MATH - QUARTER 2: EXAM REVIEWER
Review review mora japun mamatay!!!
(GIKAN RA NI SA AKONG NOTES!!! SO IF NAAY KULANG SORRY DAAN)
LINEAR INEQUALITIES IN TWO VARIABLES
Ax + By > 0
Ax + By < 0
Ax + By ≤ 0
Ax + By ≥ 0
EXAMPLE:
2x + 3y > 6
1. Change “>” “<” to “=” (2x + 3y = 6)
2. Graph using any method (recommended: intercept form)
INTERCEPT FORM
Let y = 0
2x + 3y = 6
2x + 3(0) = 6
2x = 6
2
2
Let x = 0
2x + 3y = 6
2(0) + 3y = 6
3y = 6
3
3
X=3
(3 , 0)
Y=2
(0 , 2)
Shaded
part
3. TEST POINT
(0 , 0)
2x + 3y > 6
2(0) + 3(0) > 6
0 > 6 (false)
(REMEMBER: IF THE STATEMENT IS FALSE, THE SHADED AREA WILL NOT
TOUCH THE SUBSTITUTE/the middle point)
(IF THE GIVEN IS “>” “<” THE LINES WILL BE BROKEN)
(IF THE GIVEN IS “ ≥” “≤” THE LINES WILL BE STRAIGHT)
MATHEMATICAL SYMBOLS
EXCEEDS - >
FEWER THAN - <
OVER - >
MINIMUM - ≥
MAXIMUM - ≤
UNDER - <
MORE THAN - >
BELOW - <
AT LEAST - ≥
AT MOST - ≤
RELATION AND FUNCTION
RELATION - A SET OF ORDERED PAIRS
FUNCTION - A RELATION WHOSE ELEMENT IS THE DOMAIN THAT IS
MATCHED AT EXACTLY ONE ELEMENT IN THE RANGE
DOMAIN - SET OF X-COORDINATES AKA THE VALUE OF X
RANGE - SET OF Y-COORDINATES AKA THE VALUE OF Y
“ALL FUNCTION IS A RELATION BUT NOT ALL RELATION IS A FUNCTION”
1. (2,6) (3,4) (5, -2)
D
R
2
6
3
4
5
-2
2. (4,5) (3,5) (-6,5)
D
R
4
3
5
-6
3. (2,3) (4,6) (2,5)
D
R
3
2
6
4
5
FUNCTION
FUNCTION
RELATION
(SO BASICALLY PWEDE MANGABIT SI RANGE PERO BAWAL MANGABIT SI
DOMAIN)
FINDING THE DOMAIN AND RANGE
1.
D: ( x l x e lR)
R: ( y l y e lR)
2.
D: ( x l x ≥ 0)
R: ( y l y e lR)
3.
D: ( x l x e lR)
R: ( y l y ≤ 0)
l - such as
e - is an element
lR - of real numbers
≥ - if the line is going positive
≤ - if the line is going negative
= - if the line is going straight
e lR - if the line is horizontal/vertical
FINDING THR DOMAIN OF A FUNCTION
Easy lvl:
1. f(x) = 3x/x+2
2. f(x) = √x-2
x+2=0
x ≠ -2
D: (x l x e lR, x ≠ -2)
x-2≥0
x ≥2
D: (x l x ≥ 2)
3. f(x) = x + 2
D: (x l x e lR)
(if there’s no denominator, you don’t need to solve it.)
Average lvl:
1. f(x) 4/x² + 6x + 8 (x+4)(x+2) = 0
= 4/(x+4)(x+2)
4 = -4 2= -2
D: (x l x e lR, x ≠ -4, -2)
2. f(x) 5/x² - 4x - 12 (x+6)(x-2) = 0
= 5/(x+6)(x-2)
6 = -6 -2 = 2
D: (x l x e lR, x ≠ 4, -4)
3. f(x) 5/x² - 16
(x-4)(x+4) = 0
=(x-4)(x+4)
-4 = 4 4 = -4
D: ( x l x e lR, x ≠ 4, -4)
(numbers can be at any order)
Hard lvl:
1. f(x) √x² - 6x + 8
x² - 6x + 8 ≥ 0
(x-4)(x-2) = 0
-4 = 4 -2 = 2
D: (x l x ≥ 4, x ≤ 2)
2. f(x) = √x² - 9
√x² -9
(x+3)(x-3)
3 = -3 -3 = 3
D: ( x l x ≤ -3, x ≥ 3)
3. f(x) = √x² - 14x -51
(x+3)(x-17)
3 = -3 -17 = 17
D: ( x l x ≥ 17, x ≤ -3)
(the numbers can also be at any order as long as the signs remain correct)
LOGICAL REASONING
IF - THEN STATEMENTS
conditional statement - symbolized by p --> q, is an if-then statement in which p is a hypothesis
and q is a conclusion. The logical connector in a conditional statement is denoted by the Symbol -> . The conditional is defined to be true unless a true hypothesis leads to a false conclusion
p - hypothesis
q - conclusion
EXAMPLE
Conditional Statement: If a car is in good condition, then it is safe for driving. Hypothesis: A car is in
good condition
Conclusion: It is safe for driving.
INVERSE, CONVERSE, and CONTRAPOSITIVE
Conditional Statement - if p then q
Converse - if q then p
Inverse - If not p then not q
Contrapositive - If not q then not p
DEDUCTIVE AND INDIDUCTIVE REASONING
Inductive reasoning - a kind of reasoning where the conclusion is made based upon current
knowledge, observation, examples and patterns. It uses specific examples to arrive at a general
rule, generalizations or conclusions. It is judging by experience. It involves uncertainty in making
conclusions. Inductive Reasoning is a process of observing data, recognizing patterns, and
making generalizations from observations.
EXAMPLE:
My Math teacher is strict. My previous Math teacher was strict. What can you say about all math
teachers?
All Math teachers are strict
Deductive reasoning - a type of logical reasoning that uses accepted facts to reason in a step-bystep manner until we arrive at the desired statements. From the given statement, you are to make
a sound judgment or a conclusion. For a clearer thought, let us take some laws in logic that is vital
in deduction.
Law of Detachment (Modus Ponens)
Major Premise: If p is true, then q is true.
Minor Premise: p is true.
Conclusion: Therefore, q is true.
1. Major Premise: If you are an 18-year old Filipino citizen, then you can vote.
Minor Premise: Pete is an 18-year old Filipino.
Conclusion: Therefore, Pete can vote.
Law of Syllogism (Chain Rule)
Major Premise: If p is true, then q is true.
Minor Premise: If q is true, then r is true.
Conclusion: If p, then r
1. Major Premise: If it is May, then there are many flowers.
Minor Premise: If there are many flowers, then I am happy.
Conclusion: If it is May, then I am happy.