Download Non-linear inequalities

Non-linear inequalities
Quadratic inequalities are solved in a similar way to quadratic equations. At the
final stage, we can study the signs of the factors and decide which real numbers fulfill
the condition of making the quadratic positive or negative.
Inequalities with rational expressions are to be solved in the same way after
factorising both numerator and denominator.
Examples:
formula)
x  1
 x  2
x  1x  2
x 2  3x  2  0
x 2  3x  2
the roots of
are 1 and 3
(use the quadratic
so we can write x  1x  2  0
we must find those values of “x” that make the product negative
when x<1
when x=1
when 1<x<2 when x=2
when 2<x
NEGATIVE ZERO
POSITIVE
POSITIVE POSITIVE
NEGATIVE NEGATIVE NEGATIVE ZERO
POSITIVE
POSITIVE
ZERO
NEGATIVE ZERO
POSITIVE
so those values are: numbers smaller than 1 and numbers greater
than 2 as well
we can say it more concisely (-,1)U(2,)
x3
0
2x
we must find those values of “x” that make the quotient positive
or zero
x  3
2  x
x3
2x
when x < –
3
NEGATIVE
POSITIVE
NEGATIVE
when x = –
3
ZERO
POSITIVE
ZERO
when –
3<x<2
POSITIVE
POSITIVE
POSITIVE
when x=2
when 2<x
POSITIVE POSITIVE
ZERO
NEGATIVE
does not
NEGATIVE
exist
so those values are: number – 3 and numbers between – 3 and 2
as well
we can say it more concisely [– 3 , 2)
In quadratic inequalities, a sketch graph is often helpful at the final stage.