Quadratic Inequalities

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Quadratic Inequalities

• Tidewater Community College
• Karen Overman

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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.

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Quadratic Inequalities

• What do they look like?
• Here are some examples

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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.

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Solving a quadratic inequality

• We can find the values where the quadratic equals
  zero
• by solving the equation,

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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

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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.

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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
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Solving a quadratic inequality
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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.

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Example 2

• Solve
• First find the zeros by solving the equation,

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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
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Example 2
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Example 2

• Thus the interval makes up the solution set
  for
• the inequality .

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Example 3

• Solve the inequality .
• First find the zeros.

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Example 3

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

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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.

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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.

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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.

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Practice Problems