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DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 2

Topic 2: Equations & Inequalities
Sub topics:
2
...
2
Quadratic Equations
2
...
4
Linear Inequalities
2
...

Know how to solve linear equation
2
...

3
...

Solve polynomial equations by factoring, radical equations, equations with
rational exponents, equations that are quadratic in form and equations involving
absolute value
...

Graph an inequality’s solution set, use set-builder and interval notations, use
properties of inequalities to solve inequalities, solve compound inequalities and
solve inequalities involving absolute value
...

Solve quadratic inequalities, rational inequalities and problems modeled by
nonlinear inequalities
...
1

Linear Equations

Equation: A statement where 2 expressions are equal
...
Identity: - An equation that is true for all real numbers for which both sides are
defined
...
Conditional Equation: - An equation that is true for at least one real number
...
Inconsistent equation: - An equation that is not true for even one real number
...

Linear Equations in one variable
A linear equation in one variable is equivalent to an equation of the form:
ax b  0 where a and b are real numbers and a  0
...
If necessary, clear the equation of factions by multiplying both sides’ denominators
of all fractions
...
Remove all parentheses and simplify
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Collect all terms with variable on one side and all other terms on the other side
...
Check the solution(s)
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5x  8  72

2
...


x 2x 5


4
3 6

x 3x

5
2
...

5
2
3

Linear equation involving Rational Expressions
1
...


5 17 1


2 x 18 3x

2

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2
...

Example: x  4 x  4  0
Method of solving quadratic equation:
1
...

Completing the Square
3
...

If AB = 0, then A= 0 or B= 0
Examples: 1
...


16 x( x  2)  8x  25

Square Root method
If

U2  d

Examples:


1
...
( y  4)

2

2
...


Write the equation in the form:

2
...


Factor the equation in left side:

2

x 2  mx  c
2
2
 m
m
2
 x  mx     c   
2
2

2
2
x 2  mx   m    x  m 
2
2

4
...


Solve:

6
...


Example: 1
...
3x  5x  10  0
2
3
...


x 2  8x  12  0

2
...


Discriminant

Number of solutions

b2  4ac  0
b 2  4ac  0
b 2  4ac  0

2 unequal real solution

1
...
3x

2

 4x  2  0

5

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 2

2
...

Examples: 1
...


9 y 3  8  4 y  18 y 2

Equations containing Radicals
Radical equation
– An equation in which the variable occurs in a square root, cube root, or any higher
root
...


20  8x  x

Examples: 1
...


x 5  x 3  2

Equation involving a Rational Exponent
Examples: 1
...


8x

5

3

 24  0

6

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 2

Equation that is Quadratic in Form
- One that can be expressed as a quadratic equation using an appropriate substitution
...

3
...


3x 6  4 x 3  15

6  0

Equation involving Absolute Value
-

x c

Example:

is equivalent to

1
...


2 x  3  11

7

DIM5058

2
...
Nonnegative Property, a  0
2
...
Negative Multiplication and Division:
If a < b and c is negative, then ac > bc
a b
If a < b and c is negative, then 
c c
Graphing Inequalities
Types of Interval
Closed
Open

Half-open or
Half-closed

Sign
a xb
a xb
a xb
a xb
a x
a x
 xa
 xa
 x

Solving Linear Inequalities
1
...


18x  45  12x  8

2
...


7  x  5  11

Interval Notation
[a,b]
(a,b)
(a,b]
[a,b)
[a,  )
(a,  )
(-  ,a]
(-  ,a)
(-  ,  )

8

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 2

Solving Inequalities with Absolute Value

u a

 a u  a

u a

 u  a

u a

 a u  a

u a

 u  a

1
...


3
...


Examples:

or u  a

or u  a

x 5
3( x  1)
6
4

9

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 2

2
...
(x+3) (x - 5) > 0

2
...


3
...


4

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