Question

use a graph to solve the inequality.
$x^2 - x - 12 > 0$
graph $y = x^2 - x - 12$. use the graphing tool to graph the equation.
solve the inequality. select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the solution is
(type an inequality or a compound inequality. simplify your answer. use a comma to separate answers as needed.)
b. the solution is all real numbers.
c. there is no solution.

Explanation:

Step1: Find roots of \(y = x^2 - x - 12\)

Factor the quadratic: \(x^2 - x - 12=(x - 4)(x + 3)\). Set to zero: \(x - 4 = 0\) or \(x + 3 = 0\), so roots are \(x = 4\) and \(x=-3\).

Step2: Analyze parabola direction

The coefficient of \(x^2\) is positive (\(1>0\)), so the parabola opens upward.

Step3: Determine where \(y>0\)

For an upward - opening parabola \(y = ax^2+bx + c\) (\(a>0\)), \(y>0\) when \(x <\) the smaller root or \(x>\) the larger root. Here, smaller root is \(-3\), larger is \(4\). So \(x < - 3\) or \(x>4\).

Answer:

\(x < - 3\) or \(x>4\)