Solve the following inequation : `2y - 3 lt y + 1 le 4y + 7`, if :
`y in` (Integers)

Answer

To solve the inequality \(2y - 3 < y + 1 \leq 4y + 7\) for integer values of \(y\), we can break it down into two parts and solve each part separately.

### Step 1: Solve the first part of the inequality
We start with the first part:
\[ 2y - 3 < y + 1 \]
Subtract \(y\) from both sides:
\[ 2y - y - 3 < 1 \]
This simplifies to:
\[ y - 3 < 1 \]
Now, add \(3\) to both sides:
\[ y < 4 \]

### Step 2: Solve the second part of the inequality
Now, we consider the second part:
\[ y + 1 \leq 4y + 7 \]
Subtract \(y\) from both sides:
\[ 1 \leq 4y + 7 - y \]
This simplifies to:
\[ 1 \leq 3y + 7 \]
Now, subtract \(7\) from both sides:
\[ 1 - 7 \leq 3y \]
This gives us:
\[ -6 \leq 3y \]
Now, divide both sides by \(3\):
\[ -2 \leq y \]

### Step 3: Combine the results
From the two parts, we have:
\[ -2 \leq y < 4 \]
This means \(y\) can take integer values from \(-2\) up to \(3\) (since \(4\) is not included).

### Step 4: List the integer solutions
The integer solutions for \(y\) are:
\[ y = -2, -1, 0, 1, 2, 3 \]

### Final Answer
The integer solutions for the inequality \(2y - 3 < y + 1 \leq 4y + 7\) are:
\[ y \in \{-2, -1, 0, 1, 2, 3\} \]

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