The general forms for quadratic inequalities are:
ax2+ bx + c > 0
ax2+ bx + c < 0
ax2+ bx + c >= 0
ax2+ bx + c =< 0
The number line plays an important role in solving factorable quadratic inequalities.
Consider the following :
x2+ x -12 >= 0
x2+ x -12 >= 0
( x + 4 )( x - 3 ) >= 0
( x + 4 )( x - 3 ) = 0
factor and solve for x, if the inequality is equal to zero.
x + 4 = 0 or x - 3 = 0x = -4 or x = 3
The numbers -4 and 3 are the critical numbers for the inequality, which is then plotted in the number line.
After the critical numbers are plotted, the number line is divided into three intervals;
-
, -4; -4, 3 and 3, .
Now, find test numbers in each interval to find its affects on the signs of the factors, x + 4 and x - 3, also consequently, the product of these factors;
( x + 4 )( x - 3).
The diagram shows the sign of the factors for each interval and also the sign of the products of the factors for each interval.
From the number line it can be clearly seen the solution to the inequality.
x2+ x -12 >=0
\ the solution set is (-, -4] 
[ 3, ).
Consider the following:
x2-35 =< 2x
x2-2x -35 =< 0
( x + 5 )( x - 7 ) =< 0
( x + 5 )( x - 7 ) = 0
x + 5 = 0 or x - 7 = 0
x = -5 or x = 7
( x + 5 )( x - 7 ) = < 0
\ the solution set is [ -5, 7 ]
