Questions: Solve the inequality. Write the solution set in interval notation. x^2-x-6<0 Choose the correct solution below. A. (-∞,-2) ∪(3, ∞) B. -2,3 C. (-2,3) D. No solution

Solution Steps

To solve the inequality \(x^2 - x - 6 < 0\), we first find the roots of the corresponding equation \(x^2 - x - 6 = 0\) using the quadratic formula. These roots will help us determine the intervals to test for the inequality. We then test the intervals between and outside the roots to see where the inequality holds true. Finally, we express the solution set in interval notation.

To solve the inequality \(x^2 - x - 6 < 0\), we first find the roots of the equation \(x^2 - x - 6 = 0\). Using the quadratic formula, we find the roots to be: \[ x = -2 \quad \text{and} \quad x = 3 \]

The roots divide the number line into three intervals:

1. \((-\infty, -2)\)
2. \((-2, 3)\)
3. \((3, \infty)\)

We need to test each interval to see where the inequality \(x^2 - x - 6 < 0\) holds true.

• For the interval \((-\infty, -2)\), choose a test point, e.g., \(x = -3\): \[ (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6 \quad (\text{not valid}) \]

• For the interval \((-2, 3)\), choose a test point, e.g., \(x = 0\): \[ 0^2 - 0 - 6 = -6 \quad (\text{valid}) \]

• For the interval \((3, \infty)\), choose a test point, e.g., \(x = 4\): \[ 4^2 - 4 - 6 = 16 - 4 - 6 = 6 \quad (\text{not valid}) \]

The inequality \(x^2 - x - 6 < 0\) holds true in the interval \((-2, 3)\). Therefore, the solution set in interval notation is: \[ (-2, 3) \]

Final Answer