Quadratic inequalities can be solved graphically or algebraically.
The graph of an inequality is the
collection of all solutions of the inequality.
Example 1 (one variable inequality):

The trick to solving a quadratic
inequality is to replace the inequality symbol with an equal sign
and solve the resulting equation. The solutions to the
equation will allow you to establish intervals that will let you
solve the inequality.
Plot the solutions on a number line
creating the intervals for investigation. Pick a number from
each interval and test it in the original inequality. If the
result is true, that interval is a solution to the inequality.
Change the inequality to =
and solve:
The solutions are 4 and 3.
(ONLY these values are placed on the number line to
create the intervals.) 
Prepare the number line:

Answer:


Answer in
interval notation:


This problem could also
be solved by examining the corresponding graph of
. Graph the quadratic
(parabola) by hand or with the use of a graphing calculator.

The quadratic is greater
than zero where the graph is ABOVE the xaxis.


Example 2 (two variable inequality):

Begin by graphing the corresponding
equation .
(Use a dashed line for < or > and a
solid line for < or >.)
Test a point above the parabola and
a point below the parabola into the original inequality. Shade
the entire region where the test point yields a true result.

The parabola graph was
drawn using a solid line since the inequality was
"greater than or equal to".
The point (0,0) was
tested into the inequality and found to be
true.
The point (0,2) was
tested into the inequality and found to be
false.
The graph was shaded in
the region where the true test point was located.
ANSWER: The shaded area (including the
solid line of the parabola) contains all of
the points that make this inequality true. 
Solving quadratic inequalities
algebraically can be somewhat of a challenge. Be careful to
consider all of your options.
When you solved quadratic equations,
you created factors whose product was zero, implying either one or both
of the factors must be equal to zero.
When solving a quadratic inequality,
you need to take more options into consideration.
Consider these two different problems:
Problem 1:
"less than"
Now, there are two ways this product could be
less than zero (negative)  (x + 4) < 0 and (x
+ 3) > 0 or
(x + 4) > 0 and (x + 3) < 0. One factor
must be negative and one must be positive.First
situation:
This tells you that 3 < x < 4.
Second situation:
There are NO values for which this situation is true.
Final answer:
3 < x < 4. 
Problem 2: "greater than"
Now, there are two ways that this product could be greater
than zero (positive)  both factors are positive or both
factors are negative. You must check out both
possibilities.Both positive:
The only condition that makes both true is x > 4.
Both negative:
The only condition that makes both true is x < 3.
Final answer:
x < 3 or x > 4 

How to use
your
TI83+/84+ graphing calculator with
quadratic inequalities.
Click calculator. 

