(x-3)(2-x)>0
This is NOT an 'equation;, it is an 'inequality'.
Equations have equal signs =
But inequalities do not have = signs, because the
left side does NOT equal the right side. They have
inequality signs like this: >, or <, or ≥, or ≤ .
Solve the inequality
(x-3)(2-x)>0
Find critical numbers, which are the numbers that cause
the left side to become 0. They are 3 and 2. Put
those on a number line:
-----------------o---o----------------
-2 -1 0 1 2 3 4 5 6 7
Test any number left of 2, choose x=0
Substitute it in the inequality
(x-3)(2-x) > 0
(0-3)(2-0) > 0
(-4)(2) > 0
-8 > 0
That's false, so do not shade left of 2.
We still have this:
-----------------o---o----------------
-2 -1 0 1 2 3 4 5 6 7
Test any number between 2 and 3, choose x=2.5
Substitute it in the inequality
(x-3)(2-x) > 0
(2.5-3)(2-2.5) > 0
(-0.5)(-0.5) > 0
0.28 > 0
That's true, so we shade between 2 and 3
-----------------o===o----------------
-2 -1 0 1 2 3 4 5 6 7
Test any number right of 3, choose x=4
Substitute it in the inequality
(x-3)(2-x) > 0
(4-3)(2-4) > 0
(1)(-2) > 0
-2 > 0
That's false, so do not shade right of 2.
We only have this:
-----------------o===o----------------
-2 -1 0 1 2 3 4 5 6 7
The end points 2 and 3 are not part of the solution,
since when substituted they give 0 on the left.
Therefore the solution is:
{x | 2 < x < 3} in set-builder notation or (2,3) in
interval notation.
Edwin
