Solve the inequality `10-3xltx-17,x in{1,2,3,4,5,6,7,8,9,10}`

Answer

To solve the inequality \(10 - 3x < x - 17\) where \(x\) belongs to the set \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, we can follow these steps:

### Step 1: Rearranging the inequality
Start by moving all terms involving \(x\) to one side and constant terms to the other side.

\[ 10 - 3x < x - 17 \]

Add \(3x\) to both sides:

\[ 10 < x + 3x - 17 \]

This simplifies to:

\[ 10 < 4x - 17 \]

### Step 2: Isolate \(4x\)
Next, add \(17\) to both sides to isolate the term with \(x\):

\[ 10 + 17 < 4x \]

This simplifies to:

\[ 27 < 4x \]

### Step 3: Solve for \(x\)
Now, divide both sides by \(4\) to solve for \(x\):

\[ \frac{27}{4} < x \]

Calculating \(\frac{27}{4}\):

\[ \frac{27}{4} = 6.75 \]

So we can write:

\[ x > 6.75 \]

### Step 4: Determine integer solutions
Since \(x\) must be an integer and belongs to the set \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, we need to find integers greater than \(6.75\). The integers that satisfy this condition are:

\[ x = 7, 8, 9, 10 \]

### Conclusion
Thus, the solution set for the inequality \(10 - 3x < x - 17\) is:

\[ \{7, 8, 9, 10\} \]

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