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Algebra (3/4)
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Inequalities
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Differences in manipulation inequalities vs. Equalities
"For inequalities everything is equal to equalities, except:
(1) When multiplying/dividing by a NEGATIVE - flip the sign
(2) When inverting inequalities - flip the sign
TIP: If you have an integer constraint for inequalities, you should be able to solve the questions with very little information."
"Inequalities:
Multiplying/dividing by a variable: x/y<1 -> x>/<y"
"When the denominator is
(1) negative --> flip the sign
(2) unknown --> you cannot multiply/divide"
"Inequalities:
Taking the reciprocal of x<y --> 1/x>/<1/y"
"When the signs of numerator/denominator are:
(1) the same --> flip the inequality sign
(2) different --> sign stays the same
(3) unknown --> you cannot take the reciprocal"
"Inequalities:
Squaring variables"
"When the signs of numerator/denominator are:
(1) both positive --> inequality sign stays the same
(2) both negative --> flip the inequality sign
(3) different --> you cannot square the variables without knowing their values"
"When you have to combine two inequalities...
Ex.: What is the range of xy, given that 0<=x<=3 and y<8?"
"...use extreme values to determine the new upper & lower limit:
(1) Build extreme value cases
For x: Low is 0; High is 3
For y: Low is n/a; High is LT8
(2) Build scenarios out of the cases
Lower limit of xy is 0*n/a = n/a
Upper limit of xy is 3*LT8=LT24
Attention: If x could be negative, then there would be no upper limit!
"
When you have to determine a possible range from an inequality and an equation...
...use extreme value to determine the upper and the lower limit.
"Linear (arithmetic) Sequence
(1) direct formula
(2) recursive formula"
"The DIFFERENCE between terms is always the same.
Direct: S(n) = kn+x
Recursive:
S(n-1) + k
S(1) = k + x
Where:
k = constant difference,
X = some other constant (starting point)"
"Exponential (geometric) Sequence
(1) Direct formula
(2) Recursive formula"
"The RATIO between terms is always the same.
(1) Direct: S(n) = xk^n
(2) Recursive
kS(n-1)
S(1) = xk
Where:
k = constant difference,
X = some other constant (starting point)
TIP: If you are not given successive terms (e.g. C(3) and C(5)) you have to build the ""bridge"" algebraically first."
Sequences technique
"(1) Calculate ""k"" first with the recursive formula (since you need to compare terms)
(2) Use the direct formula S(1) to calculate ""x"""
Linear (arithmetic) sequence - Formula & Technique
"Formula: Sn = kn + x
Technique:
1. Identify the rule first
2. Solve for x (plug in n=1 and solve for x)
3. Check the rule (plug in the result for x"
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