6x - 5 < 6/x
We need to get all the xs on one side
multiply both sides by x
x(6x - 5) < 6
distribute
6x^2 - 5x < 6
Looks like a quadratic. Let's bring the 6 over and set the inequality to 0
6x^2 - 5x - 6 < 0
Now we need to factor 6x^2 - 5x - 6
color(white)(..) + 36
color(white)(..) xx 5
. . . . . . . . .
color(white)(..) 1 xx 36
color(white)(..) 2 xx 18
color(white)(..) 3 xx 12
color(white)(..) color(orange)(4) xx color(white)(0)color(orange)(9)color(white)(0) => color(orange)(-9)+color(orange)(4)
Since the leading coefficient of the expression isn't 1, we need to factor by grouping
(6x^2 + color(orange)(4)x) + (color(orange)(-9)x - 6)
2x(3x + 2) + -3(3x + 2)
(2x - 3)(3x + 2)
So now we have
(2x - 3)(3x + 2) < 0
* * * * * * * * * * * * * * * * * *
Solve for x in (2x - 3):
2x - 3 < 0
2x < 3
x < 3/2
* * * * * * * * * * * * * * * * * *
Solve for x in (3x + 2):
3x + 2 < 0
3x < -2
x < -2/3
So -2/3 < x < 3/2