2.6

Solving Linear Inequalities

1. Represent solutions to inequalities graphically and using set notation.

2. Solve linear inequalities.

Inequalities < >   is less than is greater than is less than or equal to is greater than or equal to Inequality always points to the smaller number.

True or False?

4  4 True 4 > 4 False x > 4 is the same as {5, 6, 7…} False Represent inequalities: Graphically Interval Notation Set-builder Notation

Graphing Inequalities If the variable is on the left, the arrow points the

same direction

as the inequality.

Parentheses/bracket method : Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ x < 2 x ≥ 2 Open Circle/closed circle method: Open Circle: endpoint is not included <, > Closed Circle: endpoint is included ≤, ≥ x < 2 x ≥ 2

Inequalities – Interval Notation [( smallest, largest )] Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ Infinity:

always

uses a parenthesis x < 2 ( –∞, 2) x ≥ 2 [2, ∞) 4 < x < 9 3-part inequality (4, 9)

Inequalities – Set-builder Notation {variable | condition } pipe {

x

|

x

 5 } The set of all

x

such that

x

is greater than or equal to 5.

x < 2 ( –∞, 2) < 2 x ≥ 2 [2, ∞) { x | x ≥ 2} 4 < x < 9 (4, 9) { x | 4 < x < 9}

Inequalities Graph, then write interval notation and set-builder notation.

x ≥ 5

[

Interval Notation: Set-builder Notation: [ 5, ∞) { x | x ≥ 5} x < –3

)

Interval Notation: (– ∞, –3) Set-builder Notation: { x | x < –3 }

Inequalities Graph, then write interval notation and set-builder notation.

1 < a < 6

( )

Interval Notation: ( 1, 6 ) Set-builder Notation: { a | 1 < a < 6 } –7 < x ≤ 3

( ]

Interval Notation: (– 7, –3] Set-builder Notation: { x | –7 < x ≤ 3 }

Inequalities 4 < 5 4 + 1 < 5 + 1 4 < 5 4 – 1 < 5 – 1 5 < 6 3 < 4 True True

The Addition Principle of Inequality

If

a

<

b

, then

a +

c

<

b

+

c

for all real numbers

a, b,

and

c

. Also true for >,  , or  .

Inequalities 4 < 5 4 (2) < 5 (2) 8 < 10 True 4 < 5 4 (–2) < 5 (–2) –8 < –10 –8 > –10 False If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

The Multiplication Principle of Inequality

If

a < b,

then

ac

<

bc

if

c

is a positive real number.

If

a

<

b

, then

ac

>

bc

if

c

is a negative real number . The principle also holds true for >,  , and  .

Solving Inequalities If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

4

x

4  16 4

x

 4   4 4

x

 16  4

x

  4 4

x

4   16 4

x

  4

Solving Inequalities

Solve

then

graph

the solution and write it in

interval notation

and

set-builder notation

.

3  4 Don’t write = !

 3

x x

  4 3  7 4 3 3

x

 1

(

Interval Notation: ( 1, ∞ ) Set-builder Notation: { x | x > 1 }

Solving Inequalities

Solve

then

graph

the solution and write it in

interval notation

and

set-builder notation

.

4   9

k

4

k

  4  4

k k

 19  4 4  5

k

  19 4   5

k

5   15 5

k

  3

]

Interval Notation: Set-builder Notation: (– ∞, –3 ] { k | k ≤ –3 }

Solving Inequalities

Solve

then

graph

the solution and write it in

interval notation

and

set-builder notation

.

 5 3

p

  10 3    3 5

p

 3   10    5

p

5

p

  6   30 5

)

Interval Notation: (– ∞, 6 ) Set-builder Notation: { p | p < 6 }

Solving Inequalities

Solve

then

graph

the solution and write it in

interval notation

and

set-builder notation

.

1 5  6

m

 7   1 2  3

m

 1  10   2  1 5 6

m

 6

m

 7    7   5  3

m

10    1 2 1   3

m

 1   12

m

15

m

 14  15

m

 15

m

 5  3

m

  14 14    5 14

Moving variable to the right.

12

m

 14  15

m

 5  12

m

 12

m

  14 5  3

m

  5 5  9  3 3  

m

3

m

3

[

  3

m

3   9 3 Interval Notation: [– 3, ∞ ) Set-builder Notation: { m | m ≥ – 3 }

m

  3