Physics and Mathematics
Properties of Modulus Function
1. Non-Negativity Property
For any real number [x],
[|x| ≥ 0]
Meaning:
The modulus of a number is never negative because it represents distance.
2. Zero Property
[|x| = 0 ⇔ x = 0]
Meaning:
Only the number zero has zero distance from itself.
3. Even Function Property
[|−x| = |x|]
Meaning:
Modulus function is an even function and its graph is symmetric about the y-axis.
4. Identity Inequalities
For all real [x],
[|x| ≥ x]
[|x| ≥ −x]
Meaning:
Modulus value is always greater than or equal to both [x] and [−x].
5. Triangle Inequality
For any real numbers [x] and [y],
[|x + y| ≤ |x| + |y|]
Meaning:
The distance of the sum is never more than the sum of distances.
6. Reverse Triangle Inequality
[||x| − |y|| ≤ |x − y|]
Meaning:
Difference of distances is less than or equal to distance of difference.
7. Multiplicative Property
[|xy| = |x||y|]
Meaning:
Modulus of a product equals the product of moduli.
8. Division Property
[\left|\dfrac{x}{y}\right| = \dfrac{|x|}{|y|}], where [y ≠ 0]
Meaning:
Modulus of a quotient equals the quotient of moduli.
9. Distance Interpretation
[|x − a|] represents the distance between x and a on the number line.
10. Boundedness Property
If [|x| ≤ a], then
[−a ≤ x ≤ a]
Meaning:
Modulus inequality gives a closed interval.
11. Conceptual Questions with Solutions
1. Why is modulus always non-negative?
Because it represents distance, which is never negative.
2. When does modulus become zero?
Only when the number itself is zero.
3. Why is modulus function even?
Because [|−x| = |x|] for all real numbers.
4. What does [|x − a|] represent geometrically?
It represents the distance between x and a.
5. Why is triangle inequality important?
It helps in solving inequalities involving modulus.
6. Is [|x+y| = |x|+|y|] always true?
No, it holds only in special cases.
7. Does modulus change multiplication?
No, modulus preserves multiplication.
8. Can [|x|] be less than [x]?
No, because [|x| ≥ x].
9. Is reverse triangle inequality always true?
Yes, it holds for all real numbers.
10. Why is [|x| ≤ a] written as an interval?
Because it represents a range of values between [−a] and [a].
11. Does modulus function have a maximum?
No, it is unbounded above.
12. What happens if [a < 0] in [|x| ≤ a]?
There is no solution.
13. Is modulus function continuous?
Yes, it is continuous everywhere.
14. Does [|x| = a] have two solutions?
Yes, if [a > 0].
15. Why is modulus useful in inequalities?
Because it converts distance conditions into intervals.
12. FAQ / Common Misconceptions
1. [|x|] is always equal to [x].
False. It equals x only when x ≥ 0.
2. Modulus can be negative.
False. It is never negative.
3. [|x+y| = |x|+|y|] always.
False. It is true only in special cases.
4. Modulus removes the value.
False. It removes only the sign.
5. [|x| ≤ −3] has solutions.
False. There are no solutions.
6. Modulus function is odd.
False. It is even.
7. Modulus inequalities give single value.
False. They give an interval.
8. [|x| = a] has one solution.
False. It has two solutions when [a > 0].
9. Modulus is discontinuous.
False. It is continuous everywhere.
10. Modulus has a maximum value.
False. It is unbounded above.
13. Practice Questions with Step-by-Step Solutions
Question 1. Find the value of [|−9|].
Step-by-Step Solution:
Since [−9 < 0], modulus removes the negative sign.
[|−9| = −(−9)]
Conclusion:
[|−9| = 9]
Question 2. Evaluate [|5 − 11|].
Step-by-Step Solution:
First simplify inside the modulus: [5 − 11 = −6].
Take modulus of [−6].
Conclusion:
[|5 − 11| = 6]
Question 3. Verify that [|−x| = |x|] for [x = −4].
Step-by-Step Solution:
Substitute [x = −4].
LHS: [|−(−4)| = |4| = 4].
RHS: [|−4| = 4].
Conclusion:
LHS = RHS, hence verified.
Question 4. Solve [|x| = 5].
Step-by-Step Solution:
[|x| = 5] means distance of x from 0 is 5.
So, [x = 5] or [x = −5].
Conclusion:
Solution set = [{−5, 5}]
Question 5. Solve the inequality [|x| ≤ 3].
Step-by-Step Solution:
Use the property: [|x| ≤ a ⇒ −a ≤ x ≤ a].
Here, [a = 3].
Conclusion:
[−3 ≤ x ≤ 3]
Question 6. Solve the inequality [|x| > 4].
Step-by-Step Solution:
[|x| > 4] means distance from 0 is greater than 4.
So, [x > 4] or [x < −4].
Conclusion:
Solution set = [(−∞, −4) ∪ (4, ∞)]
Question 7. Evaluate [|3 × (−7)|].
Step-by-Step Solution:
First multiply: [3 × (−7) = −21].
Take modulus: [|−21|].
Conclusion:
[|3 × (−7)| = 21]
Question 8. Verify the property [|xy| = |x||y|] for [x = −2] and [y = 5].
Step-by-Step Solution:
LHS: [|−2 × 5| = |−10| = 10].
RHS: [|−2||5| = 2 × 5 = 10].
Conclusion:
LHS = RHS, property verified.
Question 9. Find the distance between the points [x = −3] and [x = 5] on the number line.
Step-by-Step Solution:
Distance between two points is given by [|x₁ − x₂|].
[|−3 − 5| = |−8|].
Conclusion:
Distance = 8 units
Question 10. Find the minimum value of [|x − 7|].
Step-by-Step Solution:
Modulus is minimum when the expression inside is zero.
Set [x − 7 = 0 ⇒ x = 7].
Conclusion:
Minimum value = 0
Sign in with Google