Download 8.6

Chapter 8
Quadratic
Functions
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-1
1
Chapter Sections
8.1 – Solving Quadratic Equations by Completing
the Square
8.2 – Solving Quadratic Equations by the
Quadratic Formulas
8.3 – Quadratic Equations: Applications and
Problem Solving
8.4 – Writing Equations in Quadratic Form
8.5 – Graphing Quadratic Functions
8.6 – Quadratic and Other Inequalities in One
Variable
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-2
2
§ 8.6
Quadratic and Other
Inequalities in One
Variable
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-3
3
Quadratic and Other Inequalities
Quadratic Inequality
A quadratic inequality is an inequality that can be
written in one of the following forms
ax2 + bx + c < 0
ax2 + bx + c > 0
ax2 + bx + c ≤ 0
ax2 + bx + c ≥ 0
where a, b, and c are real numbers, with a ≠ 0.
Solution to a Quadratic Inequality
The solution to a quadratic inequality is the set of all
values that make the inequality a true statement.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-4
4
Solve Quadratic Inequalities
To Solve Quadratic and Other Inequalities
1. Write the inequality as an equation and solve the
equation. The solutions are the boundary values.
2. Construct a number line and mark each boundary value
from step 1 as follows
• If the inequality symbol is < or >, use an open circle ◦.
• If the inequality symbol is ≤ or ≥, use a closed circle,●.
3. If solving a rational inequality, determine the values that
make any denominator 0. These values are also
boundary values. Indicate these boundary values on
your number line with an open circle.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-5
5
Solve Quadratic Inequalities
To Solve Quadratic and Other Inequalities
4. Select a test value in each interval and determine
whether it satisfies the inequality.
5. The solution is the set of points that satisfy the
inequality.
6. Write the solution in the form instructed.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-6
6
Solve Quadratic Inequalities
Example Solve the inequality x2 – 4x ≥ -4. Give the
solution a) on a number line, b) in interval notation,
and c) in set builder notation.
x 2  4 x  4
x2  4x  4  0
( x  2)( x  2)  0
x20
x2
or
x20
x2
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-7
7
Solve Quadratic Inequalities
The solution set includes both intervals and the boundary
value, 2. The solution set is the set of real numbers.
a)
b) (, )
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
c) {x |   x  }
Chapter 8-8
8
Solve Other Polynomial Inequalities
The same procedure we used to solve quadratic
inequalities can be used to solve other polynomial
inequalities, as illustrated in the following examples.
Example Solve the polynomial inequality
(3x – 2)(x + 3)(x + 5) < 0. Illustrate the solution on a
number line and write the solution in both interval
notation and set builder notation.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-9
9
Solve Other Polynomial Inequalities
We use the zero-factor property to solve the equation.
3x  2  0
2
x
3
or
x3 0
x  3
or
x5  0
x  5
The boundary values -5, -3, and 2/3 are indicated
with open circles and break the number line into
four intervals. The test values we will use are -6, -4,
0, and 1. We show the results in the following table.
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-10
10
Solve Other Polynomial Inequalities
The solutions are
2

(,5)    3, 
3

 x | x  5 or - 3  x  2 


3

Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-11
11
Solve Rational Inequalities
Rational inequalities are inequalities that contain at
least one rational expression.
Example Solve the inequality x  1  2 .
x3
Graph the solution on a number line and write
the solution in interval notation.
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-12
12
Solve Rational Inequalities
x 1
We start by solving the equation
2 .
x3
x 1
2
x3
x 1
x  3
 2( x  3)
x3
x 1  2x  6
1  x  6
7  x
Thus, -7 is a boundary value and is indicated with a
closed circle on the number line.
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-13
13
Solve Rational Inequalities
When solving rational inequalities, we also need to
determine the value or values that make the
denominator 0. We set the denominator equal to 0
and solve.
x3 0
x  3
Therefore, -3 is a boundary value. We use the
solution to the equation, -7, and the value that makes
the denominator 0, -3, as our boundary values. We
will use -8, -5, and 0 as our test values.
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-14
14
Solve Rational Inequalities
The solution written in interval notation is [-7, -3),
and the solution illustrated on the number line is
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-15
15
Solve Rational Inequalities
x 1
The graph below shows f ( x) 
and the graph
x3
of y=2.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-16
16