Linear Inequalities Solving Linear Inequalities. 7/2/2013 Linear Inequalities 2 2 Meanings and Solutions What does f(x) < g(x) mean? f(x) < g(x) is an

Linear Inequalities

SolvingLinear Inequalities

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


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


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


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


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 ?


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 ?


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 ?


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


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)


… an equivalent inequality

ORx + 3 > 4


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)


… an equivalent inequality

OR-x – 3 < -4

Question: What if c = 0 ?


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 ≥ , < , ≤


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


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


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?


Example 1: Analytical Method Solve:

Solving:

Solution Set is:

Inequalities

3 – 2x ≤ 5

– 2x ≤ 5 – 3 = 2 x ≥ – 1

or [ – 1, ){ x | x ≥ – 1 }


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


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 }


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


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


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 ?


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


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


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


Basic Absolute Value Facts

3. If x < b then –b < x < b

4. If x > b > 0

x


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


Examples

1. │ 3 │ < 7 then –7 < 3 < 7

2. │–3│ < 7 then –7 < –3 < 7


│3│0 3 7–7

7–7 –3 │–3│ 0


Examples

3. │ –7 │ > 3

Rule: Replace absolute value with each of the two forms indicated in the definition

│–7│


3–3 –7 70

then either –7 > 3 or –7 < –3


Examples

4. If │x + 2 │< 9


│x + 2 │xx –9 x + 2 9–11 70x + 2

then –9 < x + 2 < 9 and –11 < x < 7


Examples

5. If x + 2 > 7 > 0


then either x > 5 or x < –9

│x + 2 │x + 2xx –9 5–7 7x + 2 0–│x + 2│


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

[ )



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

) [



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




– 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

)(




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 !




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{ }


Think about it !

Documents

Linear Inequalities Solving Linear Inequalities. 7/2/2013 Linear Inequalities 2 2 Meanings and Solutions What does f(x) < g(x) mean? f(x) < g(x) is an