Quadratic Inequalities

[Eigenvalue]

As a way to check your work for quadratic inequalities, you can graph the function and find the length that is greater than or less than 0 (the lengths would be bound by the roots, as the value of the function is 0 at the roots)

We have the inequalities (x-a)(x-b)<0 and (x-a)(x-b)>0
For consistency, let b>a
If (x-a)(x-b)<0
x-a>0, x-b<0
x>a, x<b
x-a<0 x-b>0
x<a x>b
The overlapping values are a<x<b, or (a, b)
On the graph, the values (a,b) have y-values less than 0

If (x-a)(x-b)>0
x-a>0 x-b>0
x>a x>b
x-a<0 x-b<0
Test the value x=c, c is less than b but greater than a
(c-a)(c-b)=-(c-a)(b-c)
-(c-a)(b-c) not greater than 0, as it is a negative value
The values are (- infinity, a) U (b, infinity)
On the graph, the values (- infinity, a) U (b, infinity), have y-values greater than 0

Note: This method is only feasible if the quadratic has 2 real roots as complex roots cannot be graphed on a x-y plane; to use this for any polynomial inequality, the polynomial must have all real roots

[nycmath]

As a way to check your work for quadratic inequalities, you can graph the function and find the length that is greater than or less than 0 (the lengths would be bound by the roots, as the value of the function is 0 at the roots)

We have the inequalities (x-a)(x-b)<0 and (x-a)(x-b)>0
For consistency, let b>a
If (x-a)(x-b)<0
x-a>0, x-b<0
x>a, x<b
x-a<0 x-b>0
x<a x>b
The overlapping values are a<x<b, or (a, b)
On the graph, the values (a,b) have y-values less than 0

If (x-a)(x-b)>0
x-a>0 x-b>0
x>a x>b
x-a<0 x-b<0
Test the value x=c, c is less than b but greater than a
(c-a)(c-b)=-(c-a)(b-c)
-(c-a)(b-c) not greater than 0, as it is a negative value
The values are (- infinity, a) U (b, infinity)
On the graph, the values (- infinity, a) U (b, infinity), have y-values greater than 0

Note: This method is only feasible if the quadratic has 2 real roots as complex roots cannot be graphed on a x-y plane; to use this for any polynomial inequality, the polynomial must have all real roots

A wonderful, professional message. It is now in my math files.