Solve : x^2 - 4x + 3 < 0
I need to graph the solution also
(1) FactorizeOriginally Posted by Brian D
x^2 - 4x + 3 = (x-3)*(x-1)
(2) Think, when is (x-3)*(x-1) negative (less than zero)? When each parenthetical group has a different sign.
Possibilities:
(a) x<3 making (x-3) negative AND x>1 making (x-1) positive
(b) x>3 making (x-3) positive AND x<1 making (x-1) negative
Therefore, 1<x<3
On a number line, open circles at 1 and 3 and connect them.
The simplest way to solve "complicated" inequalities is to solve the EQUATION. Where something is EQUAL to 0 separates where it is positive from where it is negative! (If your inequality involves fractions, you want to check points where the denominator is 0 also.)
x^2- 4x+ 3= (x- 3)(x- 1)= 0 when x= 3 or x= 1.
Now check a value in each of the three intervals x< 1, 1< x< 3, 3< x.
0 is less than 1 and 0^2- 4(0)+ 3= 3 which is positive.
2 is between 1 and 3 and 2^2- 4(2)+ 3= 4- 8+ 3= -1 which is negative
4 is larger than 3 and 4^2- 4(4)+ 3= 16- 16+ 3= 3 which is positive.
We now know that x^2- 4x+ 3 is negative for ALL numbers between 1 and 3 and negative for ALL numbers less than 1 or larger than 3.