Compound linear inequalities
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Compound Linear Inequalities. Identify the Symbol. Less Than Less Than or Equal To Greater Than Greater Than or Equal To. Identify the Symbol. Less Than Less Than or Equal To Greater Than Greater Than or Equal To. Identify the Symbol. Less Than or Equal To Greater Than or Equal To

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Identify the symbol
Identify the Symbol

  • Less Than

  • Less Than or Equal To

  • Greater Than

  • Greater Than or Equal To


Identify the symbol1
Identify the Symbol

  • Less Than

  • Less Than or Equal To

  • Greater Than

  • Greater Than or Equal To


Identify the symbol2
Identify the Symbol

  • Less Than or Equal To

  • Greater Than or Equal To

  • All Real Numbers

  • Not Equal To


Identify the symbol3
Identify the Symbol

  • Less Than or Equal To

  • Greater Than or Equal To

  • All Real Numbers

  • Not Equal To


Conjunction
Conjunction

  • Mathematical sentences joined by “and”

  • Meaning: an intersection


Disjunction
Disjunction

  • Mathematical sentences joined by “or”

  • Meaning: a union


Which is it
Which is it?

  • All students who have red hair and are boys.

  • All students who have brown hair or wear glasses.


X 3 and x 1
x < -3 and x > 1

Where on the number line are both of these statements true?


Solving conjunctions
Solving Conjunctions

  • Graph both inequalities.

  • Find the intersection. (overlapping portions)

  • Write the answer as an inequality.


Conjunction case 1

-4

-3

0

-2

1

3

-6

-5

-1

2

Conjunction – Case 1

  • x < -3 and x > 1


Conjunction case 11
Conjunction – Case 1

  • No overlap and arrows going in the opposite direction

  • No solutions


Conjunction case 2

-4

-3

0

-2

1

3

-6

-5

-1

2

Conjunction – Case 2

  • x > -3 and x < 1

  • Both must be true.

  • -3 < x < 1


Conjunction case 21
Conjunction – Case 2

  • Overlapping and arrows going in the opposite direction

  • The solution is between the two numbers.


Conjunction case 3

-4

-3

0

-2

1

3

-6

-5

-1

2

Conjunction – Case 3

  • x < -3 and x < 1

  • Both must be true.

  • x < -3


Conjunction case 31
Conjunction – Case 3

  • Overlapping and arrows going in the same direction.

  • The solution will be a single greater than/less than inequality.


Conjunction case 32

-4

-3

0

-2

1

3

-6

-5

-1

2

Conjunction – Case 3

  • x > -3 and x > 1

  • Both must be true.

  • x > 1


Solving disjunctions
Solving Disjunctions

  • Graph both inequalities.

  • Find the union. (Join the two graphs)

  • Write the answer as an inequality.


Disjunction case 1

-4

-3

0

-2

1

3

-6

-5

-1

2

Disjunction – Case 1

  • x < -3 or x > 1


Disjunction case 11
Disjunction – Case 1

  • No overlap and arrows going in the opposite direction

  • The solution is the original inequalities.


Disjunction case 2

-4

-3

0

-2

1

3

-6

-5

-1

2

Disjunction – Case 2

  • x > -3 or x < 1

  • Either can be true.

  • All Real Numbers


Disjunction case 21
Disjunction – Case 2

  • Overlapping and arrows going in the opposite direction

  • If every part of the number line is covered at least once, then the solution is all real numbers.


Disjunction case 3

-4

-3

0

-2

1

3

-6

-5

-1

2

Disjunction – Case 3

  • x < -3 or x < 1

  • Either can be true.

  • x < 1


Disjunction case 31
Disjunction – Case 3

  • Overlapping and arrows going in the same direction.

  • The solution will be a single greater than/less than inequality.


Disjunction case 32

-4

-3

0

-2

1

3

-6

-5

-1

2

Disjunction – Case 3

  • x > -3 or x > 1

  • Either can be true.

  • x > -3


X 3 or x 1
x > 3 or x > 1

  • x > 1

  • x > 3

  • All real numbers

  • None of these


X 3 and x 11
x > 3 and x > 1

  • x > 1

  • x > 3

  • All real numbers

  • None of these


Disjunction1

-1

0

3

1

4

6

-3

-2

2

5

Disjunction

  • x > 3 or x  0


Conjunction1

-1

0

3

1

4

6

-3

-2

2

5

Conjunction

  • x > 3 and x  0


2x 4 or x 8 1

-7

-6

-3

-5

-2

0

-9

-8

-4

-1

-2x > 4 or x + 8 < 1

  • Solve each inequality first!

  • x < -2 or x < -7

  • x < -2


3x 5 1 and x 5 3
3x – 5 < 1 and x – 5 > -3

  • x = 2

  • x > 2

  • -2 < x < 2

  • The empty set

  • None of these


3x 5 1 or x 5 3

-2

-1

2

0

3

5

-4

-3

1

4

3x – 5 < 1 or x – 5 > -3

  • x < 2 or x > 2

  • x ≠ 2


2 x 1 5
-2 < x + 1 < 5

  • “x + 1 lies between -2 and 5.”

  • Always a conjunction.

  • Write as two separate inequalities, then solve as usual.

  • x + 1 > -2 and x + 1 < 5


15 3 x 1 12
-15 < 3(x – 1) < 12

  • x < -4

  • -4 < x < 5

  • x < 5

  • The empty set

  • None of these


4x 12 or x 6 5
4x > -12 or x + 6 < 5

  • x > -3

  • -3 < x < -1

  • All real numbers

  • The empty set

  • None of these



Page 74

-2

-1

2

0

3

5

-4

-3

1

4

Page 74

  • 6.


Page 741
Page 74

  • 8. 1 < x  7

  • 10. All real numbers


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