Question

find range of x such that (2x-1)(x+3)(2-x)(1-x)²/x⁴(x+6)(x-9)(2x²+4x+9) < 0

Answer 1

We have to find the range of x such that

(2x - 1)(x + 3)(2 - x)(1 - x)²/x⁴(x + 6)(x - 9)(2x² + 4x + 9) < 0

Solution : step 1 : equate each terms to zero and find values of x

(2x - 1) = 0 ⇒x = 1/2

(x + 3) = 0 ⇒x = -to3

(2 - x) = 0 ⇒x = 2

(1 - x) = 0 ⇒x = 1

x⁴ = 0 ⇒x = 0

(x + 6) = 0 ⇒x = - 6

(x - 9) = 0 ⇒x = 9

(2x² + 4x + 9) > 0 because D = (4)² - 4 × 2 × 9 < 0

step 2 : now put all these values of x in number line [ See figure ]

Now take any number grater than 9.

let's take x = 10

put it into inequality to check sign of it.

(+)(+)(-)(+)/(+)(+)(+)(+) = (-)

so in range 9 to ∞ , put negative.

similarly take a number between 2 to 9.

Let's take x = 5

now (+)(+)(-)(+)/(+)(+)(-)(+) = (+)

so, put positive sign in 2 to 9.

Similarly check all intervals and put the signs that you have found.

you will get,

-∞ < x < -6 ⇒+ve

-6 < x < -3 ⇒-ve

-3 < x < 0⇒+ve

0 < x < 1/2 ⇒+ve

1/2 < x < 1 ⇒-ve

1 < x < 2 ⇒-ve

2 < x < 9 ⇒+ve

9 < x < ∞ ⇒-ve

Now we have to choose the intervals in which we get negative sign because sign of inequality is less than zero.

so, -6 < x < -3, 1/2 < x < 1 , 1 < x < 2 and x > 9 are the intervals in which inequality satisfies its condition.

So the range of x is (-6,-3) U (1/2, 1) U (1, 2) U (9, ∞)