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# d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute...

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• Dr. AbdullaEidSection 1.4Absolute Value

Dr. Abdulla Eid

College of Science

MATHS 103: Mathematics for Business I

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• 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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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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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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Example

Solve |x − 3| = 2

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

|x − 3| = 2x − 3 = 2 or x − 3 = −2

x = 5 or x = 1

Solution Set = {5, 1}.

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• Dr. AbdullaEid

Exercise

Solve |7− 3x | = 5

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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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Exercise

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

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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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The Solution

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

2- Number Line notation

3- Interval notation(−2, 6)

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Exercise

Solve |3− 2x | ≤ 5

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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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Exercise

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

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Exercise

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

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Example

Solve |x + 5| ≥ 7

Solution:

|x + 5| ≥ 7x + 5 ≥ 7 or x + 5 ≤ −7

x ≥ 2 or x ≤ −12

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• 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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Exercise

Solve |3x − 4| > 1

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Example

Solve | 3x−82 | ≥ 4

Solution:

|3x − 82| ≥ 4

3x − 82≥ 4 or 3x − 8

2≤ −4

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

x ≥ 163

or x ≤ 0

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• Dr. AbdullaEid

The Solution

1- Set notation

Solution Set = {x | x ≥ 163

or x ≥ 0}

2- Number Line notation

3- Interval notation

(−∞, 0] ∪ [163,∞)

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Exercise

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

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Exercise

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

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