How do you solve (x^2-x-12)/(x^2+4)<=0 using a sign chart?

1 Answer
Jun 30, 2018

The solution is x in [-3,4]

Explanation:

Factorise the numerator

x^2-x-12=(x+3)(x-4)

Therefore,

(x^2-x-12)/(x^2+4)<=0

<=>, ((x+3)(x-4))/(x^2+4)<=0

AA x in RR, (x^2+4)>0

Let f(x)=((x+3)(x-4))/(x^2+4)

Let's build the sign chart

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaa)-3color(white)(aaaaaaa)4color(white)(aaaaa)+oo

color(white)(aaaa)x+3color(white)(aaaaaa)-color(white)(aa)0color(white)(aaa)+color(white)(aaaaaa)+

color(white)(aaaa)x-4color(white)(aaaaaa)-color(white)(aa)#color(white)(aaaa)-#color(white)(aa)0color(white)(aaa)+

color(white)(aaaa)f(x)color(white)(aaaaaaa)+color(white)(aa)0color(white)(aaa)-color(white)(aa)0color(white)(aaa)+

Therefore,

f(x)<=0 when x in [-3,4]

graph{(x^2-x-12)/(x^2+4) [-10, 10, -5, 5]}