Transcript MTH 60 Elementary Algebra I

MTH 070
Elementary Algebra
Chapter 2
Equations and Inequalities in One
Variable with Applications
2.4 – Linear Inequalities
Copyright © 2010 by Ron Wallace, all rights reserved.
Inequality
A statement that one expressions is …
<, ≤, >, or ≥
… a second expression.
2( x  3)  5
Applications involving inequalities involve terms
such as “at least” (≥) and “at most” (≤).
Solving an Inequality
Determine ALL values of the variable
that makes the inequality a true
statement.
2( x  3)  5
Try some …
0?
7?
1?
5?
-3?
20?
10?
8?
Solutions of Inequalities

Three possible forms …

Simple inequality
x2
 x < –3



Interval notation
)

[2,

(–, –3)
Graph
[
2
)
3
Review – Solving Equations



Eliminate grouping symbols.
Combine like terms.
Eliminate terms



NO
PROBLEM !
Add opposites to other side
i.e. addition property
Eliminate factors


Multiply by reciprocals on the other side
i.e. multiplication property
What happens when you do these things to inequalities?
The Addition Property
w/ Inequalities
a

b
What happens if you add the same amount
of weight to both sides of the scale?
The Addition Property
w/ Inequalities
ac

Works with
subtraction
too!
bc
The inequality relationship
remains the same.
The Addition Property
w/ Inequalities
If an expression is added to
(subtracted from) both sides of
an inequality, the result will be an
equivalent inequality (i.e. same
solutions).
The Multiplication Property
w/ Inequalities
a

b
What happens if you multiply each weight
by the same POSITIVE value?
The Multiplication Property
w/ Inequalities
NOTE:
C>0
ac

Works with
division
too!
bc
The inequality relationship
remains the same.
The Multiplication Property
w/ Inequalities
a

b
What happens if you multiply each weight
by the same NEGATIVE value?
The Multiplication Property
w/ Inequalities
NOTE:
C<0
ac

Works with
division
too!
bc
The inequality relationship
reverses its direction.
The Multiplication Property
w/ Inequalities


If both sides of an inequality are
multiplied or divided by a positive
expression, the result will be an equivalent
inequality (i.e. same solutions).
If both sides of an inequality are
multiplied or divided by a negative
expression AND the direction of the
inequality is reversed, the result will be an
equivalent inequality (i.e. same solutions).
Switching Sides
 If
a < b, then how is b related to a?
b>a
 If
a > b, then how is b related to a?
b<a
Likewise for ≤ and ≥.
Solving Inequalities – Strategy
(just like equations w/ one exception)
Basic Goal: x > ? or x < ? or x ≥ ? or x ≤ ?




Eliminate grouping symbols
Combine like terms
Addition principle for inequalities
Multiplication principle for inequalities


Careful w/ this one!
If the variable is on the right; switch sides

Don’t forget to reverse the inequality symbol.
Checking Solutions

Need to check 3 values … (okay, maybe 2 will do)

Assume the solution: x < 3


Either of
these will
do.

Check x = 3 … this should make both sides equal
Check any value less than 3 … this should make
the original inequality TRUE.
Check any value greater than 3 … this should
make the original inequality FALSE
Hint: When checking inequalities, always check the number 0.
Solving & Checking Inequalities
Example 1 of 5
Give solutions in all three forms
5x  125
Solving & Checking Inequalities
Example 2 of 5
Give solutions in all three forms
3x  12
Solving & Checking Inequalities
Example 3 of 5
Give solutions in all three forms
2x  7  3
Solving & Checking Inequalities
Example 4 of 5
Give solutions in all three forms
x  4  8x  3
Solving & Checking Inequalities
Example 5 of 5
Give solutions in all three forms
2 x 
2
3

3
4
x 1