Q:

Is the set $A =\{x: x^2 < 3x\}$ bounded, find $\sup A$ and $\inf A$ if they exist?

Is the set $A = \{x: x^2 < 3x\}$ bounded? Find $\sup A$ and $\inf A$ if they exist.

My answer is no, it is unbounded and so it has no $\sup$ and $\inf$. Am I correct?

A:

Note that$$x^2 - 3x < 0$$
$$x(x-3) < 0$$
Hence $$0<x<3$$
Note that $0$ is a lower bound. Try to find out the infimum and supremum.

A:

$$
x^2 - 3x = x(x-3) < 0
$$
So there are $3$ options.
$x<0$ then both $x$ and $x-3$ are negative so the inequality is not satisfied.
$0<x<3$ then the $x$ factor is positive but $x-3$ is negative so the inequality is satisfied.
$x>3$ then both factors are positive so the inequality is not satisfied.
$A=(0,3)$. It is bounded, has $\inf=0$ and $\sup=3$.

