If `x in{-2,-1,0,1,2,3,4,5}`, find the solution set of each of the following inequations : `(7)/(4)-3xlt(5)/(6)`

Answer

To solve the inequality \( \frac{7}{4} - 3x < \frac{5}{6} \) for \( x \in \{-2, -1, 0, 1, 2, 3, 4, 5\} \), follow these steps:

### Step 1: Rewrite the Inequality
Start with the given inequality:
\[ \frac{7}{4} - 3x < \frac{5}{6} \]

### Step 2: Eliminate the Fractions
To eliminate the fractions, find a common denominator. The least common multiple of 4 and 6 is 12. Multiply the entire inequality by 12:
\[ 12 \left( \frac{7}{4} \right) - 12(3x) < 12 \left( \frac{5}{6} \right) \]
This simplifies to:
\[ 21 - 36x < 10 \]

### Step 3: Rearrange the Inequality
Now, rearrange the inequality to isolate \( x \):
\[ 21 - 10 < 36x \]
This simplifies to:
\[ 11 < 36x \]
or equivalently,
\[ \frac{11}{36} < x \]

### Step 4: Calculate \( \frac{11}{36} \)
Now, calculate \( \frac{11}{36} \):
\[ \frac{11}{36} \approx 0.3056 \]

### Step 5: Identify the Solution Set
We need to find the values of \( x \) from the set \( \{-2, -1, 0, 1, 2, 3, 4, 5\} \) that are greater than \( 0.3056 \). The values greater than \( 0.3056 \) are:
\[ 1, 2, 3, 4, 5 \]

### Final Solution
Thus, the solution set for the inequality \( \frac{7}{4} - 3x < \frac{5}{6} \) is:
\[ \{1, 2, 3, 4, 5\} \]