If `2x + 4 ( x-3) lt - 2 - x lt 2x -1,` then find the value of x. (a) -1 (b) 1 (c) 2 (d) -2

Options

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

Answer

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

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

### Step 1: Solve the first inequality

Start with the first inequality:

\[ 2x + 4(x - 3) < -2 - x \]

Distributing the \(4\):

\[ 2x + 4x - 12 < -2 - x \]

Combine like terms on the left side:

\[ 6x - 12 < -2 - x \]

Now, add \(x\) to both sides:

\[ 6x + x - 12 < -2 \]

This simplifies to:

\[ 7x - 12 < -2 \]

Next, add \(12\) to both sides:

\[ 7x < 10 \]

Now, divide by \(7\):

\[ x < \frac{10}{7} \]

### Step 2: Solve the second inequality

Now, let's solve the second inequality:

\[ -2 - x < 2x - 1 \]

Add \(x\) to both sides:

\[ -2 < 2x + x - 1 \]

This simplifies to:

\[ -2 < 3x - 1 \]

Now, add \(1\) to both sides:

\[ -1 < 3x \]

Dividing by \(3\):

\[ -\frac{1}{3} < x \]

### Step 3: Combine the results

Now we have two inequalities:

1. \( x < \frac{10}{7} \)
2. \( x > -\frac{1}{3} \)

### Step 4: Determine the range of \(x\)

The combined result is:

\[ -\frac{1}{3} < x < \frac{10}{7} \]

### Step 5: Check the options

Now we need to check which of the given options falls within this range:

(a) -1 (not in the range)

(b) 1 (in the range)

(c) 2 (not in the range)

(d) -2 (not in the range)

The only value that satisfies the condition is:

**Answer: (b) 1**