This document discusses inequalities and modulus. It begins by explaining how to find the maximum value of expressions like xp yq when ax + by = k. It then discusses how this relates to finding minimum values. It also covers the definition and properties of modulus/absolute value, including how to solve inequalities and equations involving modulus. It gives examples of finding maximum and minimum values of functions involving multiple modulus expressions.

1. Inequalities and Modulus Session 2
1
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2. Maxima and Minima – (a+b) to ab
model
If ax + by = k, maximum value of xp yq , when a, b, x, y, p
and q are positive real numbers.
Example: If 3x + 4y = 20, find the maximum possible
value of x2 y3 , when x, y >0
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Generalization
If ax + by = k, maximum value of xpyq is when
𝑎𝑥
𝑝
=
𝑏𝑦
𝑞

3. If 5p + 6q = 42, what is the maximum possible value
of p5 q2 when p, q > 0?
1.2562
2.6523
3.3523
4.None of these
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(a+b) to ab model - Problems
Answer : Option 4

4. If xp yq = k, minimum value of ax + by , when a, b, x, y, p
and q are positive real numbers.
Example: If x5 y3 =55
28
, what is the minimum possible
value of 4x + 6y when x, y > 0?
4
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Generalization
If xpyq = k, minimum value of ax + by is when
𝑎𝑥
𝑝
=
𝑏𝑦
𝑞
Maxima and Minima – ab to (a + b)
model

5. If p q2 r3 =33
28
, what is the minimum possible value of
p + q + r when p, q > 0?
1.15
2.20
3.12
4.18
5
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ab to (a + b) model - Problems
Answer : option 3

6. Absolute Value ( Modulus)
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6

7. If x is the coordinate of a point on a real number line, then
the distance of x from the origin is represented by 𝑥 .
x is called Absolute value of x or modulus of x.
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Modulus and Distance

8. Definition of Absolute value
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Modulus and Distance

9. Properties of Modulus
1. 𝑥 ≥ 0
2. − 𝑥 ≤ 0
3. 𝑥 + 𝑦 ≤ 𝑥 + 𝑦
4. 𝑥 − 𝑦 ≤ 𝑥 − 𝑦
5. 𝑥 ∗ 𝑦 = 𝑥 ∗ 𝑦
6.
𝑥
𝑦
=
𝑥
𝑦
( as long as y≠0)
7. 𝑥 2 = 𝑥2
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Modulus

10. 𝑥 = 3
𝑥 < 3
𝑥 > 3
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Modulus and Distance

11. 11
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Modulus and Distance
What if you have to measure a distance from a point other than 0?
Distance of a coordinate x from a is given by 𝑥 − 𝑎
Then what does 𝑥 + 𝑎 represent?
Distance of x from -a

12. 𝑥 − 1 = 3
𝑥 − 1 < 3
𝑥 − 1 > 3
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Modulus and Distance

13. Find the value of x is 3𝑥 + 2 = 5
Answer : 1 or −
7
3
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Modulus

14. 14
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Modulus
General Properties of 𝑥

15. 15
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Modulus
General Properties of 𝑎𝑥 + 𝑏

16. 16
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Modulus - Problems
Solve for x if 2𝑥 − 1 < 3
Answer : (-1, 2)
Solve for x if 7 − 3𝑥 > 2
Answer : −∞,
5
3
∪ (3, ∞)

17. 17
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Modulus
How about solving for x for the expression below?
𝑥 − 4 = 3𝑥 − 8

18. 18
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Modulus - Problems
Solve for x for the expression below?
3𝑥 − 4 = 𝑥 + 5

19. 19
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Modulus – Maximum and Minimum
Values
What is the maximum value of 𝑓 𝑥 = 15 − 9 + 𝑥 ?
Answer : 15, when x = -9

20. 20
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Modulus – Maximum and Minimum
Values
Maximum and Minimum values when more than one modulus
functions are given.
What is the minimum value of the function
𝑥 − 2 + 𝑥 − 9 + 𝑥 + 4
Answer: 13 ( 9+4)

21. 21
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Modulus – Maximum and Minimum
Values
Maximum and Minimum values when more than one
modulus functions are given.
What is the minimum value of the function
𝑥 − 2 + 2 𝑥 − 9 + 3 𝑥 + 4
Answer: 32

22. 22
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Modulus – Maximum and Minimum
Values
Maximum and Minimum values when more than one modulus
functions are given.
What is the minimum value of the function
𝑥 − 2 + 3 𝑥 − 9 + 2 𝑥 + 4
Answer: 33

23. 23
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Modulus – Maximum and Minimum
Values
Maximum and Minimum values when more than one modulus functions are
given.
What is the minimum value of the function
𝑝(𝑥) = 𝑥 − 3 + 3|𝑥 − 4| + 2|𝑥 − 5| + 3|𝑥 − 7|
Answer: 11

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Modulus – Maximum and Minimum
Values
How many integer solutions are possible for the inequality
𝑥 − 6 + 𝑥 − 8 + 𝑥 + 4 < 11
1.1
2.2
3.0
4.Infinitely many
Answer : 0, as minimum value of LHS is 8+4 = 12

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Modulus – Practise Problems
Find the range of x for which
|4𝑥 + 1| < |3𝑥|
Answer :−1 < 𝑥 < −1/7

26. 26
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Modulus – Practise Problems
Find the range of x for which
|𝑥 + 3| ≥ |𝑥 − 1|
Answer :x ≥ −1

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Modulus – Practise Problems
Find the range of x for which
𝑥2 − 7𝑥 + 12 ≤ 𝑥2 − 4𝑥
Answer : [
3
2
, ∞)

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Modulus – Practise Problems
Find the range of x for which
|𝑥2 − 3𝑥 + 1| < 1
Answer : (0,1) ∪ (2,3)

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Modulus – Practise Problems
Find the range of x for which
𝑥2 − 5𝑥 + 5 ≤ 𝑥
Answer :[1,5]