The solution set of system of linear inequalities `2(x+1) le x+5, 3(x+2) gt 2 - x , x in R ` is

Options

A. `[-1,3]`
B. `(-1,3)`
C. `(-1,3]`
D. `[-1,3)`

Answer

To solve the system of linear inequalities given by:

1. \( 2(x + 1) \leq x + 5 \)
2. \( 3(x + 2) > 2 - x \)

We will solve each inequality step by step.

### Step 1: Solve the first inequality

**Inequality:**
\[ 2(x + 1) \leq x + 5 \]

**Expanding the left side:**
\[ 2x + 2 \leq x + 5 \]

**Rearranging the terms:**
Subtract \( x \) from both sides:
\[ 2x - x + 2 \leq 5 \]
This simplifies to:
\[ x + 2 \leq 5 \]

**Subtracting 2 from both sides:**
\[ x \leq 3 \]

### Step 2: Solve the second inequality

**Inequality:**
\[ 3(x + 2) > 2 - x \]

**Expanding the left side:**
\[ 3x + 6 > 2 - x \]

**Rearranging the terms:**
Add \( x \) to both sides:
\[ 3x + x + 6 > 2 \]
This simplifies to:
\[ 4x + 6 > 2 \]

**Subtracting 6 from both sides:**
\[ 4x > 2 - 6 \]
This simplifies to:
\[ 4x > -4 \]

**Dividing by 4:**
\[ x > -1 \]

### Step 3: Combine the results

From the first inequality, we have:
\[ x \leq 3 \]

From the second inequality, we have:
\[ x > -1 \]

### Final Solution Set

Combining these results, we find that:
\[ -1 < x \leq 3 \]

This can be represented in interval notation as:
\[ (-1, 3] \]