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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
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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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