Properties of Inequalities 1.7 – Linear Inequalities and Compound Inequalities If a b, then a + c...
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Transcript of Properties of Inequalities 1.7 – Linear Inequalities and Compound Inequalities If a b, then a + c...
Properties of Inequalities
1.7 – Linear Inequalities and Compound Inequalities
If a < b, then a + c < b + c or If a > b, then a + c > b + c
If a < b, then a - c < b - c or If a > b, then a - c > b - c
If a < b, then a • c < b • c or If a > b, then a • c > b • c
If a < b, then a/c < b/c or If a > b, then a/c > b/c
Addition and Subtraction Property of Inequality
Multiplication and Division Property of Inequality
c is positive:
If a < b, then a • c > b • c or If a > b, then a • c < b • c
If a < b, then a/c > b/c or If a > b, then a/c < b/c
c is negative:
Solving Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Examples:
-3 -2 -1
4 𝑥−9+3 𝑥≤2𝑥−5+7 𝑥7 𝑥−9≤9𝑥−5−2 𝑥≤ 4𝑥≥−2
[−2 , ∞ ) 9 10 11
−7 (𝑥+9 )≥ 40+3 𝑥−7 𝑥−63≥40+3 𝑥
−10 𝑥≥103𝑥≤−10.3
(−∞ ,−10.3 ]
Solving Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Examples:
-1 0 1
−2 (2−2 𝑥 )−4 (𝑥+5 )≤−24−4+4 𝑥−4 𝑥−20≤−24
−24 ≤−24(1) Lost the variable
(−∞ ,∞ ) -1 0 1
3 (1−2 𝑥 )>8−6 𝑥3−6𝑥>8−6 𝑥
3>8
∅∨{}
(2) True statementSolution: All Reals(1) Lost the variable(2) False statementSolution: the null set
Properties of Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Union and Intersection of SetsThe Union of sets A and B represents the elements that are in either set.The Intersection of sets A and B represents the elements that are common to both sets.𝐴 : {𝑥∨𝑥 ≥5 } 𝐵 : {𝑥∨3≤ 𝑥<12 } 𝐶 : {𝑥∨𝑥<−1 }
Examples: Determine the solution for each set operation.𝐴∩𝐵
5 )3 12𝐴∪𝐵
[5 , 12 )
-1)[3 , ∞ )
𝐵∩𝐶∅
𝐴∪𝐶(−∞ ,−1 )∪ [5 , ∞ )
Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Example:
7𝑣−5≥65𝑜𝑟 −3𝑣−2≥−27𝑣−5≥657𝑣 ≥70𝑣 ≥10
−3 𝑣−2≥−2−3 𝑣≥0𝑣 ≤0
(−∞ , 0 ]∪ [10 , ∞ )
Compound Inequalities
1.7 – Linear Inequalities and Compound Inequalities
Example:
8 𝑥+8≥−64 𝑎𝑛𝑑−7−8 𝑥≥−798 𝑥+8≥−64
8 𝑥≥−72𝑥≥−9
−7−8 𝑥≥−79−8𝑥 ≥−72𝑥≤9
[−9 ,9 ]
Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
Properties of Absolute Values Equations
No Solution
One solution:
Two Solutions:
|𝑢|=|𝑤|±𝑢=±𝑤
+𝑢=+𝑤 +𝑢=−𝑤 −𝑢=+𝑤 −𝑢=−𝑤𝑢=𝑤 𝑢=−𝑤 𝑢=−𝑤 𝑢=𝑤
𝑢=𝑤 𝑢=−𝑤
Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
|−2𝑛+6|=6−2𝑛+6=6 −2𝑛+6=−6
Examples:
−2𝑛=0𝑛=0
−2𝑛=−12𝑛=6
𝑛=0 ,6
|𝑥+8|−5=2
𝑥+8=−7 𝑥+8=7𝑥=−15 𝑥=−1
𝑥=−15 ,−1
|𝑥+8|=7
Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
3|3−5𝑟|−3=18
3−5𝑟=−7 3−5𝑟=7
Example:
−5𝑟=−10𝑟=2
3|3−5𝑟|=21
|3−5𝑟|=7
−5𝑟=4
𝑟=−45
𝑟=−45,2
Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
5|9−5𝑛|−7=38
9−5𝑛=−9 9−5𝑛=9
Example:
−5𝑛=−18
𝑛=185
5|9−5𝑛|=45
|9−5𝑛|=9
−5𝑛=0𝑛=0
𝑛=2 ,185
Absolute Value Equations
1.8 – Absolute Value Equations and Inequalities
|2 𝑥−1|=|4 𝑥+9|2 𝑥−1=4 𝑥+9 2 𝑥−1=− ( 4 𝑥+9 )
Example:
−2 𝑥=10𝑥=−5
2 𝑥−1=−4 𝑥−96 𝑥=−8
𝑥=−5 ,−43
𝑥=−86=−
43
Absolute Value Inequalities
1.8 – Absolute Value Equations and Inequalities
Properties of Absolute Values Inequalities
|10 𝑦−4|<3410 𝑦−4>−34 10 𝑦−4<34
10 𝑦>−30𝑦>−3
10 𝑦<38
𝑦<195𝑜𝑟 𝑦<3.8
−3<𝑦<3.8(−3 ,3.8 )
Absolute Value Inequalities
1.8 – Absolute Value Equations and Inequalities
Properties of Absolute Values Inequalities
|−8 𝑥−3|>11−8𝑥−3<−11 −8𝑥−3>11−8𝑥<−8𝑥>1
−8𝑥>14
𝑥<−148
𝑥<−74𝑜𝑟 𝑥>1
(−∞ ,− 74 )∪ (1 ,∞ )
𝑥<−74
Absolute Value Inequalities
1.8 – Absolute Value Equations and Inequalities
Properties of Absolute Values Inequalities
4|6−2𝑎|+8≤2 4
6−2𝑎≥−4 6−2𝑎≤4−2𝑎≥−10𝑎≤5
−2𝑎≤−2𝑎≥1
1≤𝑎≤5[1 ,5 ]
4|6−2𝑎|≤16|6−2𝑎|≤4
Absolute Value Inequalities
1.8 – Absolute Value Equations and Inequalities
Properties of Absolute Values Inequalities
9|𝑟 −2|−10<−73
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑣𝑎𝑙𝑢𝑒𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑒𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
9|𝑟 −2|<−63|𝑟 −2|<−7
∴𝑛𝑜𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛