The set of values of `x in R` satisfying the inequality `x^(2)-4x-21 le 0` is

Options

A. (3, 7]
B. (-3, 7]
C. [-7, 3]
D. [-7, -3]

Answer

To solve the inequality \( x^2 - 4x - 21 \leq 0 \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ x^2 - 4x - 21 \leq 0 \]

### Step 2: Factor the quadratic expression
To factor the quadratic expression \( x^2 - 4x - 21 \), we need to find two numbers that multiply to \(-21\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers \(-7\) and \(3\) satisfy this condition.

Thus, we can factor the expression as:
\[ (x - 7)(x + 3) \leq 0 \]

### Step 3: Identify the critical points
The critical points occur when the expression equals zero:
\[ (x - 7)(x + 3) = 0 \]
This gives us the points:
\[ x = 7 \quad \text{and} \quad x = -3 \]

### Step 4: Test intervals
We will test the intervals defined by the critical points \( -3 \) and \( 7 \):
1. Interval \( (-\infty, -3) \)
2. Interval \( (-3, 7) \)
3. Interval \( (7, \infty) \)

**Testing the interval \( (-\infty, -3) \)**:
Choose \( x = -4 \):
\[ (-4 - 7)(-4 + 3) = (-11)(-1) = 11 > 0 \]
This interval does not satisfy the inequality.

**Testing the interval \( (-3, 7) \)**:
Choose \( x = 0 \):
\[ (0 - 7)(0 + 3) = (-7)(3) = -21 < 0 \]
This interval satisfies the inequality.

**Testing the interval \( (7, \infty) \)**:
Choose \( x = 8 \):
\[ (8 - 7)(8 + 3) = (1)(11) = 11 > 0 \]
This interval does not satisfy the inequality.

### Step 5: Include the critical points
Since the inequality is less than or equal to zero, we include the critical points:
\[ x = -3 \quad \text{and} \quad x = 7 \]

### Conclusion
The solution set for the inequality \( x^2 - 4x - 21 \leq 0 \) is:
\[ [-3, 7] \]