If `3x - 5 (x-1) gt 2x -1 lt 2 + 4x,` then x can take which of the following values ? (a) 2 (b) 2 (c) 3 (d) 1

Options

A. 2
B. 2
C. 3
D. 1

Answer

To solve the inequality \(3x - 5(x - 1) > 2x - 1 < 2 + 4x\), we will break it down into two separate inequalities:

1. \(3x - 5(x - 1) > 2x - 1\)
2. \(2x - 1 < 2 + 4x\)

### Step 1: Solve the first inequality \(3x - 5(x - 1) > 2x - 1\)

1. Distribute the \(-5\) in the left side:
\[ 3x - 5x + 5 > 2x - 1 \]
This simplifies to:
\[ -2x + 5 > 2x - 1 \]

2. Move \(2x\) to the left side:
\[ -2x - 2x + 5 > -1 \]
This simplifies to:
\[ -4x + 5 > -1 \]

3. Subtract \(5\) from both sides:
\[ -4x > -6 \]

4. Divide by \(-4\) (remember to reverse the inequality):
\[ x < \frac{3}{2} \]

### Step 2: Solve the second inequality \(2x - 1 < 2 + 4x\)

1. Move \(4x\) to the left side:
\[ 2x - 4x - 1 < 2 \]
This simplifies to:
\[ -2x - 1 < 2 \]

2. Add \(1\) to both sides:
\[ -2x < 3 \]

3. Divide by \(-2\) (remember to reverse the inequality):
\[ x > -\frac{3}{2} \]

### Step 3: Combine the results

From the two inequalities, we have:
1. \(x < \frac{3}{2}\)
2. \(x > -\frac{3}{2}\)

Thus, the combined result is:
\[ -\frac{3}{2} < x < \frac{3}{2} \]

### Step 4: Determine which options satisfy this range

The options given are:
(a) 2
(b) 2
(c) 3
(d) 1

Only option (d) \(1\) is within the range \(-\frac{3}{2} < x < \frac{3}{2}\).

### Final Answer:
The value of \(x\) that satisfies the inequalities is \(1\).

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