22
Algebra: Equations & Inequalities Miguel Pérez Fontenla November, 2010

Algebra equations & inequalities

Embed Size (px)

Citation preview

Page 1: Algebra equations & inequalities

Algebra: Equations & Inequalities

Miguel Pérez Fontenla

November, 2010

Page 2: Algebra equations & inequalities
Page 3: Algebra equations & inequalities

Algebra: Equations & Inequalities

What is an equation?

2222 1

2 34 9

yxx

x

Example:

An equation is a mathematical statement

Page 4: Algebra equations & inequalities

Algebra: Equations & Inequalities

Properpies of equations

Property 1 - Adding or Subtracting a NumberAn equation is not changed when the same number is added or subtracted from both sides of the equality.Example: A = B (adding 4 to both sides gives)⇔ A + 4 = B + 4

Property 2 - Multiplying or dividing by a NumberAn equation is not changed if both sides are multiplied or divided by the same number.Example: A = B (Multiplying both sides by 2 gives) ⇔ 2A = 2B

A = B (Dividing both sides by 3 gives) ⇔ A/3 = B/3

Page 5: Algebra equations & inequalities

Algebra: Equations & Inequalities

Types of equations?4 1 5

3 2 2

x x bax b x

a

Linear equations

Quadratic equations

Biquadratic

Simultaneous equations

Linear

Quadratic

Rational equations

Irrational equations

Other types

22 4

02

b b acax bx c x

a

24 2 2 4

02

b b acax bx c x

a

5 8 19

2 2 10

x y

x y

2 2

3 8 8

9 28

x y

x y

2

2

3 4 1

4 2 2

x x

x x x

2 15 2 4x x

3 23 4 12 0x x x

Page 6: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving linear equations1 1 5 14 2 9 7

4 8 4 5 2 8

x x x x

1 5 14 2 9 7

4 32 40 2 8

x x x x 1. No parenthesis

2. No fractions

3. Isolate x to side one

4. Obtain x

5. Check your work

4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x

27 80 860 41 53 901x x x

53 901 90153 901 17

53 53 53

xx x

17 1 1 17 5 14 2 17 17 9 7 1 7 25 25

4 3 4 44 8 4 5 2 8 8 8 8 8

Page 7: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving quadratic equations

2 1 5 2 2

2 6 3 3

x x x 1. No parenthesis

2. No fractions

3. Isolate everythingto side one

4. Obtain x

5. Check your work

22;6;3 6 3 3 5 4 4mcm x x x

23 5 2 0x x

2

5 72

( 5) ( 5) 4 3 ( 2) 5 49 6

5 7 12 3 6

6 3

x x

2 1 2 1 2 5 2 3 3 6 3 12 1 2....

2 6 3 2 6 3 2 2

1 1 5 21

2 6 3

x x xx

Page 8: Algebra equations & inequalities

Algebra: Equations & Inequalities

What is an inequality?

SIMBOLS

= Equal to

< Less than

> Greater than

Less than or equal

Greater than or equal

Page 9: Algebra equations & inequalities

Algebra: Equations & Inequalities

What is an inequality?

2222 1

2 34 9

yxx

x

Example:

Page 10: Algebra equations & inequalities

Algebra: Equations & Inequalities

Properpies of inequalities

Property 1 - Adding or Subtracting a NumberThe sense of an inequality is not changed when the same number is added or subtracted from both sides of the inequality.Example: 9 > 6 (adding 4 to both sides gives)⇔ 9 + 4 > 6 + 4Property 2 - Multiplying by a Positive NumberThe sense of the inequality is not changed if both sides are multiplied or divided by the same positive number.Example: 8 < 15 (Multiplying both sides by 2 gives) ⇔ 8 × 2 < 15 × 2Property 3 - Multiplying by a Negative NumberThe sense of the inequality is reversed if both sides are multiplied or divided by the same negative number.Example: 4 > −2 (Multiplying both sides by -3 gives) ⇔ 4 × −3 < −2 × −3 ⇔ −12 < 6 (Note the change in the sign used)

Page 11: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving Linear inequalities1 1 5 14 2 9 7

4 8 4 5 2 8

x x x x

1 5 14 2 9 7

4 32 40 2 8

x x x x 1. No parenthesis

2. No fractions

3. Isolate x to side one

4. Obtain x

5. Check

4;32;40;2;8 160 40 40 5 25 56 8 80 720 140mcm x x x x

27 80 860 41 53 901 53 901 ...x x x x

901... 17

53x

0 1 1 0 5 14 2 0 0 9 7 1 1 51 9 7If 0

4 8 4 5 4 8 4 8 20 4 8

1 51 25 11 25

4 160 8 160 8

x

Page 12: Algebra equations & inequalities

Algebra: Equations & Inequalities

Linear inequalities: Graphic Solution

1 1 5 14 2 9 717

4 8 4 5 2 8

x x x xx

Page 13: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving quadratic inequalities

2 1 5 2 2

2 6 3 3

x x x 1. No parenthesis

2. No fractions

3. Isolate everythingto side one

4. Obtain solutions of the equation

5. Set the intervalssolution

6. Check

22;6;3 6 3 3 5 4 4mcm x x x

23 5 2 0x x

2

5 72

( 5) ( 5) 4 3 ( 2) 5 49 6

5 7 12 3 6

6 3

x x

1 1 5 21

2 6 3

x x xx

1

, 2,3

Page 14: Algebra equations & inequalities

Algebra: Equations & Inequalities

Quadratic inequalities: Graphic Solution

1 1 5 21

2 6 3

x x xx

:

1, 2,

3

1/ 2

3

Solutions

x x x

Page 15: Algebra equations & inequalities

Algebra: Equations & Inequalities

Puting into a Graph

A linear equation with two variables can be represented by a straight line in theplane.

A quadratic equation with two variables can be represented by a parabole in theplane.

Page 16: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving simultaneous linear inequalities

1 1

2 2 2 2

y x y x

y x y x

Page 17: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving simultaneous inequalities

2 2

1 1

6 6

y x y x

y x x y x x

Page 18: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving simultaneous quadratic inequalities

2 2

2 2

3 2 2 3

6 6

y x x y x x

y x x y x x

Page 19: Algebra equations & inequalities

Algebra: Equations & Inequalities

Solving simultaneous inequalities

2 22 2

12 4 2

22

2

y xy x

x y y x

x yy x

Page 20: Algebra equations & inequalities

Algebra: Equations & Inequalities

Page 21: Algebra equations & inequalities

Algebra: Equations & Inequalities

Page 22: Algebra equations & inequalities

Algebra: Equations & Inequalities