Solving Polynomial Inequalities Analytically

Exercise 4.

Find the solutions of the inequality

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

Answer.

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}

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Helmut Knaust
1998-06-12

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