How do you solve x/(x-3)<=-8/(x-6) using a sign chart?

1 Answer
Mar 15, 2017

The solution is x in [-6,3[ uu [4,6 [

Explanation:

We cannot do crossing over

x/(x-3)<=-8/(x-6)

We rearrange and factorise the inequality

x/(x-3)+8/(x-6)<=0

(x(x-6)+8(x-3))/((x-3)(x-6))<=0

(x^2-6x+8x-24)/((x-3)(x-6))<=0

(x^2+2x-24)/((x-3)(x-6))<=0

((x-4)(x+6))/((x-3)(x-6))<=0

Let f(x)=((x-4)(x+6))/((x-3)(x-6))

We can build the sign chart

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaa)-6color(white)(aaaaaaa)3color(white)(aaaaa)4color(white)(aaaaaaa)6color(white)(aaaa)+oo

color(white)(aaaa)x+6color(white)(aaaaa)-color(white)(aaaa)+color(white)(aaaa)||color(white)(aa)+color(white)(aa)+color(white)(aaaa)||color(white)(aa)+

color(white)(aaaa)x-3color(white)(aaaaa)-color(white)(aaaa)-color(white)(aaaa)||color(white)(aa)+color(white)(aa)+color(white)(aaaa)||color(white)(aa)+

color(white)(aaaa)x-4color(white)(aaaaa)-color(white)(aaaa)-color(white)(aaaa)||color(white)(aa)-color(white)(aa)+color(white)(aaaa)||color(white)(aa)+

color(white)(aaaa)x-6color(white)(aaaaa)-color(white)(aaaa)-color(white)(aaaa)||color(white)(aa)-color(white)(aa)-color(white)(aaaa)||color(white)(aa)+

color(white)(aaaa)f(x)color(white)(aaaaaa)+color(white)(aaaa)-color(white)(aaaa)||color(white)(aa)+color(white)(aa)-color(white)(aaaa)||color(white)(aa)+

Therefore,

f(x)<=0 when x in [-6,3[ uu [4,6 [