If `x +5 (x -2) lt 4 - x lt 4x - 1,` then the value of x is

Options

A. `(-3)/(2)`
B. `(3)/(2)`
C. 0
D. 3

Answer

To solve the inequality \( x + 5(x - 2) < 4 - x < 4x - 1 \), we will break it down into two parts and solve each part step by step.

### Step 1: Solve the first inequality \( x + 5(x - 2) < 4 - x \)

1. Expand the left side:
\[ x + 5(x - 2) = x + 5x - 10 = 6x - 10 \]
So the inequality becomes:
\[ 6x - 10 < 4 - x \]

2. Add \( x \) to both sides:
\[ 6x + x - 10 < 4 \]
This simplifies to:
\[ 7x - 10 < 4 \]

3. Add \( 10 \) to both sides:
\[ 7x < 14 \]

4. Divide by \( 7 \):
\[ x < 2 \]

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

1. Add \( x \) to both sides:
\[ 4 < 4x + x - 1 \]
This simplifies to:
\[ 4 < 5x - 1 \]

2. Add \( 1 \) to both sides:
\[ 5 < 5x \]

3. Divide by \( 5 \):
\[ 1 < x \quad \text{or} \quad x > 1 \]

### Step 3: Combine the results from both inequalities

From the first inequality, we found:
\[ x < 2 \]
From the second inequality, we found:
\[ x > 1 \]

### Final Result

Combining these results, we have:
\[ 1 < x < 2 \]

Thus, the value of \( x \) lies between \( 1 \) and \( 2 \). A specific value that satisfies this condition is \( x = \frac{3}{2} \) or \( 1.5 \).

### Summary

The value of \( x \) that satisfies the inequality \( x + 5(x - 2) < 4 - x < 4x - 1 \) is:
\[ x = \frac{3}{2} \]