How do you solve the inequality 6x - 5 < 6/x?

1 Answer
Mar 8, 2018

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