Given 6x + (6-x) > 2x -2 < 5x /2-2-3x/4, then x can take which of the following values ?

Options

A. `9`
B. `-1`
C. 5
D. `-9`

Answer

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

### Step 1: Solve the first inequality \( 6x + (6 - x) > 2x - 2 \)

1. Simplify the left side:
\[ 6x + 6 - x = 5x + 6 \]
So, the inequality becomes:
\[ 5x + 6 > 2x - 2 \]

2. Rearranging the inequality:
\[ 5x - 2x > -2 - 6 \]
\[ 3x > -8 \]

3. Divide both sides by 3:
\[ x > -\frac{8}{3} \]

### Step 2: Solve the second inequality \( 2x - 2 < \frac{5x}{2} - 2 - \frac{3x}{4} \)

1. First, simplify the right side. To do this, we need a common denominator. The common denominator for 2 and 4 is 4:
\[ 2x - 2 < \frac{5x}{2} - 2 - \frac{3x}{4} \]
Rewrite \( 2x \) as \( \frac{4x}{2} \):
\[ 2x - 2 < \frac{5x}{2} - 2 - \frac{3x}{4} = \frac{10x}{4} - \frac{3x}{4} - 2 \]
Combine the fractions:
\[ 2x - 2 < \frac{7x}{4} - 2 \]

2. Eliminate the -2 from both sides:
\[ 2x < \frac{7x}{4} \]

3. Multiply through by 4 to eliminate the fraction:
\[ 8x < 7x \]

4. Rearranging gives:
\[ 8x - 7x < 0 \]
\[ x < 0 \]

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

From the first inequality, we have:
\[ x > -\frac{8}{3} \]
From the second inequality, we have:
\[ x < 0 \]

### Conclusion

Combining both results, we find:
\[ -\frac{8}{3} < x < 0 \]

### Final Answer
Thus, \( x \) can take values in the interval \( (-\frac{8}{3}, 0) \).