Quadratic Inequalities
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Quadratic inequalities can be solved graphically or algebraically.

Solve Graphically:

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:
x < -3  or  x > 4

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 x-axis.
 

 

 

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.

 

 

Solve Algebraically:

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
TI-83+/84+ graphing calculator  with quadratic inequalities.
Click calculator.