Solve : `(3x)/(5) - (2(x-2))/(3) le 1`

Explanation

To solve the inequality \(\frac{3x}{5} - \frac{2(x-2)}{3} \leq 1\), we will follow these steps:

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

### Step 2: Find a common denominator
The denominators are 5 and 3. The least common multiple (LCM) of 5 and 3 is 15. We will multiply each term by 15 to eliminate the fractions:
\[ 15 \left(\frac{3x}{5}\right) - 15 \left(\frac{2(x-2)}{3}\right) \leq 15(1) \]

### Step 3: Simplify each term
Now simplify each term:
\[ 3 \times 3x - 5 \times 2(x-2) \leq 15 \]
This simplifies to:
\[ 9x - 10(x - 2) \leq 15 \]

### Step 4: Distribute the terms
Distributing the \(-10\) in the second term:
\[ 9x - 10x + 20 \leq 15 \]

### Step 5: Combine like terms
Combine the \(x\) terms:
\[ -1x + 20 \leq 15 \]

### Step 6: Isolate the variable
Subtract 20 from both sides:
\[ -1x \leq 15 - 20 \]
This simplifies to:
\[ -1x \leq -5 \]

### Step 7: Multiply by -1
When multiplying or dividing by a negative number, we must reverse the inequality sign:
\[ x \geq 5 \]

### Step 8: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ x \in [5, \infty) \]

### Final Answer
Thus, the solution to the inequality is:
\[ x \geq 5 \quad \text{or} \quad x \in [5, \infty) \]

---