Find the value interval of x for the inequality `-6lex^2-5xle6` .

Options

A. `[-1,2]uu[3,6]`
B. `(-oo,2]uu[3,oo)`
C. `[-1,6]
D. none of these

Answer

To solve the inequality \(-6 \leq x^2 - 5x \leq 6\), we will break it down into two parts and solve each inequality step by step.

### Step 1: Split the Inequality
The given inequality can be split into two separate inequalities:
1. \(-6 \leq x^2 - 5x\)
2. \(x^2 - 5x \leq 6\)

### Step 2: Solve the First Inequality
Starting with the first inequality:
\[ -6 \leq x^2 - 5x \]
Rearranging gives:
\[ x^2 - 5x + 6 \geq 0 \]
Now, we factor the quadratic:
\[ (x - 2)(x - 3) \geq 0 \]

### Step 3: Determine the Critical Points
The critical points from the factors are \(x = 2\) and \(x = 3\). We will test intervals around these points to determine where the product is non-negative.

### Step 4: Test Intervals
1. **Interval \( (-\infty, 2) \)**: Choose \(x = 0\):
\[ (0 - 2)(0 - 3) = 6 \geq 0 \quad \text{(True)} \]
2. **Interval \( (2, 3) \)**: Choose \(x = 2.5\):
\[ (2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25 < 0 \quad \text{(False)} \]
3. **Interval \( (3, \infty) \)**: Choose \(x = 4\):
\[ (4 - 2)(4 - 3) = 2 \geq 0 \quad \text{(True)} \]

### Step 5: Conclusion for the First Inequality
From the tests, the solution for the first inequality is:
\[ x \in (-\infty, 2] \cup [3, \infty) \]

### Step 6: Solve the Second Inequality
Now, we solve the second inequality:
\[ x^2 - 5x \leq 6 \]
Rearranging gives:
\[ x^2 - 5x - 6 \leq 0 \]
Factoring the quadratic:
\[ (x - 6)(x + 1) \leq 0 \]

### Step 7: Determine the Critical Points
The critical points are \(x = -1\) and \(x = 6\). We will test intervals around these points.

### Step 8: Test Intervals
1. **Interval \( (-\infty, -1) \)**: Choose \(x = -2\):
\[ (-2 - 6)(-2 + 1) = (-8)(-1) = 8 \geq 0 \quad \text{(False)} \]
2. **Interval \( (-1, 6) \)**: Choose \(x = 0\):
\[ (0 - 6)(0 + 1) = (-6)(1) = -6 \leq 0 \quad \text{(True)} \]
3. **Interval \( (6, \infty) \)**: Choose \(x = 7\):
\[ (7 - 6)(7 + 1) = (1)(8) = 8 \geq 0 \quad \text{(False)} \]

### Step 9: Conclusion for the Second Inequality
From the tests, the solution for the second inequality is:
\[ x \in [-1, 6] \]

### Step 10: Combine the Solutions
Now we combine the solutions from both inequalities:
1. From the first inequality: \(x \in (-\infty, 2] \cup [3, \infty)\)
2. From the second inequality: \(x \in [-1, 6]\)

### Step 11: Find the Intersection
The intersection of the two sets gives us the final solution:
\[ x \in [-1, 2] \cup [3, 6] \]

### Final Answer
Thus, the value interval of \(x\) for the inequality \(-6 \leq x^2 - 5x \leq 6\) is:
\[ [-1, 2] \cup [3, 6] \]