Solve the inequality x(x − 6) > 0.

So, I got the answers x >6 or x >0, but that doesn't really make sense.
Can anyone show me how to do it properly?

x(x-6)>0 means that either x and x-6 are both positive or x and x-6 are both negative.

x>0 and x-6>0 ==> x>0 and x>6 ==> x>6

x<0 and x-6<0 ==> x<0 and x<6==> x<0

So the answer is x<0 or x>6
or
(-∞, 0) U (6, ∞)

let f(x) be a general quadratic with x1 and x2 as roots, with x1 < x2


then the sign of f(x) is as follows:

1. in between x1 and x2, f(x) is >0 and elsewhere is <0

2. or the other way around: negative in between x1 and x2 and >0 elsewhere


do determine which of the 2, find the sign of f(x3), with x3 diff from x1 and x2

1. if x1 < x3 < x2 => then in between roots f(x) takes the sign of f(x3), and the opposite elsewhere

2. otherwise (x3 < x1 and x3 > x2) => f(x) takes the OPPOSITE sign of f(x3) in between roots and the actual one elsewhere


your case, f(x) = x(x - 6) and roots are x = 0 and x = 6


let's find the sign of f:

take x = 1 => f(1) = 1(1 - 6) = -5 < 0


hence, f(x) is <0 in between 0 and 6

and >0 elsewhere


in our case, f(x) needs to be positive


hence, x ∈ (-infinity, 0] ∪ [6, inifinity)

X(X-6) > 0

That is the same as (X) x (X-6) > 0

Now inorder to get 0 as an answer in a multiplication one of the values needs to be 0 (Treat the > like a = for now)

Therefor either X > 0 or X-6 > 0 which simplfies to X > 6

Thats all there is to it.

A product is positive if either both factors are positive or both are negative.

The factors here are x and x-6.

So either x>0 AND x-6>0 => x>6 which simplifies to x>6,

or x<0 AND x-6<0 => x<6 which simplifies to x<0

So the solution is x<0 or x>6

Solve this quadratic inequality by completing the square:
x(x - 6) > 0
x² - 6x > 0
(x - 3)² - 9 > 0
(x - 3)² > 9
x - 3 < -3 OR x - 3 > 3
x < 0 OR x > 6

x(x - 6) > 0

If the product of two factors is positive, then both factors are positive

x > 0

or

x - 6 > 0
x > 6

(0 < x < ∞) U (6 < x < ∞)
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

X(x-6)>0
the solution become:
x>0 &x-6>0
x>0+6
x>6 &x>0

First multiply the insides of the brackets.
x(x-6) = xx-6x
Now add x6 to both sides.
xx-6x+6x > 0+6x is the same as
xx > 6x
And divide both sides by x.
xx/x > 6x/x is the same as
x > 6

x(x − 6) > 0
Either x > 6 OR x < 0 ANSWER

I can't

x>6 and x<0