We have spent a good deal of time solving quadratic equations, where we found
the "points" that were the roots of the equations (and the zeros of the graphs).

Solving quadratic inequalities is very similar to solving quadratic equations,
except that we are now searching for "intervals" (or "regions") instead of specific points.

ladygraphpic

Like linear inequalities, quadratic inequalities may contain one variable or two variables.

Let's check out some examples.

Our solutions are going to be expressed in
"interval notation".
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ex2
Solve the quadratic inequality x2 - 3x - 10 > 0.
(one variable inequality)
The process of "solving a quadratic inequality" will find intervals of numbers as its result.

Number Line Process:
Let's take a look at the steps in this process.

1. If needed, move all of the terms to one side of the inequality, with zero on the other side.
This equation is all set to go.

2. Replace the inequality with an equal sign.
Solve for the roots of the equation.
Roots are -2 and 5. These root values become "critical" values to solving the inequality.

3. Set up a number line to investigate the intervals created by the equation solutions.
Pick a number in each interval and test it in the inequality. If the result is true, that interval is a solution to the inequality. There may be more than one interval which is a solution.

4. Graph the solution set on a number line.
Since this is a "strict" inequality, open circles are used. Arrows indicate where the inequality is true. State the solution in a form directed by your teacher or stated in the question.

Interval notation:


x2 - 3x - 10 > 0

quadeq

quadchartline

quadgraphchart2

Ways to State the Solution:
x < -2 or x > 5
quadnotation
quadnotation2

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hintgal
The number line that was used in Example 1 can be associated with the x-axis of the associated quadratic function.
Let's look at another approach to solving this quadratic inequality.

Associated Function Process:
Let's take a look at this alternative approach.

First, remember what we know about quadratic function graphs, y = ax2 + bx + c.
If a > 0, the graph opens up.
If a < 0, the graph opens down.

We start the same way:
1. If needed, move all of the terms to one side of the inequality, with zero on the other side.

2. Replace the inequality with an equal sign.
Solve for the roots of the equation (which are the zeros of the function).

3. Picture the parabola that is associated with this quadratic.
Since a > 0, the parabola opens up, as it passes through the two zeros.

Solution in interval notation:

x2 - 3x - 10 > 0

quadeq


This is the graph of the associated quadratic function.
It is not the graph of a quadratic inequality.

Since the graph opens upward,
the graph is positive to the
left of -2 and also to the right of 5.
The graph is negative between -2 and 5.

Solution: the quadratic is >0 when
x < -2 and x > 5

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hintgal
In Example 1, x was the only value that was being considered. But in Example 2, we will be "graphing the quadratic inequality", meaning we will be dealing with the relationship of x-values and y-values in the coordinate plane.

ex4
Graph the quadratic inequality  y < x2 - 3x - 10.
(two variables inequality)
The process of "graphing a quadratic inequality" will find a region in the coordinate plane.

While "solving the quadratic inequality" dealt with the quadratic in relation to "0", the graph of a quadratic inequality deals with the relationship of x-values to y-values.

1. Replace the inequality symbol with an equal sign, and graph the equation (a parabola). Use a dashed line for strict inequalities.

2. Choose a test point NOT on the parabola. Test the point in the inequality. If the result is true, shade the entire region where the test point lies. If the result is false, shade the entire region on the other side of the parabola.

3. Solution: The shaded area contains all of the points that make the inequality true.

quadgraph3The point (0,0) tests FALSE in this inequality.
0 < 0² - 3(0) - 10 is false.

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For
calculator help with graph
quadratic
inequalities
click here.
solve
For
calculator help
with solve
quadratic
inequalities
click here.
solve
ti84c
How to use your
TI-84+ graphing calculator  to graph quadratic inequalities.
Click calculator.
solve



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