Need Some Help in Inequalities?

You need to know when (x - 5) * (2 - x) * (2 - 3x) is positive. First, we find when it equals 0:

(x - 5) * (2 - x) * (2 - 3x) = 0

x = 5

x = 2

x = 2/3

So we have four domains to look at:

-infinity < x < 2/3

2/3 < x < 2

2 < x < 5

5 < x < infinity

Now we pick values from each domain and see if it's greater than 0 or less than 0:

x = 0

(0 - 5) * (2 - 0) * (2 - 3 * 0) =

(-5) * (2) * (2) =

-20

x = 1

(1 - 5) * (2 - 1) * (2 - 3 * 1) =

(-4) * (1) * (2 - 3) =

(-4) * (-1) =

4

x = 3

(3 - 5) * (2 - 3) * (2 - 3 * 3) =

(-2) * (-1) * (-7) =

-14

x = 6

(6 - 5) * (2 - 6) * (2 - 18) =

1 * (-4) * (-16) =

64

So, our solution is:

2/3 < x < 2

and

5 < x < infinity

That's when f(x) > 0

Find zeroes. Find the values of numbers between each of the zeroes and outside them. If a value is higher than zero, that means that between the two zeroes between which the value was taken (or from the outer zero to plus or minus infinity), y > 0.

Zeroes are at x=5, x=2 and x=2/3 (A*B=0 => A=0 or B=0)

So we look for the value at x=0, x=1, x=3 and x=10.

for x=0: (-5)(2)(2) = -20 < 0, therefore the statement is false for < <-, 2/3]

for x=1: (-4)(1)(-1) = 4 > 0, therefore the statement is true for <2/3 , 2>

for x=3: (-2)(-1)(-7) = -14 < 0, therefore the statement is false for [2, 5]

for x=10: (5)(-8)(-28) = 1120 > 0, therefore the statement is true for <5, -> >

So the answer is

2/3 < x < 2 OR x > 5

this is saying that in order to satisfy this equation, "x' cant be a number that will make the equation 0 or less than 0 (-).

so in this case x cant equal 5, 2, or 2/3

because if it was one of these answers it will make the equation equal zero which it states clearly that the equation is greater than zero

(x-5) (2-x) (2-3x) > 0

the expresion is zero if x = 5, x=2 and x= 2/3

if x= 0 the expression negatip

************2/3**************** 2 *****************5 ****

-------------- 0 +++++++++++ 0 ------------------- 0 +++

the solution : x {x| x> 5, 2/3 < x < 2}