solve : `-(1)/(3)le(x)/(2)-(4)/(3)lt(1)/(6)`

Explanation

To solve the inequality \(-\frac{1}{3} \leq \frac{x}{2} - \frac{4}{3} < \frac{1}{6}\), we will break it down into two parts and solve each part step by step.

### Step 1: Split the Inequality
We can rewrite the inequality as two separate inequalities:
1. \(-\frac{1}{3} \leq \frac{x}{2} - \frac{4}{3}\)
2. \(\frac{x}{2} - \frac{4}{3} < \frac{1}{6}\)

### Step 2: Solve the First Inequality
Starting with the first inequality:
\[ -\frac{1}{3} \leq \frac{x}{2} - \frac{4}{3} \]
Add \(\frac{4}{3}\) to both sides:
\[ -\frac{1}{3} + \frac{4}{3} \leq \frac{x}{2} \]
Calculating the left side:
\[ \frac{3}{3} \leq \frac{x}{2} \]
This simplifies to:
\[ 1 \leq \frac{x}{2} \]
Now, multiply both sides by 2:
\[ 2 \leq x \]
or
\[ x \geq 2 \]

### Step 3: Solve the Second Inequality
Now, for the second inequality:
\[ \frac{x}{2} - \frac{4}{3} < \frac{1}{6} \]
Again, add \(\frac{4}{3}\) to both sides:
\[ \frac{x}{2} < \frac{1}{6} + \frac{4}{3} \]
To add the fractions, we need a common denominator. The common denominator for 6 and 3 is 6:
\[ \frac{4}{3} = \frac{8}{6} \]
So we have:
\[ \frac{x}{2} < \frac{1}{6} + \frac{8}{6} \]
This simplifies to:
\[ \frac{x}{2} < \frac{9}{6} \]
or
\[ \frac{x}{2} < \frac{3}{2} \]
Now, multiply both sides by 2:
\[ x < 3 \]

### Step 4: Combine the Results
From the two inequalities, we have:
1. \(x \geq 2\)
2. \(x < 3\)

Combining these results, we find:
\[ 2 \leq x < 3 \]

### Final Answer
Thus, the solution can be expressed in interval notation as:
\[ x \in [2, 3) \]