solve the inequalities for real x :`(3(x-2))/(5) le (5(2 - x))/(3) `

Explanation

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

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ \frac{3(x-2)}{5} \leq \frac{5(2 - x)}{3} \]

### Step 2: Cross-multiply
To eliminate the fractions, we can cross-multiply. This is valid since both sides are positive for the values of \(x\) we will consider. Thus, we have:
\[ 3(x-2) \cdot 3 \leq 5(2 - x) \cdot 5 \]
This simplifies to:
\[ 9(x-2) \leq 25(2 - x) \]

### Step 3: Expand both sides
Now, we will expand both sides of the inequality:
\[ 9x - 18 \leq 50 - 25x \]

### Step 4: 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 5: Divide by 34
Now, we divide both sides by \(34\):
\[ x \leq 2 \]

### Step 6: Write the solution in interval notation
The solution to the inequality is:
\[ x \in (-\infty, 2] \]

### Summary of the solution
Thus, the solution to the inequality \(\frac{3(x-2)}{5} \leq \frac{5(2 - x)}{3}\) is:
\[ x \in (-\infty, 2] \]