1

PROPERTIES OF EQUALITY

Standard 1SOLUTION OF EQUATIONS

INCLUDING ABSOLUTE VALUE

EQUATIONS

EQUATIONS LEVEL 1

EQUATIONS LEVEL 2

EQUATIONS LEVEL 3

EQUATIONS LEVEL 4

EQUATIONS LEVEL 5

ABSOLUTE VALUE DEFINITION

ABSOLUTE VALUE EQUATIONSEND SHOW

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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:

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3

PROPERTIES OF EQUALITYStandard 1

REFLEXIVE PROPERTY OF EQUALITY:

For any real number a, a=a 5=5-10=-10

SYMMETRIC PROPERTY OF EQUALITY:

For all real numbers a and b, if a=b, then b=a

X=5 5=X

6X-12=8 8=6X-12

9Y -2Y +1= 3X2 3X= 9Y -2Y+12

SUBSTITUTION PROPERTY OF EQUALITY:

If a=b, then a may be replaced by b. b=2 and 3b +1=7If

then 3( )+1=72PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4

TRANSITIVE PROPERTY OF EQUALITY:

For all real numbers a, b, and c, if a=b, and b=c then a=c

If X=6 and Y= 6 then X=Y

If Y=2X+2 and Y=6-3X then 2X+2=6-3X

ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY:

PROPERTIES OF EQUALITYStandard 1

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

10 = 10+ 6 +616 = 16

22 = 22-5 -5 17 = 17

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5

MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY:

PROPERTIES OF EQUALITYStandard 1

For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, =ac

bc

15 = 152 230 = 30

28 = 287 74 = 4

24 = 243 372 = 72

36 = 3612 123 = 3

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6

Standard 1Solve these equations:

X + 2 = 10 X - 6 = 15 8 + Y = 11 23 = 10 + K

12 = 9 + R 7 = Z -34 -J +8 = 26 -12 = -9 -W

5X = 10 12 = 4Y 2X = -20 -3F =18

-2 -2X = 8

+6 +6

X = 21

-8 -8

Y = 3

-10 -10

13 = K

K =13

-9 -9

3 = RR = 3

+34 +34

41 = Z

Z = 41

-8 -8

-J = 18 (-1)(-1)

J = -18

+9 +9

-3 = -W(-1)(-1)

3 = W

W = 3

5 5

X = 24 4

3 = Y

Y = 3

2 2

X = -10

-3 -3

F = -6

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7

Standard 1

K 9

= 5 R

2= -8

X 4

6= Y

-5 3=

Z -3

-2 =

+2 +2

66=6X 6 6

X=11

5 + 2Z = 13 -12 = 4 -8X64 = 6X- 210X + 20 = 90-20 -20

10X = 7010 10

X=7

(9) (9)

K = 45

(2) (2)

R = -16

(4) (4)

24 =X

X = 24

(-5) (-5)

-15 = Y

Y = -15

6 = Z

(-3) (-3)

Z = 6

-5 -5

2Z = 8 2 2

Z = 4

-4 -4

-16 = -8X-8 -8

X = 2

2 = X

K 9

= 5 R 2

= -8 X

4 6= Y

-5 3= Z

-3-2 =

Solve these equations:

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8

Standard 1

– 4 = 12R3

2 + = 13Y5

+ 1 = 17X2 5 - = 23

Y4

6 -2X4

8 =5X - 32

= 6X= 33

7

-1 -1

X2

=16(2) (2)

X = 32

+4 +4

R3

=16(3) (3)

R = 48

-2 -2

Y5

=11(5) (5)

Y = 55

-5 -5

-Y 4

=18

Y = -72

(-4) (-4)

(2) (2)

5X – 3= 12+3 +3

5X = 155 5X = 3

(4) (4)

32 = 6 – 2X-6 -6

26 = -2X-2 -2

-13 = X

X = -13

R= 223

32

32

R = 3

+1 +1

X – 1 = 237

5X - 32

= 66 -2X

48 = R= 22

3

73

73

X = 7

Solve these equations:

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9

Standard 1

10X + 6 3X - 8 + = 18013X – 2 = 180

+2 +2

13X = 18213 13

X= 14

9X + 10 20X-7 + = 9029X +3 = 90

-3 -3

29X = 8729 29

X = 3

- 14 -14

- X = 8X + 1

-8X -8X

-9X = 1-9 -9

X -.11

14 – X = 8X + 15 9X + 28 = 13X -16

- 28 -28

9X = 13X - 44

-13X -13X

-4X= - 44-4 -4

X = 11

Solve these equations:

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10

- 14 -14

7Y = -12Y +76

+ 12Y + 12Y

19Y = 7619 19

Y = 4

7Y + 14 = 90 – 12Y142 + (2Y + 8) = 180

150 + 2Y = 180

-150 -150

2Y = 302 2

Y = 15

-2Y -2Y

36 = 3Y 3 3

Y=12

2Y+36=5Y 5X + 5 = 7X - 19 -5X -5X

5 = 2X - 19+19 +19

24 = 2X 2 2

X=12

Solve these equations:Standard 1

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11

100 + (7X) + (10X) + (9X-6) + (10X+3) + (7X-9) = 720

7X + 10X + 9X + 10X + 7X + 100 – 6 +3 -9 = 720

43X + 88 = 720

-88 -88

43 43

X 14.7

43X = 632

=9037-X +2X-16

+X+13

37 – 16 + 13 –X +2X +X = 90

34 +2X = 90-34 -34

2X = 562 2

X=28

Solve these equations:Standard 1

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12

10 = (10X)1

210 = 5X5 5

X = 2

= ( + )

1

23X-1 7X+110(12X+11) + (9X+3) + (7X+26) = 180

28X + 40 = 180-40 -40

28X = 14028 28

X = 5

Solve these equations:

12X + 9X + 7X + 11 + 3 + 26 = 180

2 = X

3(4X – 5)= -2(3X -3)

12X – 15 = -6X +6

+15 +15

12X = -6X + 21

+6X +6X

18X = 2118 18

21181

183

X =

1 318

....

33

=116

116

OR X 1.2

Standard 1

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13

Solve these equations:

23

=4(3X-2)5(2X+6)

23

=12X -8

10X+30

2(10X+30)=3(12X-8)

20X +60 = 36X -24

+24 +24

20X + 84 = 36X

-20X -20X

84 = 16X

16 16

84165

804

5 416

514

X = OR X 5.25

37

=X+6

X

3X = 7(X+6)3X = 7X + 42

-7X -7X

-4X = 42-4 -4

42 41

42

10 2 4

....

22

=1012

0

OR X= -10.5

Standard 1

X= 1012

-

....

44

51 4

=

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14

ABSOLUTE VALUE

|X|=2

0 1 2 3 4 5 6 7 8 9 10 11 12-1-2-3-4-5-6-7-10-11-12 -8-9

X= -2 X=2

|X|=12

0 1 2 3 4 5 6 7 8 9 10 11 12-1-2-3-4-5-6-7-10-11-12 -8-9

X= -12 X=12

For any real number a:

If a < 0, then |a|= -a;

If a > 0, then |a|= a;Absolute Value:

Standard 1

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15

X + 4 = 70

-4 -4

X=66

|X + 4|= 70

-(X + 4) = 70

+4 +4

X= -74

- X - 4 = 70

| ( ) + 4 |= 7066 | ( ) + 4 |= 70-74

|70 |= 70

70= 70

|-74 + 4 |= 70

|-70 |= 70

70= 70

66 and -74 are solutions

Standard 1

|X + 4|= 70

-X = 74(-1) (-1)

SOLVE:

Check:

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16

10X + 20 = 80

-20 -20

10X = 6010 10

X=6

|10X + 20|= 80

-(10X + 20) = 80

+20 +20

-10X = 100-10 -10

X= -10

-10X - 20 = 80

|10( ) + 20 |= 806 |10( ) + 20 |= 80-10

|60 + 20 |= 80

|80 |= 80

80= 80

|-100 + 20 |= 80

|-80 |= 80

80= 80

6 and -10 are solutions

Standard 1

|10X + 20 |= 80

SOLVE:

Check:

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17

5X + 2 = 52

-2 -2

5X = 50 5 5

X=10

10|5X + 2|= 520

-(5X + 2) = 52

+2 +2

-5X = 54-5 -5

X= -10.8

-5X - 2 = 52

10|5( ) + 2|= 52010 10|5( ) + 2 |= 520-10.8

10|50 + 2 |= 520

10|52 |= 520

520= 520

10|-54 + 2 |= 520

10|-52 |= 80

520= 520

10 and -10.8 are solutions

Standard 1

10|5X + 2 |= 520

10 10

|5X + 2|= 52

10(52)=520 10(52)=520

SOLVE:

Check:

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18

Standard 1

SOLVE:

|6X+10| = 8X+4

- 10 -10

6X = 8X - 6

-8X -8X

-2X= - 6-2 -2

X = 3

+10 +10

-6X = 8X + 14

-8X -8X

-14X= 14-14 -14

X = -1

6X + 10 = 8X + 4

-6X - 10 = 8X + 4

-(6X + 10) = 8X + 4

|6X+10| = 8X+4

|6( )+10| = 8( )+433 |6( )+10| = 8( )+4-1 -1

|18+10| = 24 + 4

|28| = 28

28 = 28

|-6+10| =-8+4

|4| = -4

4 = -4

The only solution is 3

Check:

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19

|8X -9|= -1

Standard 1

No solution because absolute value is never negative.

|6X + 7 | + 70 = 20

-70 -70

|6X + 7 | = -50

No solution because absolute value is never negative.

SOLVE:

SOLVE:

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