Question

How do you solve `x(3-x)(x-5)<=0`?

Answer 1

Solution is `0<=x<=3` or `x<=5`.

Explanation:

As `x(3-x)(x-5)<=0`, we have following options

either

(1) `x<0` - in such a case `x` and `x-5` are negative and `3-x` is positive and hence `x(3-x)(x-5)` is positive and hence `x<0` is not a solution.

(2) `0<=x<=3` - in such a case `x` and `3-x` are positive and `x-5` is negative and hence `x(3-x)(x-5)` is negative and hence `0<=x<=3` is a solution .

(3) `3< x< 5` - in such a case `x-5` and `3-x` are negative and `x` is positive and hence `x(3-x)(x-5)` is positive and hence `3< x< 5` is not a solution .

(4) `x<=5` - in such a case `x` and `x-5` are positive and `3-x` is negative and hence `x(3-x)(x-5)` is negative and hence `0<=x<=3` is a solution.

Hence solution is `0<=x<=3` or `x<=5`.

graph{x(3-x)(x-5) [-10, 10, -10, 10]}