`2(X - 1) lt X + 5, 3 (X + 2) gt 2 - X `

Explanation

To solve the given inequalities step by step, we will break down each part of the problem.

### Step 1: Write down the inequalities
We have two inequalities to solve:
1. \( 2(X - 1) < X + 5 \)
2. \( 3(X + 2) > 2 - X \)

### Step 2: Expand the first inequality
For the first inequality, we will distribute the 2:
\[ 2(X - 1) = 2X - 2 \]
So, the inequality becomes:
\[ 2X - 2 < X + 5 \]

### Step 3: Rearrange the first inequality
Now, we will move all terms involving \(X\) to one side and constant terms to the other side:
\[ 2X - X < 5 + 2 \]
This simplifies to:
\[ X < 7 \]

### Step 4: Expand the second inequality
Now, let's expand the second inequality:
\[ 3(X + 2) = 3X + 6 \]
So, the inequality becomes:
\[ 3X + 6 > 2 - X \]

### Step 5: Rearrange the second inequality
Now, we will move all terms involving \(X\) to one side and constant terms to the other side:
\[ 3X + X > 2 - 6 \]
This simplifies to:
\[ 4X > -4 \]

### Step 6: Solve for \(X\) in the second inequality
Now, divide both sides by 4:
\[ X > -1 \]

### Step 7: Combine the results
Now we have two results:
1. \(X < 7\)
2. \(X > -1\)

### Step 8: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ X \in (-1, 7) \]

### Step 9: Draw the number line (optional)
You can represent this solution on a number line, indicating that the interval does not include the endpoints -1 and 7.