Question

Solve for x under the assumption that x > 0. Enter your answer in interval notation using grouping symbols.

x - (10/x) < 9

I need someone to show me how to work through this type of problem, I don't understand it.

Answer 1

x - 10/x < 9
Since x>0, we can multiply by x and not change the direction of the inequality, so

x^2 - 10 < 9x
x^2 - 9x - 10 < 0
(x-10)(x+1) < 0

So, either
(A): (x-10)<0 and (x+1)>0
or
(B): (x-10)>0 and (x+1)<0

(A):
x<10 and x > -1 or -1 < x < 10

(B):
x > 10 and x < -1
can't happen

So, -1 < x < 10
But, we stipulated that x>0, so
0 < x < 10
x is in (0,10)

Or, if you remember what parabolas look like, you can note that since the parabola opens upward, (x-10)(x+1) < 0 when x is between the roots. -1 < x < 10

Answer 2

x-3/x<-2