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