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x^{2}-8x-9=0
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\left(-9\right)}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -8 for b, and -9 for c in the quadratic formula.
x=\frac{8±10}{2}
Do the calculations.
x=9 x=-1
Solve the equation x=\frac{8±10}{2} when ± is plus and when ± is minus.
\left(x-9\right)\left(x+1\right)>0
Rewrite the inequality by using the obtained solutions.
x-9<0 x+1<0
For the product to be positive, x-9 and x+1 have to be both negative or both positive. Consider the case when x-9 and x+1 are both negative.
x<-1
The solution satisfying both inequalities is x<-1.
x+1>0 x-9>0
Consider the case when x-9 and x+1 are both positive.
x>9
The solution satisfying both inequalities is x>9.
x<-1\text{; }x>9
The final solution is the union of the obtained solutions.