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Dr. Abdulla Eid Section 1.4 Absolute Value Dr. Abdulla Eid College of Science MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Absolute Value 1 / 21

d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

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Page 1: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEidSection 1.4

Absolute Value

Dr. Abdulla Eid

College of Science

MATHS 103: Mathematics for Business I

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Page 2: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Absolute Value

Definition

The absolute value of any number x is the distance between x and thezero. We denote it by |x |.

Example

|2| = distance between 2 and 0 = 2.

| − 3| = distance between -3 and 0 = 3.

|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.

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Page 3: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Properties of Absolute Values

1 |ab| = |a| · |b|.

2

∣∣ ab

∣∣ = |a||b| .

3 |a− b| = |b− a|.

4 |a+ b|≤|a|+ |b|.

5 −|a| ≤ a ≤ |a|.

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Page 4: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Rules

For equations (or inequalities) that involve absolute value we need to getrid of the absolute value which can be done only using the following threerules:

1 Rule 1: |X | = a→ X = a or X = −a.

2 Rule 2: |X | < a→ −a < X < a.

3 Rule 3: |X | > a→ X > a or X < −a.

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Page 5: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve |x − 3| = 2

Solution: We solve the absolute value using rule 1 to get rid of theabsolute value.

|x − 3| = 2

x − 3 = 2 or x − 3 = −2x = 5 or x = 1

Solution Set = {5, 1}.

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Page 6: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

Solve |7− 3x | = 5

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Page 7: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve |x − 4| = −3

Solution: Caution: The absolute value can never be negative, so in thisexample, we have to stop and we say there are no solution!Solution Set = {} = ∅.

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Page 8: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

(Old Exam Question) Solve |7x + 2| = 16.

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Page 9: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve |x − 2| < 4

Solution:

|x − 2| < 4

− 4 < x − 2 < 4

− 4+ 2 < x < 4+ 2

− 2 < x < 6

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Page 10: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

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The Solution

1- Set notationSolution Set = {x | − 2 < x < 6}

2- Number Line notation

3- Interval notation(−2, 6)

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Page 11: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

Solve |3− 2x | ≤ 5

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Page 12: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve |x + 5| ≤ −2

Solution: Caution: The absolute value can never be negative or less than anegative, so in this example, we have to stop and we say there are nosolution!Solution Set = {} = ∅.

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Page 13: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

(Old Final Exam Question) Solve |5− 6x | ≤ 1

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Page 14: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

(Old Final Exam Question) Solve |2x − 7| ≤ 9

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Page 15: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7

x + 5 ≥ 7 or x + 5 ≤ −7x ≥ 2 or x ≤ −12

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Page 16: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

The Solution

1- Set notation

Solution Set = {x | x ≥ 2 or x ≥ −12}

2- Number Line notation

3- Interval notation(−∞,−12]∪[2,∞)

where ∪ means union of two intervals.

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Page 17: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

Solve |3x − 4| > 1

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Page 18: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Example

Solve | 3x−82 | ≥ 4

Solution:

|3x − 8

2| ≥ 4

3x − 8

2≥ 4 or

3x − 8

2≤ −4

3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

x ≥ 16

3or x ≤ 0

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Page 19: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

The Solution

1- Set notation

Solution Set = {x | x ≥ 16

3or x ≥ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [16

3,∞)

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Page 20: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

(Old Exam Question) Solve |x + 8|+ 3 < 2

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Page 21: d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute Value 4 / 21. d Example Solve jx 3j= 2 Solution: We solve the absolute value using

Dr. AbdullaEid

Exercise

(Old Exam Question) Solve |10x − 9| ≥ 11.

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