Solve `, 4x-5 gt 10-x, x in {0,1,2,3,4,5,6,7}`

Answer

To solve the inequality \(4x - 5 > 10 - x\) where \(x\) belongs to the set \(\{0, 1, 2, 3, 4, 5, 6, 7\}\), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ 4x - 5 > 10 - x \]

### Step 2: Move all \(x\) terms to one side
Add \(x\) to both sides to get all \(x\) terms on the left:
\[ 4x + x - 5 > 10 \]
This simplifies to:
\[ 5x - 5 > 10 \]

### Step 3: Move constant terms to the other side
Now, add \(5\) to both sides to isolate the term with \(x\):
\[ 5x > 10 + 5 \]
This simplifies to:
\[ 5x > 15 \]

### Step 4: Divide by the coefficient of \(x\)
Now, divide both sides by \(5\):
\[ x > \frac{15}{5} \]
This simplifies to:
\[ x > 3 \]

### Step 5: Determine integer solutions
Since \(x\) must be an integer and belongs to the set \(\{0, 1, 2, 3, 4, 5, 6, 7\}\), the possible integer values for \(x\) that satisfy \(x > 3\) are:
\[ x = 4, 5, 6, 7 \]

### Final Solution
Thus, the solution to the inequality \(4x - 5 > 10 - x\) for \(x\) in the specified set is:
\[ \{4, 5, 6, 7\} \]

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