# How do you solve #6x - 5 < 6/x#?

##### 1 Answer

#### Explanation:

Right from the start, you know that

With that in mind, multiply the left-hand side of the inequality by

#6x * x/x - 5 * x/x < 6x#

#6x^2 - 5x < 6#

Next, add

#6x^2 - 5x - 6 < color(red)(cancel(color(black)(6))) - color(red)(cancel(color(black)(6)))#

#6x^2 - 5x - 6 < 0#

To help you determine the intervals on which this quadratic function is smaller than zero, you need to first determine its root by using the *quadratic formula*

#6x^2 - 5x - 6 = 0#

#x_(1,2) = (-(-5) +- sqrt((-5)^2 - 4 * 6 * (-6)))/(2 * 6)#

#x_(1,2) = (5 +- sqrt(169))/12#

#x_(1,2) = (5 +- 13)/12 = {(x_1 = (5 + 13)/12 = 3/2), (x_2 = (5 - 13)/12 = -2/3) :}#

You can thus rewrite the quadratic as

#6(x-3/2)(x+2/3) = 0#

So, you need this expression to be *negative*, which implies that

For **and**

#{(x-3/2 < 0), (x + 2/3 > 0) :} implies (x-3/2)(x+2/3) < 0#

Any value of *positive*, and any value of *negative*. Keeping in mind that you also need