If the replacement set s={-6,-3,0,3,6,9}, find the truth set of the following :
(i) `2x-1 gt 9` (ii) `3x+7 le 1`

Answer

To solve the problem, we need to find the truth set for the given inequalities using the replacement set \( S = \{-6, -3, 0, 3, 6, 9\} \).

### Part (i): Solve the inequality \( 2x - 1 > 9 \)

1. **Start with the inequality**:
\[ 2x - 1 > 9 \]

2. **Add 1 to both sides**:
\[ 2x - 1 + 1 > 9 + 1 \]
This simplifies to:
\[ 2x > 10 \]

3. **Divide both sides by 2**:
\[ \frac{2x}{2} > \frac{10}{2} \]
This simplifies to:
\[ x > 5 \]

4. **Determine the truth set**:
We need to find values in the replacement set \( S \) that satisfy \( x > 5 \). The values in \( S \) are:
\(-6, -3, 0, 3, 6, 9\). The values greater than 5 are \( 6 \) and \( 9 \).

Therefore, the truth set for part (i) is:
\[ T = \{6, 9\} \]

### Part (ii): Solve the inequality \( 3x + 7 \leq 1 \)

1. **Start with the inequality**:
\[ 3x + 7 \leq 1 \]

2. **Subtract 7 from both sides**:
\[ 3x + 7 - 7 \leq 1 - 7 \]
This simplifies to:
\[ 3x \leq -6 \]

3. **Divide both sides by 3**:
\[ \frac{3x}{3} \leq \frac{-6}{3} \]
This simplifies to:
\[ x \leq -2 \]

4. **Determine the truth set**:
We need to find values in the replacement set \( S \) that satisfy \( x \leq -2 \). The values in \( S \) are:
\(-6, -3, 0, 3, 6, 9\). The values less than or equal to -2 are \( -6 \) and \( -3 \).

Therefore, the truth set for part (ii) is:
\[ T = \{-6, -3\} \]

### Summary of Truth Sets
- For part (i): \( T = \{6, 9\} \)
- For part (ii): \( T = \{-6, -3\} \)