The solution set of `6 le -3 (2x-4) lt 12 , x in R ` is

Options

A. `(0,1]`
B. `[1,0)`
C. `(0,1)`
D. `[0,1]`

Answer

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

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

### Step 2: Distribute the -3
Next, we distribute \(-3\) to the terms inside the parentheses:
\[ -3(2x - 4) = -6x + 12 \]
So, we can rewrite the inequality as:
\[ 6 \leq -6x + 12 < 12 \]

### Step 3: Split the inequality
Now we will split the compound inequality into two separate inequalities:
1. \( 6 \leq -6x + 12 \)
2. \( -6x + 12 < 12 \)

### Step 4: Solve the first inequality
Let's solve the first inequality \( 6 \leq -6x + 12 \):
1. Subtract 12 from both sides:
\[ 6 - 12 \leq -6x \]
This simplifies to:
\[ -6 \leq -6x \]
2. Divide by -6 (remember to reverse the inequality sign):
\[ 1 \geq x \quad \text{or} \quad x \leq 1 \]

### Step 5: Solve the second inequality
Now, let's solve the second inequality \( -6x + 12 < 12 \):
1. Subtract 12 from both sides:
\[ -6x < 12 - 12 \]
This simplifies to:
\[ -6x < 0 \]
2. Divide by -6 (again, reverse the inequality sign):
\[ x > 0 \]

### Step 6: Combine the results
From the two inequalities we found:
1. \( x \leq 1 \)
2. \( x > 0 \)

Combining these results, we have:
\[ 0 < x \leq 1 \]

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