TASNUVA CHAUDHURY (TCY) CHAPTER 13: POWER AND POLITICS MGT 321: Organizational Behavior.
TCY - Inequalities.ppt
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1
INEQUALITIES
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INTRODUCTION
If a and b are real numbers then we can compare their positionsby the relation
Less than Less than or equal to
Greater than or equal to
For example: if x > 3 , it means x can be any value more than 3
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VARIOUS TYPESOFGRAPHS
Shadeup
Shadedown
Solidline
Dashedline
> x + 1
1
2
3
1
2
3
321 1 2 3
y > x + 1
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y < x + 1
1
2
3
1
2
3
321 1 2 3
y < x + 1
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x > 2
1
2
3
1
2
3
321 1 2 3
x > 2
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1
2
3
1
2
3
321 1 2 3
4
WRITE THE EQUATION
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1
2
3
1
2
3
321 1 2 3
4
WRITE THE EQUATION
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PROPERTIES OF INEQUALITIES
Ifa is greater than b
If we add c (any real number) then which one is greater
A + c or b + c
Solution: (a + c) is greater than b + c
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You know 8 is greater than 4 or 8 > 4
Add 2 on both sides
8 + 2 > 4 + 2
10 > 6
TRUE
EXAMPLE
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Ifa is greater than b
If we subtract c (any real number) then which one is greater
a c or b c
Solution: (a c) is greater than b c
PROPERTIES OF INEQUALITIES
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You know 8 is greater than 4 or8 > 4
Subtract
2from both sides
8 2 > 4 2
6 > 2
TRUE
EXAMPLE
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Ifa is greater than b i.e. (a > b)
If we multiply by c (any real number) then which one is
greater
ac or bc ?
Depends upon c because c can be a positive or negative real
number
PROPERTIES OF INEQUALITIES
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You know 8 is greater than 4 or
8 > 4
Multiply by
2both sides
8(2) > 4(2)
16 > 8
TRUE
EXAMPLE
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You know 8 is greater than 4 or
8 > 4
Multiply by
2both sides
8( 2) > 4( 2)
16 > 8
FALSE
EXAMPLE
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Ifa is greater than b
WHICH IS GREATER
ac or bc
If c is positive then ac > bc
If c is negative then ac < bc
REMEMBER
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If a is greater than b
WHICH IS GREATER
or
Is Greater than a
1
b
1
b
1
a
1
REMEMBER
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Transitive: If a < b and b < c, then a < c
Additionofinequalities: If a < b and
c < d, then a + c < b + d.
Addition of a constant: If a < b, then
a + c < b + c. Multiplication by a constant:
If a < b and c is positive real number, then: ac < bc
and if c is negative real number, then ac > bc
Taking Reciprocals: If a < b and a, b 0, then
PROPERTIES OF INEQUALITIES
b
1
a
1
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Inequalitynotation
Real number line graph
3x
3x
52 x
3x
3x
INTERVAL NOTATION
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You solve linear inequalities in the same way as you would solvelinear equations, but with one exception.
Property :If in the process of solving an inequality, you multiplyor divide the inequality by a negative number, then , you must
switch the direction of the inequality.
Ifx> a, then x
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Solve x+ 3 < 2.
Graphically
SOLVING LINEAR INEQUATIONS
CASE-1
When the equation was "x+ 3 = 2 type,
We normally subtract 3 from both sides.
Then the solution is: x< 1
X + 3 < 2
- 3 -3
-------------------------
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Solve 2x< 9.
Like inequality divide by 2
CASE-2
5.42
9x
2
9
2
x2
9x2
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What happens when the number is negative?
27x3
If you divide both sides by 3,
3
27
3
x3
9x (The inequality will change if we multiply or divide with a
negative number on both sides.)
CASE-3
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SOLVING QUADRATIC INEQUATIONS
When we have an inequality with "x2" as the highest-
degree term, it is called a "quadratic inequality".
Solve x2 3x+ 2 > 0
Step 1: Change the inequality to an equation. Find x- intercept
x2 3x+ 2 = 0(x 1) (x 2) = 0
x = 1 or 2
Step 2: Plot the points ( x = 1, 2) on the number line
3 2 1 0 1 2 3
The number line is divided into the intervals (- , 1), (1, 2),
and (2, ).
1 2
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x2
3x
+ 2 > 0 or(x 1) (x 2) > 0The number line is divided into the intervals (- , 1), (1, 2),
and (2, ).
1 2
Test-point method: Pick a point (any point) in each interval
x= 3
(x - 1) is positive(x - 1) (x - 2) is positive(x - 3) is positivePOSITIVE
x= 1.5
(x - 1) is negative(x - 1) (x - 2) is negative(x - 3) is positiveNEGATIVE
x= 0
(x - 1) is negative(x - 1) (x - 2) is positive(x - 2) is negativePOSITIVE
(x 1) (x 2) is positive when x > 2 or x < 1
1 2
SOLVING QUADRATIC INEQUATIONS
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x2 xor
You cant say
Lets f ind the interval
wh ere x2is g reater than
x
WHICH ONE IS GREATER?
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For what value of x ? x2 x 0 or(x) (x 1) 0
Step 1: Change the inequality to an equation. Find value of x
x = 0, 1
x2x= 0
Step 2: Plot the points
0 1Step 3: Test point method
0 1 At x = 2At x = 0.5At x = -1 +
+
0 1
Step 4: Solution x > 1 or x < 0
SOLUTION
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You can solve some absolute-value equationsusing logics. For instance, you have learned
that the equation |x| 8 has two solutions: 8and 8.
SOLVING ABSOLUTE-VALUE EQUATIONS
To solve absolute-value equations, you can use the fact
that the expression inside the absolute value symbols
can be either positive or negative.
Because I X I = + X if X > 0- X If X < 0
0 if X = 0
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SOLVING AN ABSOLUTE-VALUE EQUATION
Solve |x
2 | 5
x 2 IS POSITIVE|x 2 | 5
x 7 x3
x 2 IS NEGATIVE|x 2 | 5
| 7 2 | | 5 | 5 | 3 2 | | 5 | 5
The expressionx 2can be equal to5or5.
x 2 5
x 2 IS POSITIVE
x 2 5
Solve |x
2 |
5
The expressionx 2 can be equal to 5 or5.SOLUTION
x 2 5
x 2 IS POSITIVE|x 2 | 5
x 2 5
x 7
x2IS POSITIVE|x 2 | 5
x 2 5
x 7
x 2 IS NEGATIVE
x 2 5x3
x2IS NEGATIVE|x 2 | 5
x 2 5
The equation has two solutions: 7 and 3.
CHECK
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Recall that xis the distance betweenx and 0. If x 8, thenany number between 8 and 8 is a solution of the inequality.
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8
You can use the following properties to solve
absolute-value inequalities and equations.
Recall that |x | is the distance betweenx and 0. If |x | 8, thenany number between 8 and 8 is a solution of the inequality.
Recall that | x | is the distance between xand 0. If | x | 8,then any number between 8 and 8 is a solution of theinequality.
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SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES
ax b c and ax b c.
ax b
c and ax
b
c.
ax b c or ax b c.
ax b c or ax b c.
ax b c or ax b c.
| ax b | c
|ax
b
|
c
| ax b | c
| ax b | c
| ax b | c
means
means
means
means
means
means
means
means
means
means
When an absolute value is less thana number, theinequalities are connected by and. When an absolute
value is greater thana number, the inequalities are
connected by or.
SOLVING ABSOLUTE-VALUE INEQUALITIES
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Solve |x
4| < 3
x 4 IS POSITIVE x 4 IS NEGATIVE|x 4| 3
x 4 3x 7
|x 4| 3
x 4 3x 1
Reverseinequality symbol.
This can be written as 1 x 7.
The solution is all real numbers greater than 1 andless than 7.
EXAMPLE
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2x 1 9
| 2x 1 |3 6
| 2x 1 | 9
2x10
2x+ 1 IS NEGATIVE
x5
Solve | 2x 1| 3 6 and graph the solution.
| 2x 1 |3 6
| 2x 1 | 9
2x 1 +9
2x 8
2x+ 1 IS POSITIVE
x 4
SOLVING AN ABSOLUTE-VALUE INEQUALITY
Reverse
inequality symbol.
| 2x 1 |3 6
| 2x 1 | 9
2x 1 +9
x 4
2x 8
| 2x 1 |3 6
| 2x 1 | 9
2x 1 9
2x10
x5
2x+ 1 IS POSITIVE 2x+ 1 IS NEGATIVE
6 5 4 3 2 1 0 1 2 3 4 5 6
The solution is all real numbers greater than or equal
to4or
less than or equal to5
. This can be written asthe compound inequality x5orx 4.5 4.
SOLVING THE INEQUALITIES WITH
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SOLVING THE INEQUALITIES WITH
THE HELP OF OPTIONS
We can solve all the inequality questions by going with theoptions.
Take an example:
x2 7x + 10 < 0
(1) X < 2 (2) x > 5 (3) x < 5
(4) 2 < x < 5 (5) Both (1) and (2)
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SOLUTION
Since the first option is x < 2, we take x = 1 and check whether
the given inequality is satisfying or not.
If x = 1, 12 7(1) + 10 < 0
4 < 0 (wrong)
Option (1), (3) and (5) are wrong.
Now take x = 6,
62 7 6 + 10 < 0
4 < 0 (wrong)
So, option (2) is wrong.
So, the answer is (4).
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EXAMPLE
If |x 3| > 2, which will be greater?Column A Column B
|x| 2
|x 3| > 2 means x > 5 or x < 1
If x > 5, |x| > 2
But if x < 1, we cant say
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