1 4 Absolute Values Solving Absolute Value Equations
1. 4 Absolute Values Solving Absolute Value Equations By putting into one of 3 categories
What is the definition of “Absolute Value”? __________________________________ Mathematically, __________ For example, in -3 , where could x be? 0 3
To solve these situations, __________________________________ Consider ____________ 0
That was Category 1 : __________________________________ Example Case 1 Case 2
Absolute Value Inequalities Think logically about another situation. What does mean? For instance, in the equation , _________________ 0
How does that translate into a sentence? _________________ Now solve for x. This is Category 2: when x is less than a number
Absolute Value Inequalities What does mean? In the equation , _________________ 0
Less than = And statement Greater than = Or statement Note: is the same as ; just have the sign in the rewritten equation match the original.
Example inequality with x on other side Case 1 Case 2
Examples Case 1 Case 2
Case 1 Case 2
Case 1 Case 2
1 -4 Compound Absolute Values Equalities and Inequalities More than one absolute value in the equation
Find a number that works. We will find a more methodical approach to find all the solutions.
In your approach, think about the values of this particular mathematical statement in the 3 different areas on a number line. _____________________ -____________________ - ____________________
It now forms 3 different areas or cases on the number line. • ______________________________ • __________________
Case 2 Domain __________________
Case 3 Domain _______________
If you get an answer in any of the cases where the variable disappears and the answer is TRUE ________________________________
1 -5 Exponential Rules You know these already
Review of the Basics
Practice problems Simplify each expression
Do NOT use calculators on the homework, please!!
1 -6 Radicals (Day 1) and Rational Exponents (Day 2)
What is a Radical? In simplest term, it is a square root ( = ________ )
The Principle nth root of a: 1. • If a < 0 and n is odd then is a negative number __________ • If a < 0 and n is even, _______
Vocab
Lets Recall
Rules of Radicals
Practice problems Simplify each expression
1 -6 Radicals (Day 1) and Rational Exponents (Day 2)
What is a Rational Exponent? ____________________________________
Don’t be overwhelmed by fractions! These problems are not hard, as long as you remember what each letter means. Notice I used “p” as the numerator and “r” as the denominator. p= _______________ r= ________________________________________
Practice problems Simplify each expression
The last part of this topic What is wrong with this number? ___________________________________
Rationalizing If you see a single radical in the denominator ________________________________________
Rationalizing If you see 1 or more radicals in a binomial, what can we do?
What is a Conjugate? The conjugate of a + b ______. Why? The conjugate ___________________________________
Rationalizing Multiply top and bottom by the conjugate! Its MAGIC!!
1. 7 Fundamental Operations
Terms • Monomial _______________ • Binomial ________________ • Trinomial _______________
Standard form: ____________________ Collecting Like terms
Examples
1 -8 Factoring Patterns
Perfect Square Trinomial Factors as
Difference of squares
Sum/Difference of cubes
3 terms but not a pattern? This is where you use combinations of the first term with combinations of the third term that collect to be the middle term.
Examples
Solve using difference of Squares Solve using sum of cubes
Examples of Grouping
1. 9 Fundamental Operations What are the Fundamental Operations?
Addition, Subtraction, Multiplication and Division We will be applying these fundamental operations to rational expressions. This will all be review. We are working on the little things here.
Simplify the following Hint: _______________________________________ 2. What is the domain of problem 1?
Polynomial Long Division
Polynomial Long Division
Classwork
1 -10 Introduction to Complex Numbers What is a complex number?
To see a complex number we have to first see where it shows up Solve both of these
Um, no solution? ? does not have a real answer. It has an “imaginary” answer. To define a complex number ________________________ This new variable is “ i “
Definition: Note: _________ So, following this definition: ______________________________
And it cycles…. Do you see a pattern yet?
What is that pattern? We are looking at the remainder when the power is divided by 4. Why? ___________________________________ Try it with
Examples
OK, so what is a complex number? ______________________________ A complex number comes in the form a + bi
Lets try these 4 problems.
Try on your own
- Slides: 86