Answer:
[tex]0<x<2[/tex]
Step-by-step explanation:
Hi there!
[tex]x(x-2)<0[/tex]
Set x(x-2) equal to 0:
[tex]x(x-2)=0[/tex]
Solve for x by using the zero product property:
[tex]x(x-2)=0[/tex]
[tex]x=0[/tex] and [tex]x=2[/tex]
Now, we know that for the inequality [tex]x(x-2)<0[/tex], one of the following must be true:
1) [tex]0<x<2[/tex]
2) [tex]x<0[/tex] or [tex]x>2[/tex]
To find out which one it is, 1) or 2), we can substitute a value into [tex]x(x-2)<0[/tex] to see if the inequality is true.
For example, 1 takes place in between 0 and 2. If 1 satisfies [tex]x(x-2)<0[/tex], then [tex]0<x<2[/tex] must be true. If it does not, then [tex]x<0[/tex] or [tex]x>2[/tex] must be true:
[tex]1(1-2)<0\\1(-1)<0\\-1<0[/tex]
This inequality is true. Thus, [tex]0<x<2[/tex].
I hope this helps!
