Solve the inequation `(3(x -2))/(5) le (5(2 -x))/(3)` and represent this solution on number line.

Explanation

To solve the inequality \(\frac{3(x - 2)}{5} \leq \frac{5(2 - x)}{3}\), we will follow these steps:

### Step 1: Cross Multiply
To eliminate the fractions, we can cross-multiply. This gives us:
\[ 3(x - 2) \cdot 3 \leq 5(2 - x) \cdot 5 \]
This simplifies to:
\[ 9(x - 2) \leq 25(2 - x) \]

### Step 2: Distribute
Now we will distribute both sides:
\[ 9x - 18 \leq 50 - 25x \]

### Step 3: Move all terms involving \(x\) to one side
Next, we will add \(25x\) to both sides and add \(18\) to both sides:
\[ 9x + 25x \leq 50 + 18 \]
This simplifies to:
\[ 34x \leq 68 \]

### Step 4: Solve for \(x\)
Now, we will divide both sides by \(34\):
\[ x \leq 2 \]

### Step 5: Represent the solution on a number line
The solution \(x \leq 2\) means that \(x\) can take any value less than or equal to \(2\). On a number line, we will represent this by shading the line to the left of \(2\) and placing a closed circle at \(2\) to indicate that \(2\) is included in the solution.

### Final Solution
The solution can be written in interval notation as:
\[ x \in (-\infty, 2] \]

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