# How do you solve #z(2 - z) < z - 12#?

##### 1 Answer

#### Answer:

#### Explanation:

First, bring everything to the left side:

... subtract

... subtract

Expand the expression on the left side and simplify:

At this point, I would recommend to multiply by

Be careful though: if you multiply by a negative factor or divide by a negative factor, you need to flip the inequality sign!

Now, let's find the roots of the quadratic expression at the left side. It can be done e.g. with the quadratic formula:

So, the roots are

Now, let's think about what the inequality

- It's a quadratic function that opens up since the coefficient of
#z^2# is positive. - Further, its roots are
#z = 4# and#z = -3# which means that the value of the function is#0# for those#z# values. - This means that the function has positive values if
#z> 4# or#z< -3# hold.

It can also help seeing this if you look at the graph:

graph{x^2 - x - 12 [-10, 10, -15, 15]}

As we have a

So, the solution is

or, depending on your notation preference,