Transcript MTH 60 Elementary Algebra I

MTH 070
Elementary Algebra
Chapter 2
Equations and Inequalities in One
Variable with Applications
2.6 – Absolute Value Equations
and Inequalities
Copyright © 2010 by Ron Wallace, all rights reserved.
Absolute Value

x
The unsigned distance a quantity is
from zero.
17 
8 
0
Absolute Value

x
The unsigned distance a quantity is
from zero.
 x if x  0
x 
 x if x  0
Absolute Value Application

How do you find the distance
between two values?


Examples:
 7 & 12
 –3 & –15
 8 & –11
 –4 & 9
Pattern?

Subtract the numbers & ignore the sign of the
result.
Absolute Value Application

How do you find the distance
between two expressions?

Example:



2x & 3+x
Subtract them? But in which order to
guarantee the result is positive?
Consider with x = 4 and x = 2
Absolute Value – One more thing.
x a

What can be said about the value of a?
Absolute Value Equations
x 7

What can be said about the value of x?
Solving
Absolute Value Equations
expression1  expression2

How can this be rewritten without the
absolute value notation?
Solving
Absolute Value Equations
Strategy


Get the absolute value on one side of the
equation and everything else on the other
side. Simplify if possible.
Rewrite without the absolute value notation.



Gives 2 equations related by the word “or”
Solve the resulting two equations.
Check.
Solving
Absolute Value Equations
Examples
2x 1  7
x  5  8  2
x  3  2x  4
Absolute Value Inequalities
w/ “less than”

Quality control requires that the diameter of
a particular hole must be within 0.01” of 2”.
If x represents the diameter, how could you
express the value of x algebraically.
x  2  0.01
Absolute Value Inequalities
w/ “less than”

Find some values that would be solutions to
the following …

(graph them on a number line – notice anything?)
x 2
Absolute Value Inequalities
w/ “less than”

General result ...
x a


a  x  a
Therefore, to solve …



Rewrite as a double inequality
Solve
Check (at and around the endpoints)
Absolute Value Inequalities
w/ “less than”
Examples
6 x  18
x5  2
3x  4  7
Absolute Value Inequalities
w/ “greater than”

Find some values that would be solutions to
the following …

(graph them on a number line – notice anything?)
x 3
Absolute Value Inequalities
w/ “greater than”

General result ...
x a

 x  a or x  a
Therefore, to solve …



Rewrite as a compound inequality w/ “or”
Solve
Check (at and around the endpoints)
Absolute Value Inequalities
w/ “less than”
Examples
6 x  18
x5  2
3x  4  7