Question

solve the following inequality
70x-51<70/x

I get this--but I do not know how to write the final answer in a solution set

70x-51<70/x

70x^2/x - 51x /x - 70/x < 0

70x^2 - 51x - 70/x <0

(10x + 7) (7x - 10)/x <0

x=0
(10x + 7)= 0
(7x - 10)=0

Can you please help me to write the answer in a solution set???

Answer 1

when you get to this point:

70x^2/x - 51x /x - 70/x < 0

multiply by x to get

70x^2 - 51x - 70 < 0 and note that x≠0
since 1/x is not defined for x=0

now you have

(10x+7)(7x-10) < 0

At this point you should have some idea what you are looking for. You know it is a parabola which opens upward, and crosses the x-axis at -7/10 and 10/7.

So, the interval between the roots satisfies the original inequality:

-7/10 < x < 10/7

Algebraically, since

(10x+7)(7x-10) < 0, either

(10x+7) < 0 and (7x-10) > 0
or
(10x+7) > 0 and (7x-10) < 0

solve those and you will find a solution set which agrees to the interval above.

Answer 2

oops. because x cannot be 0, the final solution is

(-7/10,0) U (0,10/7)