Download Quadratic Inequalities

Quadratic Inequalities
Today we will learn to solve
quadratic inequalities using
algebra and graphing.
Quadratics
Before we get started let’s review.
A quadratic equation is an equation that can
2
be written in the form ax  bx  c  0,
where a, b and c are real numbers and a cannot equal
zero.
In this lesson we are going to graph & solve quadratic
inequalities.
Do Now: Graph # 38 on page 237
Quadratic Inequalities
What do they look like?
Here are some examples:
x 2  3x  7  0
3x  4 x  4  0
2
x  16
2
Quadratic Inequalities
When solving inequalities we are trying to
find all possible values of the variable
which will make the inequality true.
Consider the inequality
x2  x  6  0
We are trying to find all the values of x for which the
quadratic is greater than zero or positive.
Solving a quadratic inequality
We can find the values where the quadratic equals zero
2
by solving the equation, x  x  6  0
x  3x  2  0
x  3  0 or x  2  0
x  3 or x  2
Solving a quadratic inequality
You may recall the graph of a quadratic function is a
parabola and the values we just found are the zeros
or x-intercepts.
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The graph of y  x  x  6 is
Solving a quadratic inequality
From the graph we can see that in the intervals
around the zeros, the graph is either above the x-axis
(positive) or below the x-axis (negative). So we can
see from the graph the interval or intervals where the
inequality is positive. But how can we find this out
without graphing the quadratic?
We can simply test the intervals around the zeros in
the quadratic inequality and determine which make
the inequality true.
Solving a quadratic inequality
For the quadratic inequality, x 2  x  6  0
we found zeros 3 and –2 by solving the equation
. Put these values on a number line and
x2  x  6  0
we can see three intervals that we will test in the
inequality. We will test one value from each interval.
-2
3
Solving a quadratic inequality
Interval
 ,2
Test Point
Evaluate in the inequality
True/False
x2  x  6  0
x  3
 32   3  6  9  3  6  6  0 True
x2  x  6  0
 2, 3
 3, 
x0
0  0  6 0  0  6  6  0
2
False
x2  x  6  0
x4
42  4  6  16  4  6  6  0
True
Solving a quadratic inequality
Thus the intervals  ,2 or 3, 
make up the
solution set for the quadratic inequality, x 2  x  6  0 .
In summary, one way to solve quadratic inequalities
is to find the zeros and test a value from each of the
intervals surrounding the zeros to determine which
intervals make the inequality true.
Example 2:
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Solve 2 x  3x  1  0
2
First find the zeros by solving the equation, 2 x  3x  1  0
2 x 2  3x  1  0
2x 1x 1  0
2 x  1  0 or x  1  0
1
x  or x  1
2
Example 2:
Now consider the intervals around the zeros and test
a value from each interval in the inequality.
The intervals can be seen by putting the zeros on a
number line.
1/2
1
Example 2:
Interval
Test Point
Evaluate in Inequality
True/False
2 x 2  3x  1  0
1

  , 
2

x0
2 0  3 0 1  0  0 1  1  0
2
False
2 x 2  3x  1  0
1 
 ,1
2 
9 9
1
3 3
2   3   1    1   0
8 4
8
4 4
2
3
x
4
True
2 x 2  3x  1  0
1, 
x2
2 2  3 2 1  8  6 1  3  0
2
False
Example 2:
Thus the interval
the inequality
1 
 ,1
2 
makes up the solution set for
2 x 2  3x  1  0 .
Example 3:
Solve the inequality  2 x 2  x  1
.
First find the zeros.
 2 x 2  x  1 or  2 x 2  x  1  0
Since this quadratic doesn' t factor, use the quadratic formula .
 1   1  4 2 1
1  7
x

2 2
4
2
1 i 7

4
Example 3:
But these zeros x 
1 mi 7
, are complex numbers.
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What does this mean?
2
Let’s look at the graph of the quadratic, y  2 x  x  1
Example 3:
We can see from the graph of the quadratic that the
curve never intersects the x-axis and the parabola is
entirely below the x-axis. Thus the inequality is
always true.
Example 3:
How would you get the answer without the graph?
The complex zeros tell us that there are no REAL
zeros, so the parabola is entirely above or below the
x-axis. At this point you can test any number in the
inequality, If it is true, then the inequality is always
true. If it is false, then the inequality is always false.
We can also determine whether the parabola opens
up or down by the leading coefficient and this will tell
us if the parabola is above or below the x-axis.
Summary
In general, when solving quadratic inequalities
1. Find the zeros by solving the equation you get
when you replace the inequality symbol with an
equals.
2. Find the intervals around the zeros using a number
line and test a value from each interval in the
number line.
3. The solution is the interval or intervals which make
the inequality true.
Practice Problems
x 2  5 x  24  0
12  x  x 2  0
3x 2  5 x  2  0
5 x 2  13x  6  0
9  x2  0
2 x 2  5x  1  0
16 x 2  1  0
x 2  5 x  4
3x 2  2 x  1  0
x2  2x  4