# how to solve this inequality? (5-x)³(2+x)(2x+6)² < 0

durbanville | Certified Educator

First find the critical values of x which means use each of the factors (the brackets)  and make each one = 0 because the equation has 0 on the one side. Get values for x as follows:

(5 - x)^3 = 0 so x = 5

(2 + x) = 0 so x = -2

(2x + 6)^2 = 0 so 2x = -6 so x = -3

Now to find out which of these values give you an inequality that is < 0 :

draw up a table with a number line at the top(see below) (there are various methods but this is easy to explain) where you must write the x values you just calculated(smallest to largest) and at the side write the factors (the brackets).

In this question because the (5 - x) is to the ^3 you only need to use it once (the 2 remaining brackets containing (5 - x) will cancel out any negatives because  - x -  = +

Similarly the (2x + 6) appears twice so will also cancel out any negatives so you do not need it in your table

In the table make x smaller than -3 first.

So let's use -4 and so (5 - x) becomes (5 - (-4)) = 9 a positive answer

We are NOT interested in the answer but only the symbol (= or -)so write the symbol in the space (see below)

Then do the same for (2 + x) which becomes (2 + (-4)) = -2 which is a negative answer.

Next is a number between -3 and -2 then a number between -2 and 5 and finally a number bigger than 5 and each time substitute the value into the same factors and write only the symbol not the answer.

_<___ -3__________-2__________5____>____

(5 - x)             +              +                      +                _

(2 + x)             -              -                       +               +

________________________________________

-               -                      +                -

Finally in the vertical columns check what the symbols would be if you multiplied them. Do you see that the first column was  + with  - which means the answer is  - and so on for each vertical column.

Now when you consider that your inequality is < 0 you will have an answer that reflects that using information relating to negative symbols which mean that you will get an answer <0.

Answer:               x  <-2   and / or x > 5