ABSOLUTE VALUE
THANOMSAK LAOKUL
Mathematics Teacher
Mahidol Wittayanusorn School
MATH 30102
T.Laokul – p. 1/
Absolute Value
If a is a real number, then the absolute value of a is
a
|a| =
−a
if a ≥ 0
if a < 0.
T.Laokul – p. 2/
Absolute Value
If a is a real number, then the absolute value of a is
a
|a| =
−a
if a ≥ 0
if a < 0.
Distance Between Two Points on the Real Line
Let a and b be real numbers. The distance between
a and b is
d(a, b) = |b − a| = |a − b|.
T.Laokul – p. 2/
Properties of Absolute Values
For any real number a and b
T.Laokul – p. 3/
Properties of Absolute Values
For any real number a and b
1. |a| ≥ 0
T.Laokul – p. 3/
Properties of Absolute Values
For any real number a and b
1. |a| ≥ 0
2. | − a| = |a|
T.Laokul – p. 3/
Properties of Absolute Values
For any real number a and b
1. |a| ≥ 0
2. | − a| = |a|
3. |ab| = |a||b|
T.Laokul – p. 3/
Properties of Absolute Values
For any real number a and b
1. |a| ≥ 0
2. | − a| = |a|
3. |ab| = |a||b|
a |a|
4. =
, b 6= 0
b
|b|
T.Laokul – p. 3/
Properties of Absolute Values
For any real number a and b
1. |a| ≥ 0
2. | − a| = |a|
3. |ab| = |a||b|
a |a|
4. =
, b 6= 0
b
|b|
5. −|x| ≤ x ≤ |x| proof
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Properties of Absolute Values
For any real number a and b
T.Laokul – p. 4/
Properties of Absolute Values
For any real number a and b
6. Let a > 0 and x ∈ R, if |x| ≤ a then −a ≤ x ≤ a
T.Laokul – p. 4/
Properties of Absolute Values
For any real number a and b
6. Let a > 0 and x ∈ R, if |x| ≤ a then −a ≤ x ≤ a
7. Let a > 0 and x ∈ R, if |x| ≥ a then x ≥ a or x ≤ −a
T.Laokul – p. 4/
Properties of Absolute Values
For any real number a and b
6. Let a > 0 and x ∈ R, if |x| ≤ a then −a ≤ x ≤ a
7. Let a > 0 and x ∈ R, if |x| ≥ a then x ≥ a or x ≤ −a
8. |a + b| ≤ |a| + |b| proof
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Properties of Absolute Values
For any real number a and b
6. Let a > 0 and x ∈ R, if |x| ≤ a then −a ≤ x ≤ a
7. Let a > 0 and x ∈ R, if |x| ≥ a then x ≥ a or x ≤ −a
8. |a + b| ≤ |a| + |b| proof
9. |a − b| ≥ |a| − |b| and |a − b| ≥ |b| − |a|
T.Laokul – p. 4/
Properties of Absolute Values
For any real number a and b
6. Let a > 0 and x ∈ R, if |x| ≤ a then −a ≤ x ≤ a
7. Let a > 0 and x ∈ R, if |x| ≥ a then x ≥ a or x ≤ −a
8. |a + b| ≤ |a| + |b| proof
9. |a − b| ≥ |a| − |b| and |a − b| ≥ |b| − |a|
10. (|a|)2 = a2
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