Section 1.4 d i E Absolute Value a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Absolute Value 1 / 21
Absolute Value Definition The absolute value of any number x is the distance between x and the zero. We denote it by | x | . d i E Example a l l u | 2 | = distance between 2 and 0 = 2. d b A | − 3 | = distance between -3 and 0 = 3. . r D | 0 | = distance between 0 and 0 = 0. | − 2 | = distance between -2 and 0 = 2. Note: The absolute value | x | is always non–negative , i.e., | x | ≥ 0. Dr. Abdulla Eid (University of Bahrain) Absolute Value 2 / 21
Properties of Absolute Values 1 | ab | = | a | · | b | . d i E � = | a | � � a � | b | . 2 b a l l u 3 | a − b | = | b − a | . d b A 4 | a + b |≤| a | + | b | . . r D 5 −| a | ≤ a ≤ | a | . Dr. Abdulla Eid (University of Bahrain) Absolute Value 3 / 21
Rules For equations (or inequalities) that involve absolute value we need to get d rid of the absolute value which can be done only using the following three i E rules: a 1 Rule 1: | X | = a → X = a or X = − a . l l u d b 2 Rule 2: | X | < a → − a < X < a . A . r 3 Rule 3: | X | > a → X > a or X < − a . D Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21
Example Solve | x − 3 | = 2 d i Solution: We solve the absolute value using rule 1 to get rid of the E absolute value. a l l u | x − 3 | = 2 d b x − 3 = 2 or x − 3 = − 2 A x = 5 or x = 1 . r D Solution Set = { 5, 1 } . Dr. Abdulla Eid (University of Bahrain) Absolute Value 5 / 21
Exercise Solve | 7 − 3 x | = 5 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 6 / 21
Example Solve | x − 4 | = − 3 Solution: Caution: The absolute value can never be negative, so in this example, we have to stop and we say there are no solution! d Solution Set = {} = ∅ . i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 7 / 21
Exercise (Old Exam Question) Solve | 7 x + 2 | = 16. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 8 / 21
Example Solve | x − 2 | < 4 d i Solution: E a l l u | x − 2 | < 4 d b − 4 < x − 2 < 4 A − 4 + 2 < x < 4 + 2 . r D − 2 < x < 6 Dr. Abdulla Eid (University of Bahrain) Absolute Value 9 / 21
The Solution 1- Set notation d Solution Set = { x | − 2 < x < 6 } i E a l 2- Number Line notation l u d b A . 3- Interval notation r D ( − 2, 6 ) Dr. Abdulla Eid (University of Bahrain) Absolute Value 10 / 21
Exercise Solve | 3 − 2 x | ≤ 5 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 11 / 21
Example Solve | x + 5 | ≤ − 2 Solution: Caution: The absolute value can never be negative or less than a negative, so in this example, we have to stop and we say there are no d solution! i E Solution Set = {} = ∅ . a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 12 / 21
Exercise (Old Final Exam Question) Solve | 5 − 6 x | ≤ 1 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 13 / 21
Exercise (Old Final Exam Question) Solve | 2 x − 7 | ≤ 9 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 14 / 21
Example Solve | x + 5 | ≥ 7 d i E Solution: a l l u d | x + 5 | ≥ 7 b A x + 5 ≥ 7 or x + 5 ≤ − 7 . x ≥ 2 or x ≤ − 12 r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 15 / 21
The Solution 1- Set notation d Solution Set = { x | x ≥ 2 or x ≥ − 12 } i E a l l u 2- Number Line notation d b A . r D 3- Interval notation ( − ∞ , − 12 ] ∪ [ 2, ∞ ) where ∪ means union of two intervals. Dr. Abdulla Eid (University of Bahrain) Absolute Value 16 / 21
Exercise Solve | 3 x − 4 | > 1 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 17 / 21
Example Solve | 3 x − 8 | ≥ 4 2 Solution: d i E | 3 x − 8 a | ≥ 4 l l 2 u d 3 x − 8 ≥ 4 or 3 x − 8 ≤ − 4 b 2 2 A 3 x − 8 ≥ 8 or 3 x − 8 ≤ − 8 . r D 3 x ≥ 16 or 3 x ≤ 0 x ≥ 16 3 or x ≤ 0 Dr. Abdulla Eid (University of Bahrain) Absolute Value 18 / 21
The Solution 1- Set notation d Solution Set = { x | x ≥ 16 3 or x ≥ 0 } i E a l l u 2- Number Line notation d b A . r D 3- Interval notation ( − ∞ , 0 ] ∪ [ 16 3 , ∞ ) Dr. Abdulla Eid (University of Bahrain) Absolute Value 19 / 21
Exercise (Old Exam Question) Solve | x + 8 | + 3 < 2 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 20 / 21
Exercise (Old Exam Question) Solve | 10 x − 9 | ≥ 11. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Absolute Value 21 / 21
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