TCY - Inequalities.ppt

7/28/2019 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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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



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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



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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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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


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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TCY - Inequalities.ppt