Question

use a graph to solve the inequality.\\(x^2 - x - 12 > 0\\)\\(\dots\\)to solve the inequality, identify the values of \\(x\\) that make the value of the expression \\(x^2 - x - 12\\) \\(\boldsymbol{\bigtriangledown}\\) 0.\\(\text{less than}\\)\\(\text{greater than}\\)\\(\text{less than or equal to}\\)\\(\text{greater than or equal to}\\)\\(\dots\\)\\(\text{(graph of a coordinate plane with grid lines)}\\)

Explanation:

Step1: Analyze the quadratic function

The inequality is \(x^{2}-x - 12>0\). First, consider the quadratic function \(y=x^{2}-x - 12\). To find the roots, set \(y = 0\), so \(x^{2}-x - 12=0\). Factor the quadratic: \(x^{2}-x - 12=(x - 4)(x+ 3)=0\). Thus, the roots are \(x = 4\) and \(x=-3\).

Step2: Determine the graph's shape

Since the coefficient of \(x^{2}\) is positive (\(a = 1>0\)), the parabola opens upwards. The graph of \(y=x^{2}-x - 12\) is a parabola opening upwards with x - intercepts at \(x=-3\) and \(x = 4\).

Step3: Solve the inequality \(y>0\)

For a parabola opening upwards, the function \(y=x^{2}-x - 12\) is greater than 0 when \(x<-3\) or \(x>4\) (because the graph is above the x - axis outside the interval between the two roots). But the question here is about the values of x that make the value of the expression \(x^{2}-x - 12\) greater than 0 (since the inequality is \(x^{2}-x - 12>0\)). Wait, the dropdown is about the relationship for the roots? Wait, no, the original problem's dropdown: "To solve the inequality, identify the values of x that make the value of the expression \(x^{2}-x - 12\) [dropdown] 0". Wait, the inequality is \(x^{2}-x - 12>0\), so we need the values where the expression is greater than 0. But the dropdown options are "less than", "greater than", "less than or equal to", "greater than or equal to". Wait, maybe a mis - read. Wait, the inequality is \(x^{2}-x - 12>0\), so we first find where \(x^{2}-x - 12 = 0\) (the roots), then see where it's greater than 0. But the dropdown is to choose the relation for the roots? Wait, no, the problem says "To solve the inequality, identify the values of x that make the value of the expression \(x^{2}-x - 12\) [dropdown] 0". Wait, maybe it's a typo, but if we consider the process: we first find the roots (where the expression is equal to 0), then determine where it's greater than 0. But the dropdown is for the relation. Wait, the inequality is \(x^{2}-x - 12>0\), so we need the values of x where the expression is greater than 0. But the dropdown options: if we are talking about the roots, we find where the expression is equal to 0, then for the inequality \(>0\), the solution is outside the roots. But the question's dropdown: the correct relation for the inequality \(x^{2}-x - 12>0\) is that we want the values where the expression is greater than 0. But the options are "less than", "greater than", "less than or equal to", "greater than or equal to". So the answer for the dropdown (to solve \(x^{2}-x - 12>0\), we first find where the expression is equal to 0, then see where it's greater than 0. But the question is "identify the values of x that make the value of the expression \(x^{2}-x - 12\) [dropdown] 0" – wait, maybe it's a mistake, and it should be "greater than" 0. So the correct option from the dropdown is "greater than".

Answer:

greater than