Solve each of the following inequations and represent the solution set on the number line.
`(1)/(4)((2)/(3)x+1)ge(1)/(3)(x-2)," where " x in R`.

Explanation

To solve the inequality \(\frac{1}{4}\left(\frac{2}{3}x + 1\right) \geq \frac{1}{3}(x - 2)\), we will follow these steps:

### Step 1: Distribute the fractions
Start by distributing \(\frac{1}{4}\) on the left side and \(\frac{1}{3}\) on the right side.

\[ \frac{1}{4} \cdot \frac{2}{3}x + \frac{1}{4} \cdot 1 \geq \frac{1}{3}x - \frac{1}{3} \cdot 2 \]

This simplifies to:

\[ \frac{1}{6}x + \frac{1}{4} \geq \frac{1}{3}x - \frac{2}{3} \]

### Step 2: Eliminate fractions
To eliminate the fractions, find the least common multiple (LCM) of the denominators (which are 4, 6, and 3). The LCM is 12. Multiply the entire inequality by 12:

\[ 12 \left(\frac{1}{6}x\right) + 12 \left(\frac{1}{4}\right) \geq 12 \left(\frac{1}{3}x\right) - 12 \left(\frac{2}{3}\right) \]

This simplifies to:

\[ 2x + 3 \geq 4x - 8 \]

### Step 3: Rearrange the inequality
Now, rearrange the inequality to isolate \(x\):

\[ 2x + 3 + 8 \geq 4x \]

This simplifies to:

\[ 11 \geq 4x - 2x \]

So we have:

\[ 11 \geq 2x \]

### Step 4: Solve for \(x\)
Now, divide both sides by 2:

\[ \frac{11}{2} \geq x \]

This can also be written as:

\[ x \leq \frac{11}{2} \]

### Step 5: Represent the solution on the number line
The solution \(x \leq \frac{11}{2}\) means that \(x\) can take any value less than or equal to \(5.5\). On the number line, we will represent this by shading to the left of \(5.5\) and including the point \(5.5\) (indicated with a closed circle).

### Final Solution
The solution set is:
\[ x \in (-\infty, \frac{11}{2}] \]