If `6x + 4 (2-x) gt 3x - 2 gt 5x //2 - x//4,` then value of x is

Options

A. 0
B. 1
C. 3
D. 2

Answer

To solve the inequality \( 6x + 4(2 - x) > 3x - 2 > \frac{5x}{2} - \frac{x}{4} \), we will break it down into two separate inequalities and solve them step by step.

### Step 1: Rewrite the Inequalities
We can separate the compound inequality into two parts:
1. \( 6x + 4(2 - x) > 3x - 2 \)
2. \( 3x - 2 > \frac{5x}{2} - \frac{x}{4} \)

### Step 2: Solve the First Inequality
Let's solve the first inequality:
\[ 6x + 4(2 - x) > 3x - 2 \]

Distributing \( 4 \) in the left side:
\[ 6x + 8 - 4x > 3x - 2 \]

Combine like terms:
\[ (6x - 4x) + 8 > 3x - 2 \]
\[ 2x + 8 > 3x - 2 \]

Now, isolate \( x \):
\[ 8 + 2 > 3x - 2x \]
\[ 10 > x \]
or
\[ x < 10 \]

### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
\[ 3x - 2 > \frac{5x}{2} - \frac{x}{4} \]

To combine the terms on the right, we need a common denominator. The common denominator for \( 2 \) and \( 4 \) is \( 4 \):
\[ 3x - 2 > \frac{10x}{4} - \frac{x}{4} \]
\[ 3x - 2 > \frac{10x - x}{4} \]
\[ 3x - 2 > \frac{9x}{4} \]

Now, multiply both sides by \( 4 \) to eliminate the fraction:
\[ 4(3x - 2) > 9x \]
\[ 12x - 8 > 9x \]

Isolate \( x \):
\[ 12x - 9x > 8 \]
\[ 3x > 8 \]
\[ x > \frac{8}{3} \]
or
\[ x > 2.67 \]

### Step 4: Combine the Results
From the two inequalities, we have:
1. \( x < 10 \)
2. \( x > 2.67 \)

Thus, the solution for \( x \) is:
\[ 2.67 < x < 10 \]

### Conclusion
The value of \( x \) lies between \( 2.67 \) and \( 10 \). Based on the options provided, we can conclude that the correct answer is the option that fits within this range.