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REAL NUMBERS

The set of real numbers consists of:

N - set of natural numbers or counting numbers

- also called positive integers

- {1, 2, 3, 4, 5, …}

W - set of whole numbers

- also called the nonnegative integers

- {0, 1, 2, 3, …}

I/J - set of integers (may use “Z”)

- also called the signed whole numbers

- consists of the positive integers, zero, and the negative integers

- {…, -3, -2, -1, 0, 1, 2, 3, …}

Q - set of rational numbers

- numbers which can be expressed as a quotient a/b, b ≠ 0, of two integers

- includes fractions, repeating decimals and terminating decimals (if in decimal forms)

-since an integer can be written as the quotient of the integer and 1, every integer is a rational

number

Q’ - set of irrational numbers

- non-repeating and non-terminating decimals (in decimal forms)

-√2, √3, π, and 1.010010001…

R - set of real numbers

- union of rational and irrational numbers

Note: The set of rational numbers and irrational numbers are mutually exclusive (have no

elements in common)

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SET OPERATIONS

1. Union (union of sets)

If A and B are sets, the union A and B, denoted A U B, is the set of all elements that are either in

A, in B, or in both. In symbols, A U B= {x|x є A or x є B}

Ex. Given: A={1,2,3,4,5} and B={2,4,6,8} then

A U B = {1, 2, 3, 4, 5, 6, 7 ,8} since {x|x is in A or x is in B}

2. Intersection (intersection of sets)

If A and B are sets, the intersection of A and B, denoted A ∩ B, is the set of all elements that are

in both A and B. In symbols, A ∩ B = {x|x є A and x є B}

Ex. Given: A={1,2,3,4,5} and B={2,4,6,8} then

A ∩ B = {2, 4} since {x|x is in A and x is in B}

For SUBSETS, A is a subset of B if every of element in A is an element of B.

If A={1, 2, 3, 4, 5}and B={2, 4, 6, 8} then, A is not a subset of B

PROPERTIES OF REAL NUMBERS

Let a, b, c be any real numbers

*R means real number

ADDITION MULTIPLICATION

a)Closure Property/Axiom a)Closure Property/Axiom

If a є R and b є R then, (a+b) is a

real number

If a є R and b є R then, (ab) is a real

number

b)Commutative Property/Axiom b)Commutative Property/Axiom

a+b=b+a; ex. 5+7 = 7+5 ab=ba; ex. 6x(-4) = (-4)x6

c)Associative Property/Axiom c)Associative Property/Axiom

(a+b)+c=a+(b+c); ex. (2+5)+8=2+(5+8) (ab)c=a(bc); ex. (5x2)x3 = 5x(2x3)

d)Identity Property/Axiom d)Identity Property/Axiom

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a+0=a *0 is the additive identity

element; ex. 3+0 = 0+3 = 3

a(1)=a *1 is the multiplicative identity element; ex. 8x1 = 1x8 = 8

e)Inverse Property/Axiom e)Inverse Property/Axiom

a+(-a)=0 *-a is the additive inverse or

opposite of a; ex. 2 + (-2) = 0

a(1/a)=(1/a)a=1 *1/a is the multiplicative inverse of a; ex. 6(1/6) = 1

f)Zero Property/Axiom of Multiplication

a(0)=0

0(a)=0

Distributive Property/Axiom of Multiplication over Addition:

a(b+c)= ab+ac ; ex. 4(3+6) = (4x3) + (4x6)

ab+ac= a(b+c) ; ex. (5+2)3 = (5x3) + (2x3)

PROPERTIES OF EQUALITY

Let a, b, c be any real numbers

1. Reflexive= a=a ; ex. 1=1

2. Symmetric

If a=b then b=a ; ex. 1=x, x=1

3. Transitive Property

If a=b and b=c, then a=c

4. Addition Property of Equality (APE)

If a=b and c є R, then a+c=b+c

5. Multiplication Property of Equality (MPE)

If a=b and c є R, then ac=bc, c ≠0

6. Substitution Property of Equality

If x=a and x+b=c, then a+b=c

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PROPERTIES OF INEQUALITY

Let a, b, c be any real numbers

1. Trichotomy Property

-for any two real numbers a and b; only one of the following is true.

a<b, a>b, a=b

2. Transitive Property of Inequality

If a>b and b>c then a>c

If a<b and b<c then a<c

3. Addition Property of Inequality/ Subtraction

If a<b and c є R, then a+c < b+c

4. Multiplication Property of Inequality

If a<b and c>0 then ac<bc

If a<b and c<0 then ac>bc

If a<b and c=0 then ac=bc=0

PROPERTIES OF ZERO

Let a, b, and c be any real numbers

1. Division Property of Zero

-If zero is divided by any nonzero real number, the result is zero.

-0/a = 0 or 0/1 = 0, a ≠ 0

2. Division by Zero is Undefined

-a/0 is undefined

PROPERTIES OF NEGATION

Let a, b, and c be any real numbers

1. Multiplication by -1

-The opposite of real number a can be obtained by multiplying the real number by -1

- (-1)a = -a; (-1)(-a) = a

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2. Placement of a Minus Sign

-The opposite of the product of two numbers is equal to the product of one of the numbers and

the opposite of the other

- (-ab) = (a)(-b) = (-a)(b)

3. Product of Two Opposites

-The product of the opposites of two real numbers is equal to the product of two real numbers

- (-a)(-b) = ab

INTEGRAL EXPONENTS

Definition: Positive Integral Exponent

Let a be a nonzero real number and n is a positive integer

an = a*a*a*a… (n factors of a)

an is an exponential expression with base a and exponent or power n

ex. 22= 2*2=4, 32=3*3=9

Definition: Zero Exponent

a0=1, if a≠0

ex. (1000)0=1

Definition: Negative Integral Exponent

Let a be a nonzero real number and n a positive integer

a-n= 1/an , if a ≠ 0

RULES OF INTEGRAL EXPONENTS/ LAWS OF EXPONENTS

Let a and b be nonzero real numbers m and n are integers

1. Product Rule

an * am = an+m

ex. (3x2)(5x3) = 15x5

72m+5 * 72-3m= 7-m+7 --> same base so simply add the exponents

2. Power Rule

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(am)n = amn multiply exponents

ex. ((b2)5)3 = b(2)(5)(3) = b30

3. Powers of Products

(ab)m = am x bm

ex. (3x5y-2)4 = (34)(x5)4(y-2)4 = 81 x20 y-8 = 81 x20/ y8

4. Quotient Rule

am/ an = am-n, if m>n and am/ an = 1/ an-m, if m<n

ex. 81x5/ 9x3 = 9x2

5. Powers of Quotients

(a/c)n = an/ cn

ex. [ 2a2b4/ 6ab-3 ] -3 = [ ab7 / 3] -3 = [a-3b-21] / -27 = 27/ a3b21

ALGEBRAIC EXPRESSION

-combination of constants (arithmetic expressions and numbers), and variables; may also include

grouping symbols

-constants are fixed values (e, Δ, , 5, a –may represent a constant)

-variables are letters, and those that have an unknown value

TYPES/CLASSIFICATIONS:

One term- monomial

Two terms- binomial

Three terms- trinomial

4 or more terms- multinomial/polynomial

Standard form of a polynomial

AnXn + An-1Xn-1 + … + A1X1 + A0 , where an - leading coefficient; a0 – constant term

*there is a non-negative integral powers of variables

Remark: If all the exponents of a multinomial are non-negative integers, then we all call the

expression a polynomial.

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OPERATIONS ON POLYNOMIALS

A. Addition and Subtraction

-combine like terms of the same literal coefficient

Ex. (3x+2y)+(4x-3y)

=7x-y

(3x+2y)-(x2-2x+y)

= -x2+5x+y

B. Multiplication of Polynomials

-use of distributive property and laws of exponents

Ex. 3x(x+1)= 3x2+3x

(x+1)(2x2+x-1)

=2x3+2x2+x2+x-x-1

=2x3+3x2-1

C. Division of Polynomials

-use of distributive property and laws of exponents

Ex. 3x2y3/ 27xy2 = xy/ 9

-12x4y3/ 15x3y5 = -4x/ 5y2

SYNTHETIC DIVISION

P(x)/ x-a = Q(x) + R/ x-a

Steps:

1. Arrange P(x) in descending powers of x

2. Write the detached numerical coefficients of P(x) in a row

3. Write a to the right/left of the coefficients of P(x)

4. Bring down the first number in line 1 to line 3

5. Multiply this number to a and write the result under the second column in line 2

6. Add the numbers in column 2 of line 1 & line 2, and write the result in line 3

7. Using the result in step 6, repeat step 5 and 6 until all numbers in line 1 have been used

8. The last sum in line 3 is the remainder. All the rest are the detached coefficient of the quotient

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Ex. P(x) = x3+2x2-x+1 / x+1 -1 1 2 -1 1Line 1

-1 -1 2Line 2

1 1 -2 3Line 3

Answer: x2+x-2 + 3/ x+1

SPECIAL PRODUCTS

1. a (x + y + z)= ax + ay +az

Ex. -3x(2x2 – 4y + 1)

= -6x3+12xy-3x

2. (x + y)(x – y)= x2 – y2

Ex. (x – 4)(x +4)

= x2 – 16

3. (x ± y)2 = x2 ± 2xy + y2

Ex. (2x + 1)2

= (2x)2 + 2(2x)(1) + (1)2

= 4x2+4x+1

4. (x±y)3 = x3 ±3x2y + 3xy2 ±y3

Ex. (x – 4)3

= x3 – 3x2(4) + 3x(4)2 – (4)3

= x3 – 12x2 + 48x – 64

5. F.O.I.L Method

Ex. (x + 3)(x – 2)

= x2 – 2x + 3x – 6

= x2 +x – 6

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6. (x + y +z)2 = x2 + y2 + z2 + 2xy + 2xz +2yz

Ex. (2x – y +3z)2

= (2x + (-y) + 3z)2

= (2x)2 + (-y)2 + (3z)2 + 2(2x)(-y) + 2(2x)(3z) +2(-y)(3z)

= 4x2 + y2 + 9z2 – 4xy + 12xz – 6yz

METHODS OF FACTORING POLYNOMIALS

1. Common Factor

Formula: ax + ay = a(x+y)

Ex. 15x2y3 – 81xy2 = 3xy2 (5xy – 27)

25a2y3 – 35a6y2 = 5a2y2 (5y – 7a4)

2. Factoring Binomials

A. Difference of Two Squares

Formula: x2 – y2 = (x +y)(x – y)

Ex. x2 – 1 = (x - 1)(x +1)

4ab3 – 16a3b = 4ab (b2 – 4a2) = 4ab (b - 2a)(b + 2a)

B. Sum and Difference of Two Cubes

Formulas: x3 + y3 = (x + y)(x2 – xy + y2)

x3 - y3 = (x - y)(x2 + xy + y2)

Ex. 8x6 – 27y3 = (2x2)3 – (3y)3

= (2x2 – 3y)(4x4 + 6x2y + 9y2)

(2x – y)3 – 8 = (2x – y)3 – (2)3

= (2x – y – 2)[(2x – y)2 +2(2x – y) +4]

= (2x – y – 2)(4x2 – 4xy + y2 +4x – 2y +4)

3. Factoring a Trinomial

A. Perfect Square Trinomial

Formulas: x2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

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Ex. x2 – 6x + 9 = (x)2 -2(3)(x) +3(x)2

= (x – 3)2

16x2y4 + 40xy2z + 25z2

(4xy2)2 (5z)2 = ( 4xy2 + 5z)2

B. Factoring a Quadratic

(Trial and Error)

Ex. x2 – 5x – 36 = (x + 4)(x – 9)

5x3 – 10x2y – 75xy2 = 5x (x2 – 2xy – 15y2)

= 5x (x – 5y)(x + 3y)

4. Factoring by Grouping

A. To produce a common factor

Ex. xy + 2x + y+2 = x(y + 2) + (y+2)

= (y + 2)(x +1)

[x2 +xy – 2y2] + [2x – 2y] = [(x – y)(x + 2y)] + [2(x – y)]

= (x – y)(x + 2y +2)

B. To produce a difference of two squares

Ex. (x4 + 6x2 + 9) – 9y2

(x2 + 3)2 - (3y)2 = (x2 + 3 +3y)(x2+3 – 3y)

4x2 – y2 + 2yz – z2 = 4x2 – (y2 – 2yz + z2)

= 4x2 – (y – 2)2

= (2x + y – z)(2x – y + z)

5. Adding and Subtracting a Perfect Square

Ex. x4 + 64

(x4 + 16x2 +64) – 16x2

When you divide this by 2, and the square it, you should get the number equal to the

third coefficient (in this case, 64) and then put it on the other side of the equation by

using the opposite sign (in this case, it was positive so use negative sign for the

other side of the equation).

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= (x2 + 8)2 – 16x2 ---> it is now a difference of two squares

= (x2 + 8 + 4x)(x2 +8 – 4x)

X4 – 11x2 +1 = (x4 – 2x2 +1) – 11x2 +2x

= (x2 – 1)2 – 9x2

= (x2 -1 +3x)(x2 -1 – 3x)

RATIONAL EXPRESSIONS

-the quotient of two polynomials

-when the denominator of a rational expression is zero, the expression is undefined

D(x)

-application of factoring and laws of exponents

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Radical Expressions and Rational Exponents

• A radical expression is anything that has a radical sign in it

• The thing inside the radical sign is the radicand. For example, 5 is the radicand in 5

• If you don’t see a number on the upper left part of the radical sign called the index, that means

it’s a square root. If there is a number, you’re supposed to root it to the nth level. That means, by

default 2 is the index. You square root something.

2 4 is just the same as 4 . (The square root of 4)

3 8 the cube root of 8

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4 81 the fourth root of 81

• Rooting something and raising it to a fraction is the same thing

X1/2 = x

• So with the basics done, let’s go through some rules/laws of radical expressions

1. a1/n = n a

2. abn = n a x n b = a 1/n x b1/n = (ab)1/n

3. am/n = ( n a )m = n a m

4. n ba / = n

n

ba = (a/b)1/n = a1/n/ b1/n

5. n a n = ( n a )n = a

6. m n a = (a1/n)1/m = a 1/mn = mn a

7. am/n = 1/ n a m

• Rules for rational expressions

1. ar x as = ar+s

2. ar/ as = ar-s

3. (ar)s = arxs

4. (ab)r = ar x br

5. (a/b)r = ar/ br

• Simple Operations

o Adding or Subtracting

You can only add/subtract those with the same radicands.

o Multiplying or Dividing

Follow the rules previously discussed

*** FINAL NOTE >> you’re not supposed to leave radicals in the denominator.

EXAMPLES

1. x4/3 . x5/6 . x2/3

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Step 1 >> All the unknowns are the same, they are all x, so we can use the first rule of rational

expressions. To add the exponents, make the bases of the fraction the same.

Step 1 = x 8/6 . x5/6 . x4/6

Step 2>> Combine them under one fraction

Step 2 = x(-8+5+4 )/ 6

Step 3 >> Simplify

Step 3 = x1/6

2. x3 16 4y3 - 3 54x 4y3

Step 1 >> Simplify the expressions

Step 1 = 2xy 3 2x - 3xy 3 2x

Step 2 = Wow! We ended up the same radicand. Time to subtract

Step 2 = -xy 3 2x

Complex Numbers

• A complex number is a number consisting of a real and imaginary part. It can be written in the

form a + bi, where a and b are real numbers, and i is the standard imaginary unit

• What is i? i = 1

• Things to remember about i:

i = 1

i 2= -1

i3 = -i

i4 = 1

• Every multiple of 4 in i’s basically reverts it back the the first 4. Such that i5= i4x I = 1 x i = i.

In this sense, you only need to what the remainder of the exponent is when you divide it by 4.

That way you end up with one of the four scenarios possible.

• Simple Operations

• Addition: (a + bi) + (c+di) = (a + c) + ( b + d) i

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• Subtraction: (a + bi) - (c + di) = (a-c) + (b-d) i

• Multiplication:

• Division:

EXAMPLES:

1. i35

Step 1: =i35 = (i4)8 ・ i3

Step 2: = 1 ・ - i

Step 3: = - i

2. 4i + 14i

Step 1 : = (4 + 14) i

Step 2: = 18i

3.( -1 - 18 ) / 3

Step 1>> Convert the radicand to something we can work with, separate the REAL part from the

imaginary part

Step 1 = [-1/3] – [( 3( 2x2) / 3]

Step 2 >> Take out i and 32 from the radicand

Step 2 = [-1/3] – [(3i 2 ) / 3]

Step 3 >> Simplify

Step 3 = [-1/3] – [(3i 2 ) / 3]

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Linear Equation in Single Variables

• An EQUATION is a statement that two expressions are equal in value.

• Isolate the variable to one side and the constants to another to know the value of the variable

• Variable = Unknown

• Tip: Brush up a LOT on simplifying expressions and you should do fine

EXAMPLE

1. (3/ x2 – 4x) - (2/ 2x2 – 5x – 12) = 9/ 2x2 + 3x

Step 1: [ 3/(x)(x-4) - -2/(2x+3)(x-4) = 9/ x(2x+3) ] (x)(2x+3)(x-4)

Step 2: 3(2x+3) – 2(x) = 9(x-4)

Step 3: 6x + 9 -2x = 9x - 36

Step 4: -5x = -45

Step 5: x=9

APPLICATION OF LINEAR EQUATIONS = Computing for Interest

• Interest = Principal x rate x time

• Solving for a particular number

• Converting things (e.g. kilos to pounds, Celsius to Fahrenheit)

Quadratic Equation in Single Variables

• Basically it’s just like linear equations except now you have 2 answers instead of one, given no

restrictions.

• This is because a negative number and a positive number, once squared yield the same result.

o E.g (-2)2= (2)2= 4

• Given an equation where there is a radical sign, once you get the answer, substitute it to the

original equation. If there is any term that yields a root of a negative number, remove that from

the solution set.

• Can be done by FACTORING, COMPLETING THE SQUARES, QUADRATIC

FORMULA

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QUADRATIC FORMULA = x = (-b ± b 2- 4ac ) / 2a

SOLVING SYSTEMS OF LINEAR EQUATION

Solve using:

I. By Substitution Method

II. By Elimination Method

III. By Graphical Solution Method

Sample Equation 1:

2x + 3y = 8

3x – y = 1

I. Substitution Method:

> Using equation 2, isolate a variable to one side.

3x – y = 1

3x – 1 = y

> Using the equation attained, substitute the value of y into equation 1.

1. 2x + 3y = 8

2x + 3 (3x-1) = 8

2x + 9x – 3 = 8

11x – 3 = 8

11x = 11

x = 1

2. 3x – 1 = y ; x = 1

3 (1) – 1 = y

2 = y

y = 2

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II. Elimination Method:

> Align variables. Multiply equation 2 by 3 to cancel out the variable y.

2x + 3y = 8

(3x – y = 1) 3

2x + 3y = 8

+) 9x - 3y = 3

11x = 11

x = 1

> Substitute the value x to any of the equations to get the value of y.

2x + 3y = 8

2 (1) + 3y = 8

2 + 3y = 8

3y = 6

y = 2

III. Graphical Solution: Note: applicable only for two unknowns.

> Get x-intercept and y-intercept for both equations.

2x + 3y = 8

X 4 0

Y 0 2.7

Ordered pairs: (4,0) and (0, 2.7)

3x – y = 1

X 0.33 0

Y 0 -1

Ordered pairs: (0.33,0) and (0,-1)

> Graph the attained ordered pairs. The point of intersection of the two equations will give you

the answer of x and y

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EXERCISES:

Exponents

1. (2a2b3)3

Answer: 8a6b9

2. (2m2n2/ 4n-2)

Answer: m2n4/ 2

3. (x-3y4/ 5) -3

Answer: 125x4/ y12

4. (a2b3c)2 (abc)

Answer: a5b7c3

5. (3xyz2/ x-2y-3z)-1 (9x2y3/ z3)

Answer: 3/ xyz4

A. Solve the following polynomials

1. Add 4x3 + 7x2 – 8 to the difference when 4x3 – 3x2 +2x – 1 is subtracted from x3+ 3x2 + 2

Answer: x3 +13x2 – 2x – 5

2. (a + b + 1)(a +b – 1)

Answer: a2 + 2ab + b2 – 1

3. x4 (5x2 – 1)2

Answer: 25x8 – 10x6 + x4

4. (h2 +4k5)3

Answer: h6 +12h4k5 + 48h2k10 +64k15

5. use synthetic division

a. (x3 – 6x +8) /(x + 2)

Answer: x2-2x-2+ 12/ x-2

b. (8x5 – 6x3 +x – 8) /(x + 2)

6. (3x5+9x3-x2) + (4x5-x4+3x3)

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Answer: 7x5 – x4+ 12x3 –x2

7. (2x3 – 8x2 +9x) – (3x3 + 5x2 -10)

Answer: -x3 – 13x2 + 9x +10

8. (3x+5) (x3-2x2+9x-15)

Answer: 3x4 – x3 +17x2 -75

9. (9x3y2 + 3xy +54x2y) / 18 x2y

Answer: xy/2 + 1/6x + 3

10. Use synthetic division

(x3 + 8x2 + 10x -25) / x+5

Answer: x2 + 3x -5

B. Factor the following

1. 3a – 2ay + 6b – 4by

Answer: (a + 2b)(3 – 2y)

2. 16x11 – 49x7y9 + 3x3y13

Answer: x3y5 (4x2 – y2)(4x2 + y2)(x4 – 3y4)

3. a2 – b2 – 6a – 10b – 16

Answer: (a + b +2)(a – b – 8)

4. 1 – x2 – x3 + x5

Answer: (x – 1)(x +1)(x – 1)(x2 +x +1)

5. x3y3 + 8x3 – y3 – 8

Answer: (x – 1)(x2+x + 1)(y – 2)(y2 +2y +4)

6. y3 – 9y2 +4y – 36

Answer: (y2 + 4)(y – 9)

7. m2 – n2 – 14m +6n +40

Answer: (m + n – 10)(m – n – 4)

8. 64a3 + b9

Answer: (4a+b3)(16a2 – 4ab3 + b6)

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9. 1 – 81x8

Answer: (1 – 3x2)(1 + 3x2)(1+ 9x4)

10. 18a3 + 9a2b – 10ab2

Answer: a(6a – 5)(3a + 4)

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Radical Expressions and Rational Exponents

1. 10xy2 / 3z2 ÷ 5x2y / 6z3 (answer = 4y/ x )

2. 4 16a / 4 a 5 (answer = 2/a)

3. 3 / yx (answer = 3 xy 2 )

4. (x2 / y1/3 ) ½ (answer = y1/6 / x or 6 y / x)

5.( 2 + 3 ) / (4 - 3 ) ( answer = (11 + 6 3 ) / 13 )

D. Complex Numbers

1. 3- i / 2+ i (ANSWER = 1- i )

2. (2 + 3i ) ( 4+ 5i) (ANSWER = -7 + 22i)

E. Linear Equation in Single Variables

1. You have 115,600 pesos of which you invest in 2 banks. Bank A’s interest rate is 10%, Bank

B’s interest rate is 12%. Over the span of 1 year, the interest from Bank A is equal to twice that

of Bank B. How much money did you invest in both banks?

• Tip 1 : To solve this you have to set up a linear equation

• Tip 2: Your unknown in this case is MONEY in both banks. Both of which must total up to

115,600) Therefore this must be (x) and (115600-x)

• I = Prt

• Working Equation>> x(.10)(1) = 2(115600-x)(.12)(1)

PS. It doesn’t matter which side has x or 115600-x, the final answer should be the same

• Answer Php 81600 must be invested into the 10% bank and Php34,000 at the 12% bank

2. In a school of x number of students, 40 more than a third are students taking up science

courses. Within that, . of those are taking Math as their science course. A total of 1/8 of the

student population are Math students. How many students are there in this school?

• Tip 1: To solve this, set up a linear equation

• Tip 2: Your unknown in this scenario is the number of students.

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• Find the common ground in this problem- math students

• Working equation = (1/8)(x) = . (1/3 x + 40)

• Answer = 240 students

3. Mocha has 10,000php of which she invests 4000php at Bank A and 6000php at Bank B. The

interest rate of Bank B is two-thirds that of Bank A. What must the interest rates be for Mocha to

have earned 720php in interest after 1?

• Tip 1:The unknowns are the interest rates

• Tip 2: One of the two interest rates are expressed in terms of the other

• Working Equation: 4000x + 6000(2/3 x) = 720

• Answers: 4000 at 9% and 6000 at 6%

F. Quadratic Equation in Single Variables

1. Raynald bought some watermelons for a price of $200. After selling 30 less than what he

originally bought, he made up for his original investment already by adding $1.50 per

watermelon. How many watermelons did Raynald originally buy?

• Tip 1: Solve this by setting up a linear equation

• Tip2: The original amount of all watermelons is $200. 30 less than the original amount of the

watermelons multiplied by the price he sold those watermelons for is also equal to 200.

• Working Equation (number of watermelons sold)(price per melon divided by the number of

watermelons originally bought + 1.5) = initial investment

(x-30) (200/ x + 1.5) = 200

• Answer is 80

2. y2 – y – 4 = y 2- y - 2

• Tip 1: Let a = y2 - y - 2

• Tip 2: Simplify the equation given Tip 1 such that

• Tip 3: You end up working with an easier equation. a2 - a – 2 = 0

• Answers: x = (1± 13 ) / 2 ; -2, 3

ORG HEADER HERE

3. ( 32 y - 2y = 1y ) 2

• Answers : y = -2 and 3, but only 3 counts because once you substitute -2 into the given

equation you get an imaginary number in the terms

4. ( 23 x = x +2 )2

• Answer : x = 9 and 1, but once plugged into the original equation, only 9 satisfies it.

5. (2x – 1/x ) 2- 3( 2x – 1/x) – 4 = 0

• Use the tip in #2.

• Answers: x = -1, 1/2, (2± 3 ) / 2

G. Problem Solving

1. The tens digit of a two digit number is 1 more than the unit digit. If the number divided by the sum of the digit is 6, what is the number? 2 digit number: 54 2. A man invested part of P 15,000 at 12% of the remainder at 8%. If his annual income from the 2 investments is P 1456, how much was invested at each rate? 12% = P 6,400 8% = P 8,600 3. Find 2 consecutive integers whose product is 72. -8, -9 or 8, 9 4. Find the dimensions of a rectangle whose length is 5 less than twice it’s width and whose area is 63 square units. Width = 7 Length = 9

ORG HEADER HERE

H. Systems of Equations: Solve for the unknowns using any of the 3 solutions: Elimination,

Substitution, or Graphical Method.

1. x + 3y = 7

2x + 7y = 8

Final answer: x = 25; y= -6

2. 6x + 3y = 2

4x – y = 0

Final Answer: x =1/9; y=4/9

3. Solve for the variables x, y, z by elimination: 1. 2x +2y = 2 2. x + y + z = 4 3x + 2y = -2 2x + 3y –z = 13 3x -2y + 2z = 3 y= 5 x = -4 x= 3 y= 2 z = -1 4. Solve for the variables x, y, z by substitution: 1. x – y =1 2. x – y = 1 4x + 8 = 2y 3x – 6 = 3y x= -1 = false / no solution y= 2 5. Solve for the variables x, y, z by graphical method: 1. 3x + 2y = 3 2. x + y = 2 4x – y = -7 3x + 3y = 9 (-1 , 3) ( no solution )

ORG HEADER HERE

I. Operations on Matrices: Solve for the following:

J. Solve using Cramer’s Rule, Matrix Inversion Method and Gauss-Jordan Elimination Method.

2x – 3y + 4z = 11

3x +4y -2z = 10

x -4y +6z=14

Final answer: x=2; y=3; z=4

ORG HEADER HERE

K. Linear Inequalities. Find the solution set.

1. 3 + 4x < 7 + 8x

Final answer: SS={x l x>-1}

2. -5 < 2x-3

Final answer: SS={x l x>-1}

3. 2x – 3 < 9

Final answer: SS={x l x<6}

4. 5x – 7 > 3x + 9 Final answer: SS={x l x > 8} 5. 4 – 3x ≤ 20 Final answer: SS={x l x ≥ -16/3} 6. 18 ≥ 4(2x−3) − 9x Final answer: SS={x l x ≥ -30}

L. Graph the inequality.

1. 3x + y > 1

2. Y > 4

3. y = 2x + 3

ORG HEADER HERE

4. 2x – 3y < 6