Question

What values of 'x' will be the solution to the inequality `15x - 2/x > 1`?

Answer 1

The answer is `x in (-1/3;0) uu (2/5;+oo)`

Explanation:

We start with the inequality `15x-2/x>1`

First step in solving such inequalities is to determine the domain. We can write that the domain is: `D=RR-{0}` (all real numbers different from zero).

Next step in solving such (in)equalities is to move all terms to the left side leaving zero on the right side:

`15x-2/x-1>0`

Now we should write all terms as fractions with comon denominator:

`(15x^2)/x-2/x-x/x>0`

`(15x^2-x-2)/x>0`

Now we have to find zeros of the numerator. To do this we have to calculate the determinant:

`Delta=1-4*15*(-2)=1+120=121`

`sqrt(Delta)=11`

`x_1=(1-11)/(2*15)=-10/30=-1/3`

`x_2=(1+11)/(2*15)=12/30=-2/5`

Now we have to sketch the function to find intervals where the values are greater than zero:

graph{x(x+1/3)(x-2/5) [-0.556, 0.556, -0.1, 0.1]}

From this graph we can clearly see the siolution:

`x in (-1/3;0) uu (2/5;+oo)`