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

Answer

To solve the inequation \( 2y - 3 < y + 1 \leq 4y + 7 \), we will break it down into two parts and solve each part step by step.

### Step 1: Break down the inequation
The given inequation can be split into two separate inequalities:
1. \( 2y - 3 < y + 1 \)
2. \( y + 1 \leq 4y + 7 \)

### Step 2: Solve the first inequality \( 2y - 3 < y + 1 \)
To solve this inequality, we will isolate \( y \):
1. Subtract \( y \) from both sides:
\[ 2y - y - 3 < 1 \]
This simplifies to:
\[ y - 3 < 1 \]
2. Now, add 3 to both sides:
\[ y < 4 \]

### Step 3: Solve the second inequality \( y + 1 \leq 4y + 7 \)
Next, we will solve the second inequality:
1. Subtract \( y \) from both sides:
\[ 1 \leq 4y - y + 7 \]
This simplifies to:
\[ 1 \leq 3y + 7 \]
2. Now, subtract 7 from both sides:
\[ 1 - 7 \leq 3y \]
This simplifies to:
\[ -6 \leq 3y \]
3. Finally, divide both sides by 3:
\[ -2 \leq y \]
or equivalently:
\[ y \geq -2 \]

### Step 4: Combine the results
From the two inequalities, we have:
1. \( y < 4 \)
2. \( y \geq -2 \)

Combining these results gives us:
\[ -2 \leq y < 4 \]

### Step 5: Write the final solution
The solution in interval notation is:
\[ [-2, 4) \]