The number of integral solutions of `x^(2)-3x-4 lt 0`, is

Options

A. 3
B. 4
C. 6
D. none of these

Answer

To find the number of integral solutions for the inequality \( x^2 - 3x - 4 < 0 \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ x^2 - 3x - 4 < 0 \]

### Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression \( x^2 - 3x - 4 \). We can do this by looking for two numbers that multiply to \(-4\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-4\) and \(1\) satisfy these conditions. Thus, we can factor the expression as:
\[ (x - 4)(x + 1) < 0 \]

### Step 3: Identify Critical Points
The critical points occur where the expression equals zero. Setting each factor to zero gives us:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
Thus, the critical points are \(x = -1\) and \(x = 4\).

### Step 4: Test Intervals
We will test the sign of the product \( (x - 4)(x + 1) \) in the intervals determined by the critical points:
1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \):
\[ (-2 - 4)(-2 + 1) = (-6)(-1) = 6 \quad (\text{positive}) \]
2. **Interval \( (-1, 4) \)**: Choose \( x = 0 \):
\[ (0 - 4)(0 + 1) = (-4)(1) = -4 \quad (\text{negative}) \]
3. **Interval \( (4, \infty) \)**: Choose \( x = 5 \):
\[ (5 - 4)(5 + 1) = (1)(6) = 6 \quad (\text{positive}) \]

### Step 5: Determine the Solution Set
From our tests, we find:
- The expression is positive in the intervals \( (-\infty, -1) \) and \( (4, \infty) \).
- The expression is negative in the interval \( (-1, 4) \).

Since we are looking for where the expression is less than zero, the solution set is:
\[ -1 < x < 4 \]

### Step 6: Identify Integral Solutions
Now, we need to find the integral solutions within the interval \( (-1, 4) \). The integers that satisfy this inequality are:
\[ 0, 1, 2, 3 \]

### Step 7: Count the Integral Solutions
The integral solutions are \(0, 1, 2, 3\), which gives us a total of:
\[ \text{Number of integral solutions} = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]