If `3 (2-3x) lt 2 - 3 x lt 4 x -6,` then x can take which of the following values ?

Options

A. 2
B. `-1`
C. `-2`
D. 1

Answer

To solve the inequality \(3(2 - 3x) < 2 - 3x < 4x - 6\), we will break it down into two separate inequalities and solve each one step by step.

### Step 1: Break down the compound inequality
We can separate the compound inequality into two parts:
1. \(3(2 - 3x) < 2 - 3x\)
2. \(2 - 3x < 4x - 6\)

### Step 2: Solve the first inequality
Let's solve the first inequality:
\[ 3(2 - 3x) < 2 - 3x \]
Distributing the 3 on the left side:
\[ 6 - 9x < 2 - 3x \]
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 6 - 2 < 9x - 3x \]
This simplifies to:
\[ 4 < 6x \]
Dividing both sides by 6 gives:
\[ \frac{4}{6} < x \quad \Rightarrow \quad \frac{2}{3} < x \]

### Step 3: Solve the second inequality
Now, let's solve the second inequality:
\[ 2 - 3x < 4x - 6 \]
Again, move all terms involving \(x\) to one side and constant terms to the other:
\[ 2 + 6 < 4x + 3x \]
This simplifies to:
\[ 8 < 7x \]
Dividing both sides by 7 gives:
\[ \frac{8}{7} < x \]

### Step 4: Combine the results
From the two inequalities, we have:
1. \(x > \frac{2}{3}\)
2. \(x > \frac{8}{7}\)

Since \(\frac{8}{7} \approx 1.14\) is greater than \(\frac{2}{3} \approx 0.67\), the more restrictive condition is \(x > \frac{8}{7}\).

### Conclusion
Thus, the values of \(x\) that satisfy the original inequality are:
\[ x > \frac{8}{7} \]