Solve the inequality: x^2 - 3x < 0 . How is the answer 0 < x < 3?

x^2<3x if x cannot be zero, then

x<3
now if x had been negative, the equality x^2<3x when divideing by x would have been x>3, but that is not possible for a negative number to be >3, so 0<x<3

x (x-3) is supposed to be negative

if x >3 it is positive
if x <0 it is positive

it is only positive * negative = negative between 0 and 3

To solve the inequality \(x^2 - 3x < 0\), we need to find the values of \(x\) that make the inequality true.

Step 1: Factorize the inequality
Let's factorize the left side of the inequality:
\(x^2 - 3x < 0\) can be written as \(x(x - 3) < 0\).

Step 2: Determine the critical points
To solve this inequality, we need to find the critical points. These points are where the expression \(x(x - 3)\) is equal to zero.

Setting \(x(x - 3) = 0\) gives us two critical points:
\(x = 0\) and \(x - 3 = 0 \Rightarrow x = 3\).

Step 3: Plot the critical points on a number line
Draw a number line and mark the critical points on it. We have \(x = 0\) and \(x = 3\).

----0----3----

Step 4: Test intervals
Now, we need to test the intervals between these critical points to determine whether the inequality is true or false.

Test the interval \(x < 0\):
Choose any value \(x\) less than \(0\), for example, \(x = -1\), and substitute it into \(x(x - 3)\):
\((-1)(-1 - 3) = 4 > 0\).
Since this expression is positive, it does not satisfy the inequality \(x(x - 3) < 0\).

Test the interval \(0 < x < 3\):
Choose any value \(x\) between \(0\) and \(3\), for example, \(x = 2\), and substitute it into \(x(x - 3)\):
\((2)(2 - 3) = -2 < 0\).
Since this expression is negative, it satisfies the inequality \(x(x - 3) < 0\).

Test the interval \(x > 3\):
Choose any value \(x\) greater than \(3\), for example, \(x = 4\), and substitute it into \(x(x - 3)\):
\((4)(4 - 3) = 4 > 0\).
Since this expression is positive, it does not satisfy the inequality \(x(x - 3) < 0\).

Step 5: Determine the solution
From the tests, we see that the inequality is true only when \(0 < x < 3\). Therefore, the solution to the inequality \(x^2 - 3x < 0\) is \(0 < x < 3\).