Find all values of x for which f(x) satisfies the indicated conditions: f(x)= x^2-5x, f(x)<0

Answer 1

x^2 -5x < 0

x(x-5) < 0

x and x-5 must be opposite signs

either x<0 and x-5 >0 or x>0 and x-5 <0

for x-5>0, x>5 which is inconsistent with x<0

for x-5<0, x<5 and x>0

only solutions are 0<x<5

or

x is in the interval (0,5)

You could have just set x^2-5x=0

and solved for x

x(x-5) =0

x =0 or 5

0 and 5 are the end points of the solution (0,5)

just check a few x values

x = 2, then (2)(2-5) = -6 <0

x= 6, then 6(6-5) = 6>0

x =-1 then -1(-1-5) = 6>0

only 0<x<5 satisfies the inequality