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Dr. AbdullaEidSection 1.4
Absolute Value
Dr. Abdulla Eid
College of Science
MATHS 103: Mathematics for Business I
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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| = 2
x − 3 = 2 or x − 3 = −2x = 5 or x = 1
Solution Set = {5, 1}.
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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| ≥ 7
x + 5 ≥ 7 or x + 5 ≤ −7x ≥ 2 or x ≤ −12
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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 − 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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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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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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