All the integral values of `x` for which `7x-3gt(x+1)^(2)gtx+3` lie in the interval

Options

A. `[1,2]`
B. `[2,3]`
C. `[3,4]`
D. `(1,4)`

Answer

To solve the inequality \( 7x - 3 > (x + 1)^2 > x + 3 \) and find all integral values of \( x \), we can break it down into two parts:

1. Solve \( 7x - 3 > (x + 1)^2 \)
2. Solve \( (x + 1)^2 > x + 3 \)

### Step 1: Solve \( 7x - 3 > (x + 1)^2 \)

1. **Expand the right side:**
\[ 7x - 3 > x^2 + 2x + 1 \]

2. **Rearrange the inequality:**
\[ 7x - 3 - 2x - 1 > x^2 \]
\[ 5x - 4 > x^2 \]

3. **Rearranging gives:**
\[ x^2 - 5x + 4 < 0 \]

4. **Factor the quadratic:**
\[ (x - 4)(x - 1) < 0 \]

5. **Find the critical points:**
The critical points are \( x = 1 \) and \( x = 4 \).

6. **Test intervals on the number line:**
- For \( x < 1 \): Choose \( x = 0 \) → \( (0 - 4)(0 - 1) = 4 > 0 \)
- For \( 1 < x < 4 \): Choose \( x = 2 \) → \( (2 - 4)(2 - 1) = -2 < 0 \)
- For \( x > 4 \): Choose \( x = 5 \) → \( (5 - 4)(5 - 1) = 4 > 0 \)

7. **Solution for this part:**
\[ 1 < x < 4 \]

### Step 2: Solve \( (x + 1)^2 > x + 3 \)

1. **Expand the left side:**
\[ x^2 + 2x + 1 > x + 3 \]

2. **Rearranging gives:**
\[ x^2 + 2x + 1 - x - 3 > 0 \]
\[ x^2 + x - 2 > 0 \]

3. **Factor the quadratic:**
\[ (x + 2)(x - 1) > 0 \]

4. **Find the critical points:**
The critical points are \( x = -2 \) and \( x = 1 \).

5. **Test intervals on the number line:**
- For \( x < -2 \): Choose \( x = -3 \) → \( (-3 + 2)(-3 - 1) = 4 > 0 \)
- For \( -2 < x < 1 \): Choose \( x = 0 \) → \( (0 + 2)(0 - 1) = -2 < 0 \)
- For \( x > 1 \): Choose \( x = 2 \) → \( (2 + 2)(2 - 1) = 4 > 0 \)

6. **Solution for this part:**
\[ x < -2 \quad \text{or} \quad x > 1 \]

### Step 3: Find the intersection of the two solutions

1. **From the first inequality:** \( 1 < x < 4 \)
2. **From the second inequality:** \( x < -2 \quad \text{or} \quad x > 1 \)

3. **Intersection of the two solutions:**
The only overlapping part is \( 1 < x < 4 \) and \( x > 1 \), which gives:
\[ 1 < x < 4 \]

### Step 4: Determine integral values of \( x \)

The integral values of \( x \) in the interval \( (1, 4) \) are:
\[ x = 2, 3 \]

### Final Answer:
The integral values of \( x \) are \( 2 \) and \( 3 \).