algebra equations & inequalities
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Algebra: Equations & Inequalities
Miguel Pérez Fontenla
November, 2010
Algebra: Equations & Inequalities
What is an equation?
2222 1
2 34 9
yxx
x
Example:
An equation is a mathematical statement
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
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
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
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
Algebra: Equations & Inequalities
What is an inequality?
SIMBOLS
= Equal to
< Less than
> Greater than
Less than or equal
Greater than or equal
Algebra: Equations & Inequalities
What is an inequality?
2222 1
2 34 9
yxx
x
Example:
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)
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
Algebra: Equations & Inequalities
Linear inequalities: Graphic Solution
1 1 5 14 2 9 717
4 8 4 5 2 8
x x x xx
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
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
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.
Algebra: Equations & Inequalities
Solving simultaneous linear inequalities
1 1
2 2 2 2
y x y x
y x y x
Algebra: Equations & Inequalities
Solving simultaneous inequalities
2 2
1 1
6 6
y x y x
y x x y x x
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
Algebra: Equations & Inequalities
Solving simultaneous inequalities
2 22 2
12 4 2
22
2
y xy x
x y y x
x yy x
Algebra: Equations & Inequalities
Algebra: Equations & Inequalities
Algebra: Equations & Inequalities
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