If both the inequalities give below are true, then what are the values of x and y which satisfy the inequations ?
`(x)/(4) + (2)/(3) lt (2x)/(3) - (1)/(6), (1)/(6) + (3)/(4y) gt (7)/(8)`

Options

A. A) `x gt2 , y lt 2`
B. B) `x lt 2 lt y`
C. C) `x gt 2, y gt 2`
D. D) `x lt 2, y lt 2`

Answer

To solve the given inequalities step by step, we will break down each inequality and find the values of \(x\) and \(y\).

### Step 1: Solve the first inequality
The first inequality is:
\[ \frac{x}{4} + \frac{2}{3} < \frac{2x}{3} - \frac{1}{6} \]

**Hint:** Start by eliminating the fractions by multiplying through by the least common multiple (LCM) of the denominators.

**Solution:**
1. The LCM of 4, 3, and 6 is 12. Multiply the entire inequality by 12:
\[ 12 \left( \frac{x}{4} \right) + 12 \left( \frac{2}{3} \right) < 12 \left( \frac{2x}{3} \right) - 12 \left( \frac{1}{6} \right) \]
This simplifies to:
\[ 3x + 8 < 8x - 2 \]

2. Rearranging the inequality to isolate \(x\):
\[ 3x + 8 + 2 < 8x \]
\[ 10 < 8x - 3x \]
\[ 10 < 5x \]
Dividing both sides by 5:
\[ 2 < x \quad \text{or} \quad x > 2 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ \frac{1}{6} + \frac{3}{4y} > \frac{7}{8} \]

**Hint:** Again, isolate the term with \(y\) by moving other terms to the right side.

**Solution:**
1. Move \(\frac{1}{6}\) to the right side:
\[ \frac{3}{4y} > \frac{7}{8} - \frac{1}{6} \]

2. Find the LCM of 8 and 6, which is 24. Rewrite the fractions:
\[ \frac{7}{8} = \frac{21}{24}, \quad \frac{1}{6} = \frac{4}{24} \]
Thus:
\[ \frac{3}{4y} > \frac{21}{24} - \frac{4}{24} \]
\[ \frac{3}{4y} > \frac{17}{24} \]

3. Cross-multiply to eliminate the fraction:
\[ 3 \cdot 24 > 4y \cdot 17 \]
\[ 72 > 68y \]
Dividing both sides by 68:
\[ \frac{72}{68} > y \]
Simplifying:
\[ \frac{18}{17} > y \quad \text{or} \quad y < \frac{18}{17} \]

### Final Results
- From the first inequality, we found \(x > 2\).
- From the second inequality, we found \(y < \frac{18}{17}\).

### Summary
The values that satisfy both inequalities are:
\[ x > 2 \quad \text{and} \quad y < \frac{18}{17} \]