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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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
• All Real Numbers
• Not Equal To
Identify the Symbol
• Less Than or Equal To
• Greater Than or Equal To
• All Real Numbers
• Not Equal To
Conjunction
• Mathematical sentences joined by “and”
• Meaning: an intersection
Disjunction
• Mathematical sentences joined by “or”
• Meaning: a union
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

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

Solving Conjunctions
• Graph both inequalities.
• Find the intersection. (overlapping portions)
• Write the answer as an inequality.
-4

-3

0

-2

1

3

-6

-5

-1

2

Conjunction – Case 1
• x < -3 and x > 1
Conjunction – Case 1
• No overlap and arrows going in the opposite direction
• No solutions
-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 2
• Overlapping and arrows going in the opposite direction
• The solution is between the two numbers.
-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 3
• Overlapping and arrows going in the same direction.
• The solution will be a single greater than/less than inequality.
-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
• Graph both inequalities.
• Find the union. (Join the two graphs)
• Write the answer as an inequality.
-4

-3

0

-2

1

3

-6

-5

-1

2

Disjunction – Case 1
• x < -3 or x > 1
Disjunction – Case 1
• No overlap and arrows going in the opposite direction
• The solution is the original inequalities.
-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 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.
-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 3
• Overlapping and arrows going in the same direction.
• The solution will be a single greater than/less than inequality.
-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 > 1
• x > 3
• All real numbers
• None of these
x > 3 and x > 1
• x > 1
• x > 3
• All real numbers
• None of these
-1

0

3

1

4

6

-3

-2

2

5

Disjunction
• x > 3 or x  0
-1

0

3

1

4

6

-3

-2

2

5

Conjunction
• x > 3 and x  0
-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
• x = 2
• x > 2
• -2 < x < 2
• The empty set
• None of these
-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
• “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
• x < -4
• -4 < x < 5
• x < 5
• The empty set
• None of these
4x > -12 or x + 6 < 5
• x > -3
• -3 < x < -1
• All real numbers
• The empty set
• None of these
-2

-1

2

0

3

5

-4

-3

1

4

Page 74
• 6.
Page 74
• 8. 1 < x  7
• 10. All real numbers