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Linear Inequalities
SolvingLinear Inequalities
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7/2/2013 Linear Inequalities 22
Meanings and Solutions What does f(x) < g(x) mean?
f(x) < g(x) is an inequality
It says f(x) is less than g(x)(algebraically smaller than g(x))
… for certain values of x
This may be TRUE for some values of x and FALSE for others
Inequalities
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7/2/2013 Linear Inequalities 33
Meanings and Solutions Other inequality forms include ≤ , > ,
and ≥
Now what do we do with inequalities ?
We solve them, i.e. find their solutions
Convert to equivalent inequality, i.e. one with same solutions
Inequalities
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7/2/2013 Linear Inequalities 44
Finding Solutions
What is a solution for an inequality?
A solution for f(x) < g(x) is a value of x that makes the inequality TRUE
Any x that makes the inequality false is not a solution
Inequalities
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7/2/2013 Linear Inequalities 55
Finding Solutions
What is the solution set for an inequality?
The solution set for the inequality is the set of all solutions
The solution set might be in several discrete pieces
The solution set might be empty
Inequalities
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7/2/2013 Linear Inequalities 66
Examples
1. 3(2x + 3) < 4x + 1
2. – x(x – 4) ≥ x – 4
Inequalities
TRUE for all x < –4 , FALSE otherwise
TRUE for –1 ≤ x ≤ 4 , FALSE otherwise
Question:
What are the solution sets for the above ?
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7/2/2013 Linear Inequalities 77
Examples
3. 2y – 4 ≤ x
Inequalities
TRUE for all (x,y) where y ≤ x + 2 , 12
FALSE otherwise
Question:What are the solution sets for the above ?
4. 3 > 4 + 2x, for x > 0
Logically FALSE WHY ?
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7/2/2013 Linear Inequalities 88
Examples
5. 3 + 4 ≤ 7
Inequalities
Question:What are the solution sets for the above ?
6. (x – 2)2 + 3 ≥ 1, for x ≥ 1
Logically TRUE WHY ?
Logically TRUE WHY ?
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7/2/2013 Linear Inequalities 99
Addition Rule:
If a > b then a + c > b + c for any real c Examples
If 7 > 3 then 7 + 4 > 3 + 4 If 7 > 3 then 7 – 9 > 3 – 9 If 2x + 5 > 3 then (2x + 5) – 5 > 3 – 5
Inequalities: Rules of the Road
OR2x > -2 … an equivalent inequality
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7/2/2013 Linear Inequalities 1010
Multiplication Rule 1:
If a > b and c > 0 then ac > bc
Examples
If 7 > 3 and 5 > 0 then 7(5) > 3(5)
If 2x + 6 > 8 then ½(2x + 6) > ½(8)
Inequalities: Rules of the Road
… an equivalent inequality
ORx + 3 > 4
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7/2/2013 Linear Inequalities 1111
Multiplication Rule 2:
If a > b and c < 0 then ac < bc
Examples
If 7 > 3 and -5 < 0 then 7(-5) < 3(-5)
If 2x + 6 > 8 then -½(2x + 6) < -½(8)
Inequalities: Rules of the Road
… an equivalent inequality
OR-x – 3 < -4
Question: What if c = 0 ?
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7/2/2013 Linear Inequalities 1212
Definition A linear inequality in one variable is one
that can be written
ax + b > 0
in standard form, with a ≠ 0
Linear Inequalities
… also includes forms with ≥ , < , ≤
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7/2/2013 Linear Inequalities 1313
Examples
1. 2x – 5 < 1
2. 3 – 4x ≥ 5
3. 2x + 1 ≤ 3x + 7
Linear Inequalities
Question: Can each of these be written in standard form ?
2x – 6 < 0
-4x – 2 ≥ 0
-x – 6 ≤ 0
4x + 2 ≤ 0
x + 6 ≥ 0
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7/2/2013 Linear Inequalities 1414
Definition
A solution for a linear inequality in one variable is a value of the variable that makes the inequality TRUE
The set of all solutions for an inequality is the solution set for the inequality
Solutions of Linear Inequalities
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7/2/2013 Linear Inequalities 1515
Example 2x – 5 < 1 Some solutions are: 2, 2.5, 1, 0, -5.4, …
Solution Set Notation Set notation: { x x < 3 } Interval notation: ( – , 3)
Solutions of Linear Inequalities
Question: Does this interval include 3 ?
How many solutions?
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7/2/2013 Linear Inequalities 1616
Example 1: Analytical Method Solve:
Solving:
Solution Set is:
Inequalities
3 – 2x ≤ 5
– 2x ≤ 5 – 3 = 2 x ≥ – 1
or [ – 1, ){ x | x ≥ – 1 }
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7/2/2013 Linear Inequalities 1717
Example 1: Graphical Method Solve:
Solutions:
Solution Set is:
Inequalities
3 – 2x ≤ 5
or [ – 1, )
y
x
y2 = 5
y1 = 3 – 2x
(-1, 5)
y1 = y2
0 1 3 5-1-3-5– [
{ x | x ≥ – 1 }
-1
y2 = 5y1 = 3 – 2x
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7/2/2013 Linear Inequalities 1818
Example 2: Analytical Method Solve:
Solving:
Solution Set is:
Inequalities
5x – 1 < 2x + 11
5x – 2x < 11 + 1 3x < 12 x < 4
or ( – , 4 )
{ x | x < 4 }
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7/2/2013 Linear Inequalities 1919
Example 2: Graphical Method Solve:
Graphically:
Solution Set is:
y
x
Inequalities
5x – 1 < 2x + 11
y 2 =
2x
+ 11
(4, 19)
or ( – , 4 )y 1
= 5
x –
1
y1 < y2
4{ x | x < 4 }
0 2 4-2-4– )
y1 = 5x – 1 y2 = 2x + 11
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7/2/2013 Linear Inequalities 2020
Notes on Notation
Symbols Sometimes the same symbols are used
to mean different things – just as words in English can have different meanings
Points and Intervals The point (2, 3) and the open interval
(2, 3) use the same notation The difference is determined by context
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7/2/2013 Linear Inequalities 2121
Notes on Notation
Points of Confusion Do not write (3, 7) or [3, 7] when you
mean { 3, 7 }
Never write [–, 7 ] , [ , 7 ) , (3, ] , or [3, ]
WHY ?
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7/2/2013 Linear Inequalities 2222
Rewrite inequality in form y < 0 (or ≤ , ≥ , > )
Example
5x – 2x – 1 – 11 < 0
3x – 12 < 0
Let y = 3x – 12
y = 3x – 12 = 0 for x = 4 (4, 0)
y
x
Horizontal Intercept Method
y = 3x – 12
y = 3x – 12 < 0
5x – 1 < 2x + 11
< 0
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7/2/2013 Linear Inequalities 2323
Rewrite inequality in form y < 0 (or ≤ , ≥ , > )
Example
y = 3x – 12 = 0 for x = 4
For y < 0 , we have x < 4
Solution set:
{ x x < 4 } or (– , 4)
(4, 0)
y
x
y = 3x – 12
y = 3x – 12 < 0
Horizontal Intercept Method
– 0 2 4-2-4
)
5x – 1 < 2x + 11
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7/2/2013 Linear Inequalities 2424
Recall:
Basic Absolute Value Facts
1. x ≥ 0 for all real x
2. x = –x for all real x
Absolute Values in Inequalities
a
= –a , for a < 0a , for a ≥ 0
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7/2/2013 Linear Inequalities 2525
Basic Absolute Value Facts
3. If x < b then –b < x < b
4. If x > b > 0
x
Absolute Values in Inequalities
WHY ?
0 x b–b –x x –x –x
x
WHY ?
0 b–b x x –x –x
then either x > b or x < -b
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7/2/2013 Linear Inequalities 2626
Examples
1. │ 3 │ < 7 then –7 < 3 < 7
2. │–3│ < 7 then –7 < –3 < 7
Absolute Values in Inequalities
│3│0 3 7–7
7–7 –3 │–3│ 0
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7/2/2013 Linear Inequalities 2727
Examples
3. │ –7 │ > 3
Rule: Replace absolute value with each of the two forms indicated in the definition
│–7│
Absolute Values in Inequalities
3–3 –7 70
then either –7 > 3 or –7 < –3
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7/2/2013 Linear Inequalities 2828
Examples
4. If │x + 2 │< 9
Absolute Values in Inequalities
│x + 2 │xx –9 x + 2 9–11 70x + 2
then –9 < x + 2 < 9 and –11 < x < 7
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7/2/2013 Linear Inequalities 2929
Examples
5. If x + 2 > 7 > 0
Absolute Values in Inequalities
then either x > 5 or x < –9
│x + 2 │x + 2xx –9 5–7 7x + 2 0–│x + 2│
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7/2/2013 Linear Inequalities 3030
Compound Inequalities
More Than One Inequality
1. Find the solution set for x ≥ –3 and x < 7
Rewriting: –3 ≤ x and x < 7
Solution set is { x –3 ≤ x < 7 }
OR [ –3, 7 )
OR compounding: –3 ≤ x < 7
– 0 1 3 5-1-3-5 7 9 11-7-9
[ )
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7/2/2013 Linear Inequalities 3131
Compound Inequalities
More Than One Inequality
2. Find the solution set for x < –3 OR x ≥ 7
Can’t rewrite as 7 ≤ x < –3
Solution set is { x x < –3 } { x x ≥ 7 }
WHY ?
OR ( – , –3) [ 7, )
– 0 1 3 5-1-3-5 7 9 11-7-9
) [
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7/2/2013 Linear Inequalities 3232
Compound Inequalities
Example Find the solution set for
8 – 4t
<–15 < 15
– 7 4
23 4
>> tNote:
Clear the fractions:
Simplify:
–23 < –4t < 7
– 2 – t 5
3 4
<<3 4
– 3 4
20( ) <2 – t
5 20( ) <
3 4
20( )
Inequalities reversed
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7/2/2013 Linear Inequalities 3333
Compound Inequalities
Example Find the solution set for
– 7 4
23 4
>> t
– 2 – t 5
3 4
<<3 4
Solution set
OR
Rewriting – 7 4
23 4
<< t
– 7 4
23 4
<< t{ t | } 23 4
,– 7 4 ( )
– 0 2 4-2-4 6 8 10-6
)(
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7/2/2013 Linear Inequalities 3434
Compound Inequalities
Example Find the solution set for
10x + 2(x – 4) < 12x – 10
–8 < –10
Simplifying: 10x + 2x – 8 < 12x – 10
A logically FALSE statement !
There is no value of x … that will make this true !
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7/2/2013 Linear Inequalities 3535
Compound Inequalities
Example Find the solution set for
10x + 2(x – 4) < 12x – 10
There is no value of x … that will make this true !
… OR just { }
Note: WHY ?
We say the solution set is EMPTY !
We write: Solution set is O
O ≠ O{ }
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7/2/2013 Linear Inequalities 3636
Think about it !