Solving Inequalities

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

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

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

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

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

Inequalities transform like equations except...

3x 6

3 3

3x 6

reverse

x 2

x 2

( 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

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

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

Objective- To solve and graph multi-step

inequalities.

Solve and graph.

2x 5 11

5 5

2x 16

2 2

x 80 8

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.

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

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

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

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.

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.

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

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

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

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

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)

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

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

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

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

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”

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

9 3x 3

3 3 3

Solve and graph.

5 3x 4 7

4 4 4

3 x 1

-3 0 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

12 2 4x 2 2 2

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

6 2x

-6 0 2

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

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

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

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

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.