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Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Jan 18, 2016

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Melvyn Sparks
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Page 1: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solving Inequalities

Page 2: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Objective- To solve and graph simple inequalities involving addition/subtraction.

Equations InequalitiesSolve and graph. Solve and graph.

x 2 5

2 2

x 3

-3 -2 -1 0 1 2 3 4

One Solution

x 2 5

2 2

x 3

-3 -2 -1 0 1 2 3 4

Infinite Solutions

Page 3: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Graph the following inequalities.

1)

2)

3)

4)

x 2

x 3

x 2

x 1

-3 -2 -1 0 1 2 3 4

-3 -2 -1 0 1 2 3 4

-3 -2 -1 0 1 2 3 4

-3 -2 -1 0 1 2 3 4

Page 4: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph the inequalities.

1) 2)

3 x 7

4 x 6

3 3

x 10

-10 0 10-1 0 1 2 3

6 6

2 x

x 2

Page 5: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph the inequalities.

3) 4)

5 8 k

3 p 9

8 8

13 k

-13 0 0 12

3 3

p 12

k 13

Page 6: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Objective- To solve and graph simple inequalities involving

multiplication/division.

4x 20

Solve and graph the inequalities.

1) 2)

4x 20

x3

2

x 5

0 5

-6 0 6

(3)x3

2(3)

x 6

4 4

Page 7: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Inequalities transform like equations except...

3x 6

3 3

3x 6

reverse

x 2

x 2

Page 8: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

( 1)3 2( 1) reverse

Inequalities transform like equations except...

When multiplying or dividing by a negativenumber you must reverse (flip) the inequality.

-4 -3 -2 -1 0 1 2 3 4

Positive sideNegative side

Large is largeLarge is small3 2 3 2

3 2

3 2

Page 9: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

4x 24

Solve and graph the inequalities.

1) 2)

4x 24

y 3

7

x 6

-6 0

-21 0

( 3)y 3

7( 3)

y 21

4 4

Page 10: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph the inequalities.

3) 4)

3x 18

m2

6

x 6

-6 0

-12 0

( 2)m 2

6( 2)

m 12

3 3

3x 18

Page 11: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Objective- To solve and graph multi-step

inequalities.

Solve and graph.

2x 5 11

5 5

2x 16

2 2

x 80 8

Page 12: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

1) 2)

3x 4 17

4 4

3x 21

3 3

x 7

0 7

x2

5 3

5 5

x2

8

(2)x2

8(2)

x 160 16

Solve and graph.

Page 13: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

3) 4)

7 x 4

7 7

x 3

0 3

3(x 5) 12

3x 15 12

3x 27

x 9

0 9

Solve and graph.

( 1)( x) ( 3)( 1)

x 3

15 15

3 3

Page 14: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

5) 6)

3(x 4) 9

12 12

3x 21

0 7

4x 3 2x 11

6x 3 11

x 73

Solve and graph.

3x 3

21 3

x 7

2x 2x

6 6

3x 12 9

3 3

6x 14

0

213

Page 15: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Joe is saving for a new $500 bike. He currentlyhas $125. If he saves $15 per week, how long must he wait to save at least $600 to cover taxand extras.

Let x = # of weeks

125 15x 600

125 125

15x 475

15 15

x 3123

At least 32 weeks

Page 16: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

A water tank contains 500 gallons of water andbegins leaking at a rate of 4 gallons per minute.Because the tank must be repaired from the inside, it can retain at most 60 gallons of water. How long must the repairman wait to fix the tank?

Let x = # of minutes tank leaks

500 4x 60

500 500

4x 440

4x 4

440 4

x 110

At least 110 min.

Page 17: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Objective- To solve compound inequalities involving “or”.

Write a compound inequality that describes all real numbers less than -2 or greater than 5.

x 2 or

x 5

-3 -2 -1 0 1 2 3 4 5 6

Graph.

Page 18: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Graphing Unions (OR)

1. Graph both inequalities on the same line.

2. If graphs overlap, then solution is all real numbers (whole # line).

all real numbersR

Page 19: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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 20: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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 21: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph.

4( 5) 4x or 2( 4) 16x

20 204 16x 4 4

4x

8 82 24x2 2

12x

4 12 x or x

0 4 12

4 20 4x 2 8 16x

Page 22: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph.

3 5 7x or 8 14 66x 5 5

3 12x 3 3

4x

14 148 80x8 8

10x4 10 x or x

0 4 10

x = All Real #’s (R)

Page 23: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Objective- To solve compound inequalities involving “and”.

Write a compound inequality that describes allthe real numbers greater than -2 and less than 5.

x 2 and

x 5

-3 -2 -1 0 1 2 3 4 5 6

Page 24: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Graphing Intersections (AND)

1. Graph the two inequalities separately (lightly).

2. Find where the graphs overlap.

3. Graph only the overlapping part.

4. If there’s no overlap, then there’s no solution.

= No Solution

Page 25: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph.

3 5 7x and 8 14 66x 5 5

3 12x 3 3

4x

14 148 80x8 8

10x

4 10 x and x

0 4 10

Page 26: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Solve and graph.

2 1 3x and 3 4 17x 1 1

2 2x2 2

1x

4 43 21x3 3

7x

1 7 x and x

0 1 7

Page 27: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Another Way to Write “AND”

x 2 and

x 5

-3 -2 -1 0 1 2 3 4 5 6

2 x 5 “In-between statement”

Page 28: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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 29: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

9 3x 3

3 3 3

Solve and graph.

5 3x 4 7

4 4 4

3 x 1

-3 0 1

Page 30: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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 31: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

12 2 4x 2 2 2

Solve and graph.7 2 5 9x 5 5 5

6 2x

-6 0 2

Page 32: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

A racquetball club charges a $20 membershipand $2 per hour. How many hours per monthcan be played on a budget of $50 to $70?

Let h = # of hours

Cost = 20 + 2h

15 25h

From 15 to 25 hours

50 20 2 70h

30 2 50h 2 2 2

20 20 20

50 Cost 70

Page 33: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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 < 3 x < 5

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

3 < x < 5

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

x = Any Real Number

Page 34: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

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

Not possible x < 3 or x > 5

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

x > 5

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

x > 3

Intersections UnionsNumber Line Graphs of Inequalities

Page 35: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Number Line Graphs of InequalitiesIntersections Unions

x < 4 x < 7and x < 4 x < 7or 0 2 4 6 8 10 12 0 2 4 6 8 10 12

x < 4 x < 7

x < 4 x > 7and 0 2 4 6 8 10 12

Not possible

x > 4 x < 7or 0 1 2 3 4 5 6

x = Any Real Number

Page 36: Solving Inequalities Objective- To solve and graph simple inequalities involving addition/subtraction. EquationsInequalities Solve and graph. -3 -2 -1.

Equalities Inequalities

= Equals- is the same as

Congruent- same size and shape

Similar- same shape

< Is less than

> Is greater than

Is less than or equal to

Approx. equal to

= Not equal to

~

Greater than or equal to.