How do solve (x^2+5x)/(x-3)>=0 and write the answer as a inequality and interval notation?

1 Answer

The answer is x in [-5,0 ] uu ]3, +oo[

Explanation:

Let f(x)=(x^2+5x)/(x-3)=(x(x+5))/(x-3)

The domain of f(x) is D_(f(x))=RR-{3}

Let's do a sign chart to solve this inequality

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaa)-5color(white)(aaaa)0color(white)(aaaa)3color(white)(aaaa)+oo

color(white)(aaaa)x+5color(white)(aaaaa)-color(white)(aaaa)+color(white)(aaa)+color(white)(aaa)+

color(white)(aaaa)xcolor(white)(aaaaaaaaa)-color(white)(aaaa)-color(white)(aa)+color(white)(aaa)+

color(white)(aaaa)x-3color(white)(aaaaa)-color(white)(aaaa)-color(white)(aaa)-color(white)(aaa)+

color(white)(aaaa)f(x)color(white)(aaaaaa)-color(white)(aaaa)+color(white)(aaa)-color(white)(aaa)+

Therefore f(x)>=0 when x in [-5,0 ] uu ]3, +oo[

graph{(x^2+5x)/(x-3) [-72.6, 75.47, -19.23, 54.86]}