1.7 – Linear Inequalities and Compound Inequalities Properties of Inequalities Addition and Subtraction Property of Inequality 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 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: 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

1.7 – Linear Inequalities and Compound Inequalities Solving Inequalities Examples: 4𝑥−9+3𝑥≤2𝑥−5+7𝑥 −7 𝑥+9 ≥40+3𝑥 7𝑥−9≤9𝑥−5 −7𝑥−63≥40+3𝑥 −2𝑥≤4 −10𝑥≥103 𝑥≥−2 𝑥≤−10.3 -3 -2 -1 9 10 11 −2, ∞ −∞, −10.3

1.7 – Linear Inequalities and Compound Inequalities Solving Inequalities Examples: −2 2−2𝑥 −4 𝑥+5 ≤−24 3 1−2𝑥 >8−6𝑥 −4+4𝑥−4𝑥−20≤−24 3−6𝑥>8−6𝑥 −24≤−24 3>8 (1) Lost the variable (1) Lost the variable (2) True statement (2) False statement ∴ Solution: All Reals ∴ Solution: the null set -1 1 -1 1 −∞, ∞ ∅ or { }

Properties of Inequalities Union and Intersection of Sets 1.7 – Linear Inequalities and Compound Inequalities Properties of Inequalities Union and Intersection of Sets The Union of sets A and B 𝐴∪𝐵 represents the elements that are in either set. The Intersection of sets A and B 𝐴∩𝐵 represents the elements that are common to both sets. Examples: Determine the solution for each set operation. 𝐴: 𝑥|𝑥≥5 𝐵: 𝑥|3≤𝑥<12 𝐶: 𝑥|𝑥<−1 ) ) 5 3 12 -1 𝐴∩𝐵 𝐴∪𝐵 𝐵∩𝐶 𝐴∪𝐶 5, 12 3, ∞ ∅ −∞, −1 ∪ 5, ∞

Compound Inequalities 1.7 – Linear Inequalities and Compound Inequalities Compound Inequalities Example: 7𝑣−5≥65 𝑜𝑟 −3𝑣−2≥−2 7𝑣−5≥65 −3𝑣−2≥−2 7𝑣≥70 −3𝑣≥0 𝑣≥10 𝑣≤0 −∞, 0 ∪ 10, ∞

Compound Inequalities 1.7 – Linear Inequalities and Compound Inequalities Compound Inequalities Example: 8𝑥+8≥−64 𝑎𝑛𝑑 −7−8𝑥≥−79 8𝑥+8≥−64 −7−8𝑥≥−79 8𝑥≥−72 −8𝑥≥−72 𝑥≥−9 𝑥≤9 −9, 9

Absolute Value Equations Properties of Absolute Values Equations 1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Properties of Absolute Values Equations 𝒖 =𝒂 𝑎<0 No Solution 𝑎=0 One solution: 𝑢=0 𝑎>0 Two Solutions: 𝑢=𝑎 𝑜𝑟 𝑢=−𝑎 𝑢 = 𝑚 𝑢=𝑤 or u=−w 𝑢 = 𝑤 ±𝑢=±𝑤 +𝑢=+𝑤 +𝑢=−𝑤 −𝑢=+𝑤 −𝑢=−𝑤 𝑢=𝑤 𝑢=−𝑤 𝑢=−𝑤 𝑢=𝑤 𝑢=𝑤 𝑢=−𝑤

Absolute Value Equations 1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Examples: −2𝑛+6 =6 −2𝑛+6=6 −2𝑛+6=−6 −2𝑛=0 −2𝑛=−12 𝑛=0 𝑛=6 𝑛=0, 6 𝑥+8 −5=2 𝑥+8 =7 𝑥+8=−7 𝑥+8=7 𝑥=−15 𝑥=−1 𝑥=−15, −1

Absolute Value Equations 1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 3 3−5𝑟 −3=18 3 3−5𝑟 =21 3−5𝑟 =7 3−5𝑟=−7 3−5𝑟=7 −5𝑟=−10 −5𝑟=4 𝑟=2 𝑟=− 4 5 𝑟=− 4 5 , 2

Absolute Value Equations 1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 5 9−5𝑛 −7=38 5 9−5𝑛 =45 9−5𝑛 =9 9−5𝑛=−9 9−5𝑛=9 −5𝑛=−18 −5𝑛=0 𝑛= 18 5 𝑛=0 𝑛=2, 18 5

Absolute Value Equations 1.8 – Absolute Value Equations and Inequalities Absolute Value Equations Example: 2𝑥−1 = 4𝑥+9 2𝑥−1=4𝑥+9 2𝑥−1=− 4𝑥+9 −2𝑥=10 2𝑥−1=−4𝑥−9 𝑥=−5 6𝑥=−8 𝑥=− 8 6 =− 4 3 𝑥=−5, − 4 3

Absolute Value Inequalities Properties of Absolute Values Inequalities 1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑢 <𝑎 −𝑎<𝑢<𝑎 𝑢 ≤𝑎 −𝑎≤𝑢≤𝑎 𝑢 >𝑎 𝑢<−𝑎 𝑜𝑟 𝑢>𝑎 𝑢 ≥𝑎 𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎 10𝑦−4 <34 10𝑦−4>−34 10𝑦−4<34 10𝑦>−30 10𝑦<38 𝑦>−3 𝑦< 19 5 𝑜𝑟 𝑦<3.8 −3<𝑦<3.8 −3, 3.8

Absolute Value Inequalities Properties of Absolute Values Inequalities 1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑢 <𝑎 −𝑎<𝑢<𝑎 𝑢 ≤𝑎 −𝑎≤𝑢≤𝑎 𝑢 >𝑎 𝑢<−𝑎 𝑜𝑟 𝑢>𝑎 𝑢 ≥𝑎 𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎 −8𝑥−3 >11 −8𝑥−3<−11 −8𝑥−3>11 −8𝑥<−8 −8𝑥>14 𝑥<− 7 4 𝑜𝑟 𝑥>1 𝑥>1 𝑥<− 14 8 −∞, − 7 4 ∪ 1, ∞ 𝑥<− 7 4

Absolute Value Inequalities Properties of Absolute Values Inequalities 1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑢 <𝑎 −𝑎<𝑢<𝑎 𝑢 ≤𝑎 −𝑎≤𝑢≤𝑎 𝑢 >𝑎 𝑢<−𝑎 𝑜𝑟 𝑢>𝑎 𝑢 ≥𝑎 𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎 4 6−2𝑎 +8≤24 4 6−2𝑎 ≤16 6−2𝑎 ≤4 6−2𝑎≥−4 6−2𝑎≤4 −2𝑎≥−10 −2𝑎≤−2 𝑎≤5 𝑎≥1 1≤𝑎≤5 [1, 5]

Absolute Value Inequalities Properties of Absolute Values Inequalities 1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑢 <𝑎 −𝑎<𝑢<𝑎 𝑢 ≤𝑎 −𝑎≤𝑢≤𝑎 𝑢 >𝑎 𝑢<−𝑎 𝑜𝑟 𝑢>𝑎 𝑢 ≥𝑎 𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎 9 𝑟−2 −10<−73 9 𝑟−2 <−63 𝑟−2 <−7 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ∴𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

Absolute Value Inequalities Properties of Absolute Values Inequalities 1.8 – Absolute Value Equations and Inequalities Absolute Value Inequalities Properties of Absolute Values Inequalities 𝑢 <𝑎 −𝑎<𝑢<𝑎 𝑢 ≤𝑎 −𝑎≤𝑢≤𝑎 𝑢 >𝑎 𝑢<−𝑎 𝑜𝑟 𝑢>𝑎 𝑢 ≥𝑎 𝑢≤−𝑎 𝑜𝑟 𝑢≥𝑎 5 3𝑛−2 +6≥51 5 3𝑛−2 ≥45 3𝑛−2 ≥9 3𝑛−2≤−9 3𝑛−2≥9 3𝑛≤−7 3𝑛≥11 𝑛≤− 7 3 𝑛≥ 11 3 −∞, − 7 3 ∪ 11 3 , ∞ 𝑛≤− 7 3 𝑜𝑟 𝑛≥ 11 3