Linear Inequalities
SolvingLinear Inequalities
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Think about it !
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