1 properties of inequality standard 1 solution of inequalities including absolute value inequalities...
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1
PROPERTIES OF INEQUALITY
Standard 1SOLUTION OF INEQUALITIES
INCLUDING ABSOLUTE VALUE
INEQUALITIES
INEQUALITIES LEVEL 1
INEQUALITIES LEVEL 2
INEQUALITIES LEVEL 3
INEQUALITIES LEVEL 4
COMPOUND INEQUALITIES “OR”
ABSOLUTE VALUE INEQUALITIES “AND”
ABSOLUTE VALUE INEQUALITIES “OR”
COMPOUND INEQUALITIES “AND”
END 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
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
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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 <
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5
MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY
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
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6
MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY
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
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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 <
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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
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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
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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
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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
-
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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>
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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
Solve these compound inequalities:Standard 1
12
-4
11+X 15>
11+X 15>
X 4>
4
X | ORX<-4 X 4>
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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
Solve these compound inequalities:Standard 1
14
4
16+X 18>
16+X 18>
X 2>
2
X | X 2>
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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
Solve these compound inequalities:Standard 1
14
4
16+X 18>
16+X 18>
X 2>
2
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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
Solve these compound inequalities:Standard 1
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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
Solve these compound inequalities:Standard 1
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20
6+5 < X 17+X < 9
11 < X
X > 1117+X < 9
-17 -17
X < -8
-9 -9
x
x
x
ANDAND
Solve these compound inequalities:Standard 1
11
-8
9+X 16>
9+X 16>
X 7>
7
0 5 10 15 20 25x
-5-10-15-20
NO SOLUTION!
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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
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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
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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
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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
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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
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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
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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
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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
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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!
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