Solve the inequalities for real x : `3(x-1)lt=2(x-3)`

Explanation

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

### Step 1: Expand both sides of the inequality
We start by distributing the terms on both sides:

\[ 3(x - 1) \leq 2(x - 3) \]

This simplifies to:

\[ 3x - 3 \leq 2x - 6 \]

### Step 2: Rearrange the inequality
Next, we want to get all the variable terms on one side and the constant terms on the other side. We can do this by subtracting \( 2x \) from both sides:

\[ 3x - 3 - 2x \leq 2x - 6 - 2x \]

This simplifies to:

\[ x - 3 \leq -6 \]

### Step 3: Isolate the variable
Now, we will isolate \( x \) by adding 3 to both sides:

\[ x - 3 + 3 \leq -6 + 3 \]

This gives us:

\[ x \leq -3 \]

### Step 4: Write the solution in interval notation
The solution to the inequality indicates that \( x \) can take any value less than or equal to -3. In interval notation, this is expressed as:

\[ (-\infty, -3] \]

### Final Answer
Thus, the solution to the inequality \( 3(x - 1) \leq 2(x - 3) \) is:

\[ x \leq -3 \]

---