If `2x - 2 (4-x) lt 2x -3 lt 3x + 3,` then x can take which of the following values ? (a) 2 (b) 3 (c) 4 (d) 5

Options

A. 2
B. 3
C. 4
D. 5

Answer

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

### Step 1: Solve the first part of the inequality
We start with the first part:
\[ 2x - 2(4 - x) < 2x - 3 \]

Distributing the \(-2\):
\[ 2x - 8 + 2x < 2x - 3 \]

Combine like terms:
\[ 4x - 8 < 2x - 3 \]

Now, subtract \(2x\) from both sides:
\[ 4x - 2x - 8 < -3 \]
\[ 2x - 8 < -3 \]

Next, add \(8\) to both sides:
\[ 2x < 5 \]

Finally, divide by \(2\):
\[ x < \frac{5}{2} \]
\[ x < 2.5 \]

### Step 2: Solve the second part of the inequality
Now we solve the second part:
\[ 2x - 3 < 3x + 3 \]

Subtract \(2x\) from both sides:
\[ -3 < x + 3 \]

Next, subtract \(3\) from both sides:
\[ -6 < x \]
or
\[ x > -6 \]

### Step 3: Combine the results
Now we have two inequalities:
1. \( x < \frac{5}{2} \) (or \( x < 2.5 \))
2. \( x > -6 \)

This means:
\[ -6 < x < \frac{5}{2} \]

### Step 4: Determine valid integer values
The integer values that satisfy this inequality are \( -5, -4, -3, -2, -1, 0, 1, 2 \).

### Step 5: Check the options provided
The options given are:
(a) 2
(b) 3
(c) 4
(d) 5

Among these options, the only value that satisfies \( -6 < x < 2.5 \) is \( 2 \).

### Final Answer
Thus, the value of \( x \) can take is:
**(a) 2**

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