Obj. 4 Inequalities and Absolute Value Equations (Presentation)

8/4/2019 Obj. 4 Inequalities and Absolute Value Equations (Presentation)

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Obj. 4 Inequalities & Absolute

Value Equations

Unit 1 Functions and Relations


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Concepts and Objectives

Inequalities and Absolute Value Equations

Solve linear and quadratic inequalities Solve absolute value equalities and inequalities


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Linear and Quadratic Inequalities

An inequalitystates that one expression is greater than,

greater than or equal to, less than, or less than or equalto another expression.

As with equations, a value of the variable for which the

all solutions is the solution set of the inequality.

Inequalities are solved in the same manner equations

are solved with one differenceyou must reverse the

direction of the symbol when multiplying or dividing bya negative number.


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Example: Solve + < 2 7 5x

+ < 2 7 5x

+ < 77 72 5x < 2 12x

The solution set is {x|x> 6}. Graphically, the solution is

>

2 2

x

> 6x


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Three-part or Compound Inequalities are solved by

working with all three expressions at the same time. The middle expression is between the outer expressions.

xamp e: o ve x


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Example: Solve 1 6 8 4x

+ + + 1 6 8 8 848 x

9 6 12x

9 6 12x

The solution set is the interval

6 6 6

3

22

3

,22


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To solve a quadratic inequality:

Solve the corresponding quadratic equation. Identify the intervals determined by the solutions of

the equation.

se a test va ue rom eac nterva to eterm newhich intervals form the solution set.


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Example: Solve >23 11 4 0x x

=23 11 4 0x x

+ =23 12 4 0 x x x

( ) ( ) + =3 4 1 4 0 x x x

( )( )+ =3 1 4 0x x

= =1

or 43

x x

1,

3

1,4

3( )4,


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Example: Solve >23 11 4 0x x

Interval Test Value True or False?

1,

31 10 > 0 True

1

,43

( )4,

0

5

4 > 0 False

16 > 0 True

( )

1, 4,3


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Absolute Value

The solution set for the equation must include

both a and a. Example: Solve

=x a

=9 4 7x

=9 4 7x = 9 4 7x

The solution set is

= 4 2 = 4 16x

=1

2x = 4

or

1,4

2


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Absolute Value

For absolute value inequalities, we make use of the

following two properties: |a| < b if and only if b < a < b.

|a| > b if and only ifa < b or a > b.

Example: Solve + 5 8 6 14x


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Absolute Value

Example: Solve + 5 8 6 14x

5 8 8x

5 8 8x 5 8 8x 8 13x 8 3

The solution set is

or 138

x 38

x

3 13, ,

8 8


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Special Cases

Since an absolute value expression is always

nonnegative: Expressions such as |2 5x| > 4 are always true. Its

solution set includes all real numbers, that is, (, ).

that is, it has no solution.

The absolute value of 0 is equal to 0, so you can solve

it as a regular equation.


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Homework & Practice Problems

Page 155: 20-50 (5s)

HW: 20, 30, 40 Page 163: 10-50 (5s)

HW: 10, 20, 30, 40, 50

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Obj. 4 Inequalities and Absolute Value Equations (Presentation)