Question

use a graph to solve the inequality: ( x^2 - x - 12 > 0 ) to solve the inequality, identify the values of ( x ) that make the value of the expression ( x^2 - x - 12 ) greater than 0. graph the equation ( y = x^2 - x - 12 ) by plotting points. complete the following table.

( x )	( -2 )	( -1 )	( 0 )	( 1 )	( 2 )

use the points to sketch the right. to solve the inequality, dropdown for value of the function is: positive, negative, positive or 0, negative or 0 where the correct graph on

Explanation:

Step1: Analyze the function type

The function \( y = x^2 - x - 12 \) is a quadratic function, and its graph is a parabola. For a quadratic function \( y = ax^2+bx + c \) (\( a = 1>0 \) here), the parabola opens upward.

Step2: Find the roots of the equation \( x^2 - x - 12=0 \)

We can factor the quadratic expression: \( x^2 - x - 12=(x - 4)(x + 3) \). Setting it equal to zero, we get \( (x - 4)(x + 3)=0 \), so the roots are \( x = 4 \) and \( x=- 3 \).

Step3: Analyze the inequality \( x^2 - x - 12>0 \)

Since the parabola opens upward (because \( a = 1>0 \)), the function \( y=x^2 - x - 12 \) is greater than 0 when \( x \) is less than the smaller root or greater than the larger root. The smaller root is \( x=-3 \) and the larger root is \( x = 4 \). So the solution to the inequality \( x^2 - x - 12>0 \) is \( x < - 3 \) or \( x>4 \).

(If we use the table to sketch the graph: when \( x=-2 \), \( y=-6 \); \( x = - 1 \), \( y=-10 \); \( x = 0 \), \( y=-12 \); \( x = 1 \), \( y=-12 \); \( x = 2 \), \( y=-10 \). As \( x \) moves away from the vertex (the vertex of \( y=x^2 - x - 12 \) has \( x=-\frac{b}{2a}=\frac{1}{2} \), and \( y = (\frac{1}{2})^2-\frac{1}{2}-12=\frac{1}{4}-\frac{1}{2}-12=-\frac{1}{4}-12=-\frac{49}{4}\)), when \( x < - 3 \) or \( x>4 \), the value of \( y=x^2 - x - 12 \) will be positive (greater than 0).)

Answer:

The solution to the inequality \( x^2 - x - 12>0 \) is \( x < - 3 \) or \( x>4 \)