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Quadratic Inequalities

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Title: Quadratic Inequalities


1
Quadratic Inequalities
  • Tidewater Community College
  • Karen Overman

2
Quadratics
  • Before we get started lets review.
  • A quadratic equation is an equation that can
  • be written in the form ,
  • where a, b and c are real numbers and a cannot
    equal
  • zero.
  • In this lesson we are going to discuss quadratic
  • inequalities.

3
Quadratic Inequalities
  • What do they look like?
  • Here are some examples

4
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
  • We are trying to find all the values of x for
    which the
  • quadratic is greater than zero or positive.

5
Solving a quadratic inequality
  • We can find the values where the quadratic equals
    zero
  • by solving the equation,

6
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.
  • The graph of is

7
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.

8
Solving a quadratic inequality
  • For the quadratic inequality,
  • we found zeros 3 and 2 by solving the equation
  • . Put these values on a number line and we can
    see three intervals that we will test in the
    inequality. We will test one value from each
    interval.

-2
3
9
Solving a quadratic inequality
10
Solving a quadratic inequality
  • Thus the intervals make up the solution set
    for the quadratic inequality, .
  • 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.

11
Example 2
  • Solve
  • First find the zeros by solving the equation,

12
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.

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1
13
Example 2
14
Example 2
  • Thus the interval makes up the solution set
    for
  • the inequality .

15
Example 3
  • Solve the inequality .
  • First find the zeros.

16
Example 3
  • But these zeros , are complex numbers.
  • What does this mean?
  • Lets look at the graph of the quadratic,

17
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.

18
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.

19
Summary
  • In general, when solving quadratic inequalities
  • Find the zeros by solving the equation you get
    when you replace the inequality symbol with an
    equals.
  • Find the intervals around the zeros using a
    number line and test a value from each interval
    in the number line.
  • The solution is the interval or intervals which
    make the inequality true.

20
Practice Problems
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