Let's solve the equation step by step:
[tex]\[
\frac{x}{2} + \frac{1}{3} \left(\frac{x}{2}\right) = 120
\][/tex]
### Step 1: Simplify the terms
First, simplify the second term:
[tex]\[
\frac{1}{3} \left(\frac{x}{2}\right) = \frac{x}{6}
\][/tex]
### Step 2: Combine like terms
Add the fractions on the left-hand side:
[tex]\[
\frac{x}{2} + \frac{x}{6}
\][/tex]
To combine these fractions, find a common denominator. The common denominator of 2 and 6 is 6:
[tex]\[
\frac{x}{2} = \frac{3x}{6}
\][/tex]
Now the equation looks like this:
[tex]\[
\frac{3x}{6} + \frac{x}{6} = 120
\][/tex]
### Step 3: Add the fractions
Combine the fractions:
[tex]\[
\frac{3x + x}{6} = 120
\][/tex]
[tex]\[
\frac{4x}{6} = 120
\][/tex]
### Step 4: Simplify the fraction
Simplify [tex]\(\frac{4x}{6}\)[/tex]:
[tex]\[
\frac{2x}{3} = 120
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], multiply both sides of the equation by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x = 120 \times \frac{3}{2}
\][/tex]
[tex]\[
x = 180
\][/tex]
So, the value of [tex]\(x\)[/tex] is 180.
### Verification
To verify, substitute [tex]\(x = 180\)[/tex] back into the original equation:
[tex]\[
\frac{180}{2} + \frac{1}{3} \left(\frac{180}{2}\right) = 120
\][/tex]
[tex]\[
90 + \frac{1}{3} \times 90 = 120
\][/tex]
[tex]\[
90 + 30 = 120
\][/tex]
[tex]\[
120 = 120
\][/tex]
The left-hand side equals the right-hand side, confirming that our solution is correct.
Thus, the solution to the equation is:
[tex]\[\boxed{180}\][/tex]
