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

Absolute Value Inequalities

1.8 – Absolute Value Equations and Inequalities

Properties of Absolute Values Inequalities

5|3𝑛−2|+6 ≥51

3𝑛−2≤−9 3𝑛−2≥93𝑛≤−7

𝑛≤−73

3𝑛≥11

𝑛≥113

(−∞ , − 73 ]∪[ 11

3, ∞ )

5|3𝑛−2|≥45|3𝑛−2|≥9

𝑛≤−73𝑜𝑟 𝑛≥

113