Question

which graph represents the solution for $x^{2}+x - 12>0$?

Explanation:

Step1: Factor the quadratic inequality

Factor $x^{2}+x - 12$ to get $(x + 4)(x-3)>0$.

Step2: Find the roots

Set $(x + 4)(x - 3)=0$. The roots are $x=-4$ and $x = 3$.

Step3: Test intervals

Test the intervals $x<-4$, $-43$. For $x<-4$, let $x=-5$, then $(-5 + 4)(-5-3)=(-1)\times(-8)=8>0$. For $-43$, let $x=4$, then $(4 + 4)(4-3)=8\times1=8>0$.

Answer:

The solution is $x<-4$ or $x>3$, which is represented by a number - line with open - circles at $x=-4$ and $x = 3$ and shading to the left of $x=-4$ and to the right of $x = 3$.