How do you solve the inequality 5+1/x>16/x5+1x>16x?

1 Answer
Jan 18, 2017

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

Explanation:

Let's rewrite the inequality

5+1/x-16/x>05+1x16x>0

5-15/x>0515x>0

(5(x-3))/x>05(x3)x>0

Let f(x)=(5(x-3))/xf(x)=5(x3)x

The domain of f(x)f(x) is D_f(x)=RR-{0}

Now we can do a sign chart

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

color(white)(aaaa)xcolor(white)(aaaaaaaa)-color(white)(aa)color(white)(aa)+color(white)(aaaa)+

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

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

Therefore,

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