In high school, absolute value is precisely defined by cases. Let us also learn to handle absolute values containing variables.
Definition of Absolute Value
The absolute value |a| of a real number a is its distance from the origin.
When a ≧ 0, |a| = a
When a ＜ 0, |a| = −a
- |7| = 7、|−3| = 3、|0| = 0
- |√2 − 1| = √2 − 1 (positive because √2 > 1)
- |1 − √2| = √2 − 1 (reverse the sign because 1 − √2 < 0)
Example ①
Find the following absolute values.
(1) |−2/3| (2) |3 − π| (3) |√5 − 2|
Answer:
(1) 2/3 (2) Since π > 3, π − 3 (3) Since √5 > 2, √5 − 2
Find the following absolute values.
(1) |−2/3| (2) |3 − π| (3) |√5 − 2|
Answer:
(1) 2/3 (2) Since π > 3, π − 3 (3) Since √5 > 2, √5 − 2
Distance between Two Points
The distance between two points A(a) and B(b) on a number line is |b − a|.
Example ②
For A(−3), B(5), and C(−7), find the following distances.
(1) AB (2) BC
Answer: (1) |5 − (−3)| = 8 (2) |−7 − 5| = 12
For A(−3), B(5), and C(−7), find the following distances.
(1) AB (2) BC
Answer: (1) |5 − (−3)| = 8 (2) |−7 − 5| = 12
Properties of Absolute Value
- |a| ≧ 0
- |a| = |−a|
- |ab| = |a||b|
- |a|² = a²
Key Point
- Definition of absolute value: a if a ≧ 0, and −a if a < 0
- For expressions containing irrational numbers, determine the sign of the contents before removing the absolute value
- The distance between two points is |b − a|
- |a|² = a² is frequently used
Practice Problems
- Find the following. (1) |−9| (2) |2 − √7| (3) |3 − √5|
- Find the distance between A(7) and B(−4).
- Find all x satisfying |x| = 5.
Answers and Explanations
- Answer: (1) 9 (2) √7 − 2 (3) 3 − √5
Explanation: √7 ≈ 2.65 > 2 and √5 ≈ 2.24 < 3. - Answer: 11
Explanation: |7 − (−4)| = 11. - Answer: x = 5, −5
Explanation: The points at a distance of 5 from the origin are ±5.