Solving Polynomial Inequalities Analytically

Find the solutions of the inequality

$\begin{displaymath}x^2+5x\geq 3.\end{displaymath}$

By using the quadratic formula, we find that the equation x²+5x-3=0 has as its roots

$\begin{displaymath}x=-\frac{5\pm\sqrt{37}}{2}.\end{displaymath}$

Note that

$\begin{displaymath}-\frac{5-\sqrt{37}}{2}>0,\end{displaymath}$

while

$\begin{displaymath}-\frac{5+\sqrt{37}}{2}<0.\end{displaymath}$

The solutions are all points in the set

$\begin{displaymath}(-\infty, -\frac{5+\sqrt{37}}{2})\cup (-\frac{5-\sqrt{37}}{2},\infty).\end{displaymath}$

1998-06-12