Answer
$(-\infty,\frac{3-\sqrt {29}}{2})\cup(\frac{3+\sqrt {29}}{2},\infty)$.
Work Step by Step
Step 1. Rewrite and factor the inequality to get $x^2-3x-5\geq0\longrightarrow (x-\frac{3+\sqrt {29}}{2})(x-\frac{3-\sqrt {29}}{2})\geq0$
Step 2. Identify the boundary points $x_1=\frac{3-\sqrt {29}}{2}\approx-1.2$, $x_2=\frac{3+\sqrt {29}}{2}\approx4.2$ and separate the number line into three intervals $(-\infty,x_1)$, $(x_1,x_2)$ and $(x_2,\infty)$
Step 3. Use test points $x=-2,0,5$ to get the signs of the left side of the inequality as $+,-,+$
Step 4. Based on the signs and consider the boundary points, we have the solution as $(-\infty,\frac{3-\sqrt {29}}{2})\cup(\frac{3+\sqrt {29}}{2},\infty)$.
