Simplify : - `(1)/(3) le (x)/(2) - 1(1)/(3) lt (1)/(6) : x in R`.
Graph the values of x on the real number line.

Answer

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

### Step 1: Rewrite the Mixed Fraction
First, we convert the mixed fraction \(1 \frac{1}{3}\) into an improper fraction:
\[ 1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]
So, the inequality can be rewritten as:
\[ -\frac{1}{3} \leq \frac{x}{2} - \frac{4}{3} < \frac{1}{6} \]

### Step 2: Solve the Left Inequality
Now, we will solve the left part of the inequality:
\[ -\frac{1}{3} \leq \frac{x}{2} - \frac{4}{3} \]
To isolate \(\frac{x}{2}\), we 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 Right Inequality
Next, we solve the right part of the inequality:
\[ \frac{x}{2} - \frac{4}{3} < \frac{1}{6} \]
Again, we add \(\frac{4}{3}\) to both sides:
\[ \frac{x}{2} < \frac{1}{6} + \frac{4}{3} \]
To add these fractions, we need a common denominator. The least common multiple of 6 and 3 is 6:
\[ \frac{4}{3} = \frac{8}{6} \]
So, we have:
\[ \frac{x}{2} < \frac{1}{6} + \frac{8}{6} = \frac{9}{6} \]
This simplifies to:
\[ \frac{x}{2} < \frac{3}{2} \]
Now, multiply both sides by 2:
\[ x < 3 \]

### Step 4: Combine the Results
From the two parts, we have:
\[ x \geq 2 \quad \text{and} \quad x < 3 \]
This can be combined into a single inequality:
\[ 2 \leq x < 3 \]

### Step 5: Graph the Solution on the Real Number Line
To graph the solution \(2 \leq x < 3\) on the real number line:
- Draw a closed circle at 2 (indicating that 2 is included).
- Draw an open circle at 3 (indicating that 3 is not included).
- Shade the region between 2 and 3.