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How do we solve Compound Inequalitie Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

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Page 1: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Aim: How do we solve Compound Inequalities?

Do Now: Solve the following inequalities

1. 2x + 3 > 2

2. 5x < 10

How do we put two inequalities together?

Page 2: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Definition

A compound inequality consists of two inequalities

connected by the word and or the word or.

Examples

-7 < x < 10 x < 8 or x > 27

x < - 4 or x > 4 12 x and x 30

Page 3: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

●Example:

●This is a conjunction because the two inequality statements are joined by the word “and”.

●You must solve each part of the inequality.

●The graph of the solution of the conjunction is the intersection of the two inequalities. Both conditions of the inequalities must be met.

●In other words, the solution is wherever the two inequalities overlap.

●If the solution does not overlap, there is no solution.

2 3 2 and 5 10x x

Page 4: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

“and’’ Statements can be Written in Two Different Ways

●1. 8 < m + 6 < 14

●2. 8 < m+6 and m+6 < 14

These inequalities can be solved using two methods.

Page 5: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Method One

Example : 8 < m + 6 < 14 Rewrite the compound inequality using the word

“and”, then solve each inequality.8 < m + 6 and m + 6 < 142 < m m < 8

m >2 and m < 8 2 < m < 8

Graph the solution:

8 2

Page 6: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Example: 8 < m + 6 < 14

To solve the inequality, isolate the variable by subtracting 6 from all 3 parts.

8 < m + 6 < 14 -6 -6 -6

2 < m < 8 Graph the solution.

8 2

Method Two

Page 7: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

●Example: ●This is a disjunction because the two inequality

statements are joined by the word “or”.●You must solve each part of the inequality.●The graph of the solution of the disjunction is the union of

the two inequalities. Only one condition of the inequality must be met. ●In other words, the solution will include each of the

graphed lines. The graphs can go in opposite directions or towards each other, thus overlapping.

●If the inequalities do overlap, the solution is all reals.

3 15 or -2 +1 0 x x

Review of the Steps to Solve a Compound Inequality:

Page 8: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

‘or’ Statements

Example: x - 1 > 2 or x + 3 < -1 x > 3 x < -4

x < -4 or x >3 Graph the solution.

3-4

Page 9: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Solve and graph the compound inequality.

4 x 3 7

4 x 3 7

4 x 3 and

x 3 7

3 3

7 x

3 3

x 4

x 7 and

x 4

7 x 4-7 0 4

4 x 3 7

3 3 3

7 x 4

Page 10: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

9 3x 3

3 3 3

Solve and graph.

5 3x 4 7

4 4 4

3 x 1

-3 0 1

Page 11: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

10 x 2

1 1 1

Solve and graph.

4 6 x 8

6 6 6

-2 0 10

10 x 2

10 x 2

2 x 10

Page 12: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

2 2

4 4

Solve and graph the compound inequality.

2x 3 7

4x 7 33or

3 3

2x 10

x 5 or

7 7

4x 40

x 10

x 5 or

x 10

0 5 10

Page 13: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

2x 24

5x 35

5 5

2 2

Solve and graph the compound inequality.

5x 35

1 2x 23or

x 7

or

1 1

2x 24

x 12

-7 0 12

x 7

x 12

Page 14: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

Number Line Graphs of InequalitiesIntersections Unions

x < 5 x < 3and x < 5 x < 3or 0 1 2 3 4 5 6 0 1 2 3 4 5 6

{ x : x < 3 } { x : x < 5 }

x < 5 x > 3and 0 1 2 3 4 5 6

{ x : 3 < x < 5 }

x < 5 x > 3or 0 1 2 3 4 5 6

{ x : x = Any Real Number }

Page 15: Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?

x > 5 x < 3and x > 5 x < 3or 0 1 2 3 4 5 6 0 1 2 3 4 5 6

{ } { x : x < 3 or x > 5 }

x > 5 x > 3and 0 1 2 3 4 5 6

{ x : x > 5 }

x > 5 x > 3or 0 1 2 3 4 5 6

{ x : x > 3 }

Intersections UnionsNumber Line Graphs of Inequalities