So, the solution to the equation is:
[tex]\[
x = \frac{50}{3}
\][/tex]
6. Verify the solution:
Insert [tex]\( x = \frac{50}{3} \)[/tex] back into the original equation to verify it satisfies both sides.
We conclude that the correct choice is:
A. The solution is [tex]\( \frac{50}{3} \)[/tex]
