The number of intergal values of x if `5x -1 lt (x+1)^2 lt 7 x-3 ` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the inequality \( 5x - 1 < (x + 1)^2 < 7x - 3 \), we will break it down into two parts and analyze each part separately.

### Step 1: Solve the first inequality \( (x + 1)^2 > 5x - 1 \)

1. Start with the inequality:
\[ (x + 1)^2 > 5x - 1 \]
2. Expand the left side:
\[ x^2 + 2x + 1 > 5x - 1 \]
3. Rearrange the inequality:
\[ x^2 + 2x + 1 - 5x + 1 > 0 \]
\[ x^2 - 3x + 2 > 0 \]
4. Factor the quadratic:
\[ (x - 1)(x - 2) > 0 \]
5. Determine the intervals where the product is positive. The critical points are \( x = 1 \) and \( x = 2 \). Test the intervals:
- For \( x < 1 \): Choose \( x = 0 \) → \( (0 - 1)(0 - 2) = 2 > 0 \) (True)
- For \( 1 < x < 2 \): Choose \( x = 1.5 \) → \( (1.5 - 1)(1.5 - 2) = -0.25 < 0 \) (False)
- For \( x > 2 \): Choose \( x = 3 \) → \( (3 - 1)(3 - 2) = 2 > 0 \) (True)

Thus, the solution for this part is:
\[ x < 1 \quad \text{or} \quad x > 2 \]

### Step 2: Solve the second inequality \( (x + 1)^2 < 7x - 3 \)

1. Start with the inequality:
\[ (x + 1)^2 < 7x - 3 \]
2. Expand the left side:
\[ x^2 + 2x + 1 < 7x - 3 \]
3. Rearrange the inequality:
\[ x^2 + 2x + 1 - 7x + 3 < 0 \]
\[ x^2 - 5x + 4 < 0 \]
4. Factor the quadratic:
\[ (x - 1)(x - 4) < 0 \]
5. Determine the intervals where the product is negative. The critical points are \( x = 1 \) and \( x = 4 \). Test the intervals:
- For \( x < 1 \): Choose \( x = 0 \) → \( (0 - 1)(0 - 4) = 4 > 0 \) (False)
- For \( 1 < x < 4 \): Choose \( x = 2 \) → \( (2 - 1)(2 - 4) = -2 < 0 \) (True)
- For \( x > 4 \): Choose \( x = 5 \) → \( (5 - 1)(5 - 4) = 4 > 0 \) (False)

Thus, the solution for this part is:
\[ 1 < x < 4 \]

### Step 3: Combine the results

We have two conditions:
1. From the first inequality: \( x < 1 \) or \( x > 2 \)
2. From the second inequality: \( 1 < x < 4 \)

To find the intersection of these conditions:
- The interval \( x < 1 \) does not overlap with \( 1 < x < 4 \).
- The interval \( x > 2 \) overlaps with \( 1 < x < 4 \), giving us \( 2 < x < 4 \).

### Step 4: Find the integral values

The integral values of \( x \) that satisfy \( 2 < x < 4 \) are:
- \( x = 3 \)

### Conclusion

Thus, the number of integral values of \( x \) that satisfy the original inequality is:
\[ \text{Number of integral values} = 1 \]