Transcript Absolute Value Inequalities

Absolute Value Problems
 Why do we create two problems when
solving an absolute value problem?
 Let's first return to the original definition
of absolute value:
 "| x | is the distance of x from zero."
 For instance, since both –2 and 2 are two
units from zero, we have | –2 | = | 2 | = 2:
 Since there are two answers to every abs
problem there must be two equations to
solve for both answers!
Absolute Value Inequalities
Stephan’s question:
Why do we flip the sign for the
negative part of the equation?
Absolute Value Inequalities
"| x | is the distance of x from zero."
For instance, since both –2 and 2 are two units from zero,
we have | –2 | = | 2 | = 2:
For the inequality: | x | < 3.
All the points between –3 and 3, but not actually including–
3 or 3, will work in this inequality.
For the inequality: | x | > 2. The solution will be all points
that are more than two units away from zero.
Multiplying Polynomials
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FOIL is the same as distributing
A*A = A2 but A + A = 2A
Exponent Rules:
A2* A3 = A5
(A2)3 = A6
Factoring
 ALWAYS LOOK FOR GCF FIRST!!!!
 Difference of 2 perfect squares
 Trinomial factoring
Factoring By Grouping
 Multiply the coefficient of A by C
 Find two numbers that multiply to
this new number and add to B
 Break the trinomial into two
binomials, using these two factors
 Factor the GCF out of each binomial
 Re-group the factors
Rational Expressions Chapter 2
 In order to add/subtract fractions
denominators must match exactly
 When multiplying…do top*top and
bottom*bottom
Complex Rational Expressions
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First add the two terms in the top
Then add the terms in the bottom
Then flip 2nd fraction and multiply
reduce
Rational Equations
 Always remember the difference
between an EXPRESSION and an
EQUATION… for expressions
denominators do NOT cancel away
 For equations… make all
denominators equal and then remove
them from the equation
Rational Inequalities
 You MUST make a number line to solve
inequalities with variables in the
denominator
 Find what number makes the denominator
undefined…add it to the #line
 Solve the inequality…add answer to the
number line
 Choose a # between the 2 #’s on the #line
and shade in if the number works in the
original equation and shade out if it doesn’t
Chapter 3- Radicals
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A square root in the calc is ^(1/2)
A cube root in the calc is ^(1/3)
A nth root in the calc is ^(1/n)
You cannot have negative numbers
under the radical sign if the index
(small #) is even
Multiplying and Dividing
 You can multiply and divide any
numbers or variables as long as the
indexes are the same
 √a*√b = √a*b
 √(a/b) = √a /√b
Adding/Subtracting and Simplifying
Radicals
 You can only add/subtract radicals
that have the same index, variables,
exponents and numbers under the
radical sign
 To simplify a number under a radical
sign re-write it as multiplication of its
factors…pick factors that are perfect
squares/ cubes/ etc.
Simplifying Variables in Radicals
 For square roots divide even
exponents by 2 and move outside the
radical
 For square roots with odd exponents
divide by 2 and leave remainder
under the radical
 For cubes or any root higher divide by
the index and leave the remainder
under the radical
Rationalizing Denominators
 Never leave a radical in a
denominator
 Multiply monomials by the radical
divided by the radical
 Multiply binomials by the conjugate
Radical Equations
Isolate the radical
Square both sides of the equation
Solve and check answer
If there is more than one radical…
separate so that each is on its own
side of the equation, then repeat the
first 3 steps until all radical are gone
 CHECK ANSWERS
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