1 PROPERTIES OF INEQUALITY Standard 1 SOLUTION OF INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES INEQUALITIES LEVEL 1 INEQUALITIES LEVEL 2 INEQUALITIES

1

PROPERTIES OF INEQUALITY

Standard 1SOLUTION OF INEQUALITIES

INCLUDING ABSOLUTE VALUE

INEQUALITIES

INEQUALITIES LEVEL 1




COMPOUND INEQUALITIES “OR”

ABSOLUTE VALUE INEQUALITIES “AND”

ABSOLUTE VALUE INEQUALITIES “OR”

COMPOUND INEQUALITIES “AND”

END SHOW

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

2

Standard 1:

Students solve equations and inequalities involving absolute value.

Estándar 6:

Los estudiantes resuelven ecuaciones y desigualdades que involucran valor absoluto.

ALGEBRA II STANDARDS THIS LESSON AIMS:



3

ADDITION AND SUBTRACTION PROPERTIES OF INEQUALITY:

PROPERTIES OF INEQUALITY

Standard 1

For any numbers a, b, and c, if a>b then a+c>b+c and a-c>b-c

20 > 10+ 6 +626 > 16

12 < 22+5 +5 17 < 27

For any numbers a, b, and c, if a<b then a+c<b+c and a-c<b-c

40 > 30- 7 -733 > 23

34 < 50-2 -2 32 < 48



4

MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY

PROPERTIES OF INEQUALITYStandard 1

18 > 152 236 > 30

42 > 287 76 > 4

40 > 243 3120> 72

36 > 1212 123 > 1

If c>0 AND a > b then:ac

bcac>bc and >

22 < 304 488 < 120

55 < 11011 115 < 10

10 < 205 550< 100

40 < 7010 104 < 7

If c>0 AND a < b then:ac

bcac<bc and <



5


PROPERTIES OF INEQUALITY Standard 1

18 > 15-2 -2-36 < -30

42 > 28-7 -7-6 < -4

40 > 24-3 -3-120 < -72

36 > 12-12 -12-3 < -1

If c<0 AND a > b then:ac

bcac<bc and <

0 25 50X

-25-50

0 100 200X

-100-200

x42 6-2-4-6 8 10-8-10 0

F-2-4-6-8-10 0



6


PROPERTIES OF INEQUALITY Standard 1

22 < 30-4 -4-88 > -120

55 < 110-11 -11-5 > -10

10 < 20-5 -5-50> -100

40 < 70-10 -10-4 > -7

If c<0 AND a < b then:ac

bcac>bc and >

0 100 200X

-100-200

F-2-4-6-8-10 0

F-2-4-6-8-10 0

0 100 200X

-100-200



7

Standard 1Solve these inequalities then graph the solution:

X + 4 < 12 25 > 12 + K-4 -4

-12 -12

13 > K

K <13

10 10

6 6

X < -10

-9 -9X 2< F -6<

X < 8X

X < 8

K < 13K

X 2X <

X< -10X

0 5 10 15 20 25K

-5-10-15-20

6X < -60

10X 20< -9F 54>

F-2-4-6-8-10 0

x42 6-2-4-6 8 10-8-10 0

x42 6-2-4-6 8 10-8-10 0

x42 6-2-4-6 8 10-8-10 0

F -6F <



Standard 1

+1 +1

33< 3X 3 3

X>11

-24 > 8 -16X-40 -40

20 20

-10 -10

4 4

-8 -8

-32 > -16X-16 -16

X > 2

2 < X

Solve these inequalities and graph the solution:

20X + 40 180>

20X 140>

X 7>

32 < 3X- 1 10 + 4Z 26<

4Z 16<

Z 4<

X 7

X > X > 2X

X >11X Z 4Z <

x42 6-2-4-6 8 10-8-10 0

X42 6-2-4-6 8 10-8-10 0

0 5 10 15 20 25X

-5-10-15-20Z

42 6-2-4-6 8 10-8-10 0

Standard 1

+ 2 < 34X4

-2 -2

X4

<32(4) (4)

X < 128

-10 -10

(-8) (-8)

R< 223

32

32

R < 3

+1 +1

73

73

Solve these inequalities and graph the solution:

10- 46Y8

>

-Y 8

36>

Y -288<

R< 223

X – 1 237

>

X 337

>

X 7>

X < 128X Y -288Y <

R < 3R X 7

X >

0 100 200X

-100-200

R42 6-2-4-6 8 10-8-10 0

x42 6-2-4-6 8 10-8-10 0

128 Y-100-200-300 0

-288

10

12-4X8

8 >10X - 64

<12(4) (4)

10X – 6< 48+6 +6

10X < 5410 10

(8) (8)

64 > 12– 4X-12 -12

52 > -4X-4 -4

-13 < X

X > -13

10X - 64

<1212 -4X

88 >

Standard 1Solve these inequalities and graph the solution:

54105

504

5 410

5 2 5

X <X

X < 5 2 5

X > -13X

X42 6-2-4-6 8 10-8-10 0

0 5 10 15 20 25X

-5-10-15-20

....

22= 5

25



11

Standard 1

+4 +4

26 26

Solve this inequality and graph it:

26X – 4 360>

26X 364>

20X +12 6X-16+ 360>

X 14>

X 14 X >

0 5 10 15 20 25X

-5-10-15-20



12

- 28 -28

14Y < -24Y +152

+ 24Y + 24Y

38Y < 15238 38

Y < 4

14Y + 28 < 180 – 24Y 71 + ( Y + 4) > 90

75 + Y > 90

- 75 -75

Y > 15

Solve these inequalities and graph them:

Standard 1

Y < 4Y

Y > 15Y

Y42 6-2-4-6 8 10-8-10 0

0 5 10 15 20 25Y

-5-10-15-20



13

Solve this inequality and graph it:

37

>2X+12

2X

6X > 7(2X+12)6X > 14X + 84

-14X -14X

-8X > 84-8 -8

84 81

804

10 4 8

....

44=10

12

0

Standard 1

0 5 10 15 20 25X

-5-10-15-20

X<

1012

-

X <X 1012

-



14

6+5 < X 17+X < 9

11 < X

X > 1117+X < 9

-17 -17

X < -8

-9 -9

0 5 10 15 20 25x

-5-10-15-20

x

x

x

-8 7

OROR

Solve these compound inequalities:Standard 1

11

-8

9+X 16>

9+X 16>

X 7>

7

X | ORX<-8 X 7>



15

10+2 < X 21+X < 17

12 < X

X > 1221+X < 17

-21 -21

X < -4

-11 -11

0 5 10 15 20 25x

-5-10-15-20

x

x

x

-4 4

OROR


12

-4

11+X 15>

11+X 15>

X 4>

4

X | ORX<-4 X 4>



16

11+3 < X 15+X > 19

14 < X

X > 1415+X > 19

-15 -15

X > 4

-16 -16

0 5 10 15 20 25x

-5-10-15-20

x

x

x

2

OROR


14

4

16+X 18>

16+X 18>

X 2>

2

X | X 2>



17

11+3 >X 15+X > 19

14 > X

X < 1415+X > 19

-15 -15

X > 4

-16 -16

0 5 10 15 20 25x

-5-10-15-20

x

x

x

OROR


14

4

16+X 18>

16+X 18>

X 2>

2

X | RPRESENTATION CREATED BY SIMON PEREZ. All rights reserved


18

15+8 > X 15+X > 8 8+X > 15

23 > X

X < 2315+X > 8

-15 -15

X > -7

8+X > 15-8 -8

X > 7

0 5 10 15 20 25x

-5-10-15-20

x

x

x

-7

7

23

X | 7<X<23

7 23

ANDAND




19

(X+4)+(3X-5) > X (X+4 )+X > (3X-5) 4X -1 >X

-4X -4X

-1 >-3X-3 -3

.33 <X

X>.33

2X +4 > 3X-5-2X -2X

4 > X-5

+5 +5

9 > X

X < 9

0 5 10 15 20 25x

-5-10-15-20

x

x

x

9

3

.33

(3X-5) +X > (X+4 )4X – 5 > X +4 -X -X3X – 5 > 4

+5 +5

3X > 93 3

X > 3

3 9

X | 3<X< 9

AND AND




20

6+5 < X 17+X < 9

11 < X

X > 1117+X < 9

-17 -17

X < -8

-9 -9

x

x

x

ANDAND


11

-8

9+X 16>

9+X 16>

X 7>

7

0 5 10 15 20 25x

-5-10-15-20

NO SOLUTION!



21

ABSOLUTE VALUE INEQUALITIES

|X|<4

X> -4X<4

Standard 1

x

x

0 2 4 6 8 10x

-2 -4 -6 -8

AND

X | -4<X<4



22

X + 6 < 70

-6 -6

X<64

|X + 6|< 70

X + 6 > -70

-6 -6

Standard 1

X > -76

SOLVE:

x

x64

-76

-76 64

X | -76<X< 64

0 100 200X

-100-200

AND



23

5X + 10 < 40

-10 -10

5X < 305 5

X < 6

|5X + 10|<40

5X+10 >-40-10 -10

5X > -505 5

X > -10

Standard 1

SOLVE:

x

x6

-10

-10 6

X | -10<X< 6

0 10 20X

-10-20

AND


24

10X + 2 < 52

-2 -2

10X < 50 10 10

X < 5

20|10X + 2|< 1040

10X + 2 > -52

-2 -2

10X > -5410 10

X > -5.4

Standard 1

20 20

|10X + 2|< 52

SOLVE:

x

x5

-5.4

-54 5

X | -5.4<X< 5

0 5 10X

-5-10

AND


25

Standard 1

SOLVE:|12X+20| < 16X+8

- 20 -20

12X < 16X - 12

-16X -16X

-4X< - 12-4 -4

X > 3

-20 -20

12X > -16X - 28

+16X +16X

28X > -2828 28

X > -1

12X + 20 < 16X + 8

12X + 20 > -16X - 8

12X + 20 > -(16X + 8)AND

0 2 4 6 8 10x

-2 -4 -6 -8

x

x

3

3

-1

X | X>3PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26

ABSOLUTE VALUE INEQUALITIES

|X|>6

X< -6X>6

Standard 1

x

x

0 2 4 6 8 10x

-2 -4 -6 -8

OR

X | ORX<-6 X>6


27

X + 5 > 50

-5 -5

X > 45

|X + 5|> 50

X + 5 < -50

-5 -5

Standard 1

X < -55

SOLVE:

x

x45

-55

-55 450 100 200

X-100-200

OR

X | ORX<-55 X>45


28

15X + 30 > 120

-30 -30

15X > 9015 15

X > 6

|15X + 30|>120

15X+30 <-120-30 -30

15X < -15015 15

X < -10

Standard 1

SOLVE:

x

x6

-10 60 20 40

X -20-40

OR

X | ORX<-10 X>6

-10


29

|5X -1|> -3

Standard 1

Solution are all reals because absolute value is always positive and greater than any negative value.

|2X + 7 | + 76 < 20

-76 -76

|2X + 7 | < -56

No solution because absolute value is always positive and never less than a negative value.

SOLVE:

SOLVE:

SPECIAL CASES TO PAY ATTENTION!


Documents

1 PROPERTIES OF INEQUALITY Standard 1 SOLUTION OF INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES INEQUALITIES LEVEL 1 INEQUALITIES LEVEL 2 INEQUALITIES