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Algebra – Absolute ValueProblem Solving – Question 10
Question
If a, b, and c are not equal to zero, what is the difference between the maximum and
minimum value of S?
S=1+a
a+2
b
b+3
ab
ab-4
c
c
A. 12
B. 14
C. 22
D. 20
E. 18
◴What is the approach?
What is the difference between the maximum and minimum
value of S?
1. When maximum?
Identify when will the value of the expression
be maximum and compute it.
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
1. When maximum?
Identify when will the value of the expression
be maximum and compute it.
2. When minimum?
Identify when will the value of the expression
be minimum and compute it.
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
1. When maximum?
Identify when will the value of the expression
be maximum and compute it.
2. When minimum?
Identify when will the value of the expression
be minimum and compute it.
3. Compute the difference
Compute the difference to find the answer.
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
1. When maximum?
Identify when will the value of the expression
be maximum and compute it.
2. When minimum?
Identify when will the value of the expression
be minimum and compute it.
3. Compute the difference
Compute the difference to find the answer.
4. Traps to watch out
Where could one have gone wrong?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
The value of xx
will either be 1 or -1. xx
= 1when x is positive. xx
= -1 when x is negative.
01 When will the value be maximum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of xx
will either be 1 or -1. xx
= 1when x is positive. xx
= -1 when x is negative.
The value of the expression is maximized if
we can make all the terms of the expression
positive.
01 When will the value be maximum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of xx
will either be 1 or -1. xx
= 1when x is positive. xx
= -1 when x is negative.
The value of the expression is maximized if
we can make all the terms of the expression
positive.
01 When will the value be maximum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
aa
, b
b,
ab
abwill each be 1 when a > 0 and b > 0.
When c < 0, cc
= -1 and -4cc
will be maximized.
The value of xx
will either be 1 or -1. xx
= 1when x is positive. xx
= -1 when x is negative.
The value of the expression is maximized if
we can make all the terms of the expression
positive.
01 When will the value be maximum? What is it?
Therefore, the maximum value of the expression is 1 + 1 + 2 + 3 – (-4) = 11
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
aa
, b
b,
ab
abwill each be 1 when a > 0 and b > 0.
When c < 0, cc
= -1 and -4cc
will be maximized.
02 When will the value be minimum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.
c has to be positive for the expression to be
minimum.
02 When will the value be minimum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.
c has to be positive for the expression to be
minimum.
If a > 0 and b < 0, the value is 1 + 1 -2 – 3 – 4 = -7
02 When will the value be minimum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.
The value of the expression reduces when ab is
negative.
If ab has to be negative, one of a or b has to be
negative and the other has to be positive.
c has to be positive for the expression to be
minimum.
If a > 0 and b < 0, the value is 1 + 1 -2 – 3 – 4 = -7
02 When will the value be minimum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.
The value of the expression reduces when ab is
negative.
If ab has to be negative, one of a or b has to be
negative and the other has to be positive.
If a < 0 and b > 0, the value is 1 - 1 + 2 – 3 – 4 = -5
c has to be positive for the expression to be
minimum.
If a > 0 and b < 0, the value is 1 + 1 -2 – 3 – 4 = -7
Therefore, the minimum value of S is -7 when a > 0, b < 0 and c > 0
02 When will the value be minimum? What is it?
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.
The value of the expression reduces when ab is
negative.
If ab has to be negative, one of a or b has to be
negative and the other has to be positive.
If a < 0 and b > 0, the value is 1 - 1 + 2 – 3 – 4 = -5
Maximum value of S = 11
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
03 Last step: Compute the difference
Minimum value of S = -7
Maximum value of S = 11
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
The difference is 11 – (-7) = 18
03 Last step: Compute the difference
Choice E is the correct answer
Minimum value of S = -7
◴What should one have watched out for?
While computing the minimum value, we would have estimated that if all the terms other than the first term
‘1’ is negative, the value of the expression will be minimum.
04 Choice D is the trap
·
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
While computing the minimum value, we would have estimated that if all the terms other than the first term
‘1’ is negative, the value of the expression will be minimum.
04 Choice D is the trap
·
However, if ab is negative ONLY ONE of a or b can be negative. The other HAS to be positive.·
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
While computing the minimum value, we would have estimated that if all the terms other than the first term
‘1’ is negative, the value of the expression will be minimum.
04 Choice D is the trap
·
However, if ab is negative ONLY ONE of a or b can be negative. The other HAS to be positive.·
If you had made all the terms negative, then the minimum value would have been 1 – 1 – 2 – 3 - 4 = -9·
And the answer would have been 11 – (-9) = 20. That is INCORRECT.·
S = 1+aa
+2b
b+3
ab
ab-4
cc; a, b, and c are not zero
What is the difference between the maximum and minimum
value of S?
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