Sure, let's solve the given equation step-by-step to find the correct answer.
We are given the equation:
[tex]\[
\frac{x}{2} + \frac{1}{2} = \frac{x}{3} - \frac{1}{3}
\][/tex]
First, let's eliminate the fractions by finding a common denominator. The common denominator for the fractions involving [tex]\( x \)[/tex] and constant terms is 6.
1. Multiply every term by 6:
[tex]\[
6 \left( \frac{x}{2} + \frac{1}{2} \right) = 6 \left( \frac{x}{3} - \frac{1}{3} \right)
\][/tex]
2. This simplifies to:
[tex]\[
3x + 3 = 2x - 2
\][/tex]
3. Next, isolate the variable [tex]\( x \)[/tex] on one side. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
3x - 2x + 3 = -2
\][/tex]
4. This further simplifies to:
[tex]\[
x + 3 = -2
\][/tex]
5. Subtract 3 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = -2 - 3
\][/tex]
6. Simplify the right-hand side:
[tex]\[
x = -5
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = -5
\][/tex]
So, the correct choice is:
(a) [tex]\( x = -5 \)[/tex]
