1-3 equations and inequalities (presentation)

8/8/2019 1-3 Equations and Inequalities (Presentation)

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13 Equations and Inequalities

Unit 1 Functions and Relations


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Concepts and Objectives

Equations and Inequalities (Obj. #5)

Solve quadratic equations, finding all solutions.

Solve cubic equations (+/ of cubes)

Solve rational equations

Solve linear and quadratic inequalities Solve absolute value equalities and inequalities


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Complex Numbers

As you should recall, there is no real number solution to

x2

= 1so the number i has been defined so that

i2 = 1

which means that . Complexnumbers arenumbers in the form a + bi, where a and b are realnumbers.

For any positive real number a,

= 1i

=a i a


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Complex Numbers

Example: Simplify (a) (b) (c)16 48 ( )( ) 4 9


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Complex Numbers

Example: Simplify (a) (b) (c)

(a)

(b)

(c)

16 48 ( )( ) 4 9

= =16 16 4i i

= =48 48 4 3i i

( )( )

= =

i4 9 4 9

36 6


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Quadratic Equations

A quadratic equation is an equation that can be written

in the form

where a, b, and c are real numbers, with a 0. This isstandard form.

A quadratic equation can be solved by factoring,

graphing, completing the square, or by using the

quadratic formula.

Graphing and factoring dont always work, but

completing the square and the quadratic formula will

always provide the solution(s).

+ + =2 0ax bx c


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Factoring Quadratic Equations

Factoring works because of the zero-factorproperty:

Ifa and b are complex numbers with ab = 0, then a = 0 or

b = 0 or both.

To solve a quadratic equation by factoring:

Put the equation into standard form (=0).

Find two numbers which multiply to ac and add to c.

Split the b term using these two numbers.

Find the GCF of each pair.

Use the distributive property to rewrite the equation into

the factors.

Set each factor equal to zero and solve.


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Example: Solve by factoring. =22 15 0x


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Example: Solve by factoring.

The solution set is

=22 15 0x x

= = = 2, 1, 15a b c 30

1

6 5 + =22 6 5 15 0 x x x

( ) ( ) + =2 3 5 3 0 x x x

( )( )+ =2 5 3 0x x

+ = =2 5 0 or 3 0x x

= 5 , 32x

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Quadratic Formula

The solutions of the quadratic equation ,

where a 0, are

Example: Solve

+ + =2 0ax bx c

=

2 4

2

b b acx

a

= 22 4x x


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Quadratic Formula

Example: Solve

The solution set is

= 22 4x x

+ =

2

2 4 0x= = =2, 1, 4a b c

( ) ( ) ( )( )

( )

=

21 1 4 2 4

2 2x

= =

1 1 32 1 31

4 4

= 1 314i

1 314 4

i


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Cubic Equations

A cubicequation is an equation of degree 3. We will

mainly be working with cubic equations that are the sum

or differenceof

two

cubes:

a3 b3 = 0

Equations of this form factor as

To solve this, set each factor equal to zero and solve.

(Use the Quadratic Formula for the quadratic factor.)

( )( ) + =2 2 0a b a ab b


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Cubic Equations

Example: Solve 8x3 + 125 = 0


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Cubic Equations

Example: Solve 8x3 + 125 = 0 ( ) + =3 32 5 0x

( )( )+ + =2

2 5 4 10 25 0 x x x + = + =22 5 0 or 4 10 25 0 x x x

+ =

=

=

2 5 0

2 5

5

2

x

x

x

( ) ( )( )( )

=

2

10 10 4 4 252 4

x

= = =10 300 10 10 3 5 5 3

8 8 4

i i

5 5 5 3,

2 4 4

iThe solution set is


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Rational Equations

A rationalequation is an equation that has a rational

expression for one or more terms.

When solving a rational equation, always be sure to

check that possible solutions do not produce a

denominator of 0.

The easiest way to solve a rational equation is to

multiply the equation by the least common denominator.


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Rational Equations

Example: Solve+

+ = 2

3 2 1 2

2 2

x

x x x x


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Rational Equations

Example: Solve+

+ = 2

3 2 1 2

2 2

x

x x x x

( )+

+ =

3 2 1 2

2 2

x

x x x x The LCD is ( ) 2x x

( ) ( ) ( )( )

+ + =

3 2 1 2

22 2

22 x x x x x x x

x x x x

( ) ( )+ + = 3 2 2 2 x x x

+ + = 23 2 2 2 x x x + =23 3 0x x

( )+ =3 1 0x x


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Rational Equations

Example: Solve

Now, when we factored the equation, we had

Based on this,xcannot be 0, so our only solution isx= 1.

+ + =

23 2 1 2

2 2

x

x x x x

( )

+ + =

3 2 1 2

2 2

x

x x x x

( )+ =3 1 0x x= + =3 0 or 1 0x= = 0 or 1x x


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Linear and Quadratic Inequalities

An inequalitystates that one expression is greater than,

greater than or equal to, less than, or less than or equal

to another expression.

As with equations, a value of the variable for which the

inequality is true is a solution of the inequality; the set or

all solutions is the solution set of the inequality.

Inequalities are solved in the same manner equalities

are solved with one differenceyou must reverse the

direction of the symbol when multiplying or dividing bya negative number.


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Example: Solve + < 2 7 5x


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Example: Solve

The solution set is {x|x> 6}. Graphically, the solution is

+ < 2 7 5x

+ < 2 7 5x + < 77 72 5x < 2 12

> 2 12

2 2

x

> 6x


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Threepart or Compound Inequalities are solved by

working with all three expressions at the same time.

The middle expression is between the outer expressions.

Example: Solve 1 6 8 4x


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Example: Solve

The solution set is the interval

1 6 8 4x

+ + + 1 6 8 8 848 9 6 12x

6 6 6

9 6 12

3

22

x

3 ,22


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To solve a quadratic inequality:

Solve the corresponding quadratic equation.

Identify the intervals determined by the solutions of the

equation.

Use a test value from each interval to determine which

intervals form the solution set.


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Example: Solve >23 11 4 0x x


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=2

3 11 4 0x x + =23 12 4 0 x x x ( ) ( ) + =3 4 1 4 0 x x x

( )( )+ =3 1 4 0x x

= =1

or 43

x x

1,

3

1,4

3( )4,


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Interval Test Value TrueorFalse?

1,

3

1

,43

( )4,

1

0

5

10 > 0 True

4 > 0 False

16 > 0 True

( )

1, 4,

3


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Absolute Value

The solution set for the equation must include

both a and a.

Example: Solve

= a

=9 4 7x


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Absolute Value

The solution set for the equation must include

both a and a.

Example: Solve

The solution set is

= a

=9 4 7x

=9 4 7x = 9 4 7

= 4 2 = 4 16x

=1

2x = 4x

or

1,4

2


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Absolute Value

For inequalities, we make use of the following two

properties:

|a| < b if and only if b < a < b.

|a| > b if and only ifa < b or a > b.

Example: Solve + 5 8 6 14x

b l l


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Absolute Value

Example: Solve

The solution set is

+ 5 8 6 14x

or

5 8 8x 5 8 8x 5 8 8x 8 13x 8 3x

138

x 38

x

3 13, ,

8 8

S i l C


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Special Cases

Since an absolute value expression is always

nonnegative:

Expressions such as |2 5x| > 4 are always true. Its

solution set includes all real numbers, that is, (, ).

Expressions such as |4x 7| < 3 are always falsethat is,

it has no solution. The absolute value of 0 is equal to 0, so you can solve it as

a regular equation.

1-3 equations and inequalities (presentation)

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