Obj. 4 Inequalities and Absolute Value Equations (Presentation)
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Transcript of Obj. 4 Inequalities and Absolute Value Equations (Presentation)
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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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Linear and Quadratic Inequalities
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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Linear and Quadratic Inequalities
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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Linear and Quadratic Inequalities
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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Linear and Quadratic Inequalities
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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Linear and Quadratic Inequalities
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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Linear and Quadratic Inequalities
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