Solve : ` 6 le -3 (2x -4) lt 12 `

Explanation

To solve the inequality \( 6 \leq -3(2x - 4) < 12 \), we will break it down into steps.

### Step 1: Rewrite the inequality
We start with the inequality:
\[ 6 \leq -3(2x - 4) < 12 \]

### Step 2: Divide the entire inequality by -3
When we divide by a negative number, we must reverse the inequality signs:
\[ \frac{6}{-3} \geq 2x - 4 > \frac{12}{-3} \]

This simplifies to:
\[ -2 \geq 2x - 4 > -4 \]

### Step 3: Split the compound inequality
We can split this into two inequalities:
1. \( -2 \geq 2x - 4 \)
2. \( 2x - 4 > -4 \)

### Step 4: Solve the first inequality
Starting with the first inequality:
\[ -2 \geq 2x - 4 \]
Add 4 to both sides:
\[ -2 + 4 \geq 2x \]
\[ 2 \geq 2x \]
Now, divide by 2:
\[ 1 \geq x \]
or
\[ x \leq 1 \]

### Step 5: Solve the second inequality
Now, let's solve the second inequality:
\[ 2x - 4 > -4 \]
Add 4 to both sides:
\[ 2x > -4 + 4 \]
\[ 2x > 0 \]
Now, divide by 2:
\[ x > 0 \]

### Step 6: Combine the results
From the two inequalities, we have:
\[ 0 < x \leq 1 \]

### Step 7: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ (0, 1] \]

### Final Answer
Thus, the solution to the inequality \( 6 \leq -3(2x - 4) < 12 \) is:
\[ (0, 1] \]

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