Math 80 Lecture - Chapter 2 Merced College; Don Power

2.1 Introduction to Integers (Signed Numbers)

Objective: Integers and the Number Line

Graph signed nrs on a nr line.

Ex: some pos, neg, 0, fractions

Use the < and > symbols to compare integers

Trickiest: comparing two negs

Rule: smaller nr is to left on nr line, large nr is to the right

Objective: Find the opposite of a signed number.

Def: The two numbers on a number line that are the same distance from 0, but on the opposite sides of 0

To write the opposite of any number, you can either add or delete a negative sign

So the opp of -8 can be written as either 8 or -(-8).

You may have been taught to use a raised symbol for a negative, so -(-8) is the same as -^-8

(Note the raised minus symbol is used for negative on many calculators)

Ex: Write the opposite of ...

Objective: Absolute Value

Notation

Meaning: Distance from 0 on a number line

Ex: pos and neg nrs, incl decimals and fractions

Challenge: abs val in a larger expression: Find 28 - |-3|

Applic: We'll use "abs val" to refer to the size of a nr independent of its sign

Ex: (1) Which is larger, -7 or 3?

(2) does -7 or 3 have the larger abs value?

Objective: Applications

Write pos and neg nrs used in everyday situations. They exist!

Examples of nr lines in real world: thermometer, altitude legend on map

Ex of integers in the real world

Altitudes in feet: Sea of Galilee (-686), Death Valley (-280), Dead Sea (-1286),

Mt Whitney (14,494), Mt McKinley (20320)

Temperatures: Absolute zero: -273.16°C = -459.69°F

Bank balances

If you have money

If you are overdrawn

2.2 Adding and Subtracting Integers

Number line: think direction and distance:

The sign [first sign] gives the direction;

[If the second sign is neg, it reverses the direction, but a pos does nothing].

+ and - without a number line (Assuming you have cleared the parentheses):

Same signs: Add absolute values, copy the sign.

Opposite signs: Subtract absolute values, take sign of larger (i.e. the number with the larger absolute value)

To combine adding and subtracting of integers (signed numbers):

I recommend clearing double signs and parentheses before you add or subtract:

The key is to count the negative signs!

One - : write - Ex: 4+(-3), Rewrite as 4 − 3 [One neg makes a neg]

Two -'s : write + Ex: 4-(-3), Rewrite as 4 + 3 [Two negs make a pos.]

Odd number of −'s: Ex: 5−(−(−3)), Same as one −, Result is always −

Even number of −'s: Ex: 5−(−(−(−3))), Same as two −'s, Result is always +

Identify properties of addition

Addition property of 0

Commutative -- changing order (from L to R) doesn't change sum

(Commuting is moving: numbers move)

Associative -- changing grouping doesn't change sum

(What two numbers associate with each other first?)

(Numbers don't move, only the parentheses do)

Use them together for long addition problems -- don't rewrite unnecessarily.

2.3 Multiplying and Dividing Integers

Multiplying

Use raised dot or paren

Sign rules are same as for clearing double signs:

One neg (or odd nr. of negs) makes a neg

Two negs (or even nr. of negs) make a pos.

(+/-) key on calculator

Identify properties

Commut

Assoc

Distrib

Estimate answers to app probs involving mult.

Dividing Integers

Obj: Divide intg

Sign rules -- same as mult or clearing double signs

Ex's are w/o remainders

Forms: Frac, /, horiz frac bar, div frame )^--

Obj: Identify properties of div

num / itself = ?

num / 1 = ?

0 / number

number / 0 Distinguish the last two by checking: Ex: 6/3 = 2

Obj: Est answers to app probs involving div

Front-end rounding

Obj: Interpret remainders in div app probs:

Round normally: X62 What is avg cost?

Round up always: X70 How many rooms are needed for ___ people?

Round down always: X72 How many scholarships of $___ can be given from a fixed fund?

2.4 Solving Equations with Integers

Obj: Determine whether a given number is a solution of an equation.

Ex: Is 2 a solution of 4x^3+18=7x? [No] What about -2? [yes]

So a solution is any value of the variable (letter) that makes the equation true.

Solving Equations Using Addition/Subtraction

Obj: Solve equations, using the addition property of equality:

First: Principle is that you can do the same thing to both sides of equation and it's still the same equation

Second: Some things you do to both sides help you find the solution, and others don't --

so you need a strategy to determine what to add or subtract.

The strategy is to separate the variable terms from the constant terms on opposite sides of the equal sign. Steps:

1. Decide which side of the equation to put the variables on,

and which side of the equation to put the constants on.

2. Pick one term that's on the wrong side

If it's +, subtract it from both sides.

If it's −, add it to both sides.

Result (after collecting terms) is one less term.

3. Repeat step 2 until you're done

Solving Equations Using Division

Obj: Solve eqns using the division property of equality

As before, the principle is that you can do the same thing to both sides of equation and it's still the same equation

As before, some things you could do to both sides will help you find the solution, and others don't -- so you need a strategy:

The strategy is first to finish steps 1-3 (the same as we did above): always do these steps before doing the new division step

Then, after separating the terms, do a new step:

Divide [by the coefficient of the variable]. This will finish the solution.

EX: 10, 12

Obj: Simplify equations and then use the division property of equality

Ex: 24, 38, 22

Obj: Solve equations such as -x = 5

Treat the coefficient as -1, so divide by -1.

Summary example: X54

Complete Procedure for Solving Linear Equations -- [We will not get any more steps]:

Basic procedure for solving first degree equations in one variable:

Floating step: Collect like terms on each side of the equation.

Clear parentheses

Clear fractions (optional). Multiply every term by the LCD of all the denominators.

Separate Terms. Technique: Move terms using the Addition/Subtraction Property

Strategy Decision: Variable terms to right or to left? Then constants go to the other side.

Important definition: Terms are elements that are added or subtracted (not multiplied or divided)

Collect like terms again. You should end up with exactly one term on each side of the equation.

Divide by coefficient of x (or multiply by the reciprocal if you did not clear fractions)

Check : Substitute the solution in place of the variable in the original equation.

Abbreviated procedure -- LEARN

[collect]

parentheses

fractions

separate terms (+ or −)

collect -- get down to one term on each side

divide

check

2.5 Exponents and Order of Operations

Write repeated factors, using exponents.

Def as repeated factors [by example] for pos integer exponents.

Vocab: to the ____th power. (or squared or cubed)

Task: exponent form to expanded form and vice versa

Ex: 2⁵ Q: How many times does 5 show up in the list of factors? result ¹ 10

Simplify expressions containing exponents DO THIS AFTER ORDER OF OPS

(-3)² vs (-3)³ and -3² and -3³ or following signs:

1+(-3)² vs 1+(-3)³ and 1+(-3²) and 1+(-3³) or

1-(-3)² vs 1-(-3)³ and 1-(-3²) and 1-(-3³)

Use the order of operations: Please Excuse My Demented Aunt Sally

P: ops inside paren

E: apply exp

M,D: same precedence, L to R

Also clear paren and apply neg signs (esp double signs) here

A,S: same precedence, L to R

Ex: 10-30/2 without exponents

9+-5+2*-2

4+3(8-3)

3(2-7)-(-5+1)

Ex: 5-5² with expo

1--10*(-3)³

5*4²-6(1+4)--3

(-5)²*(9-17)²/(-10)²

Ex: 4²|13-17|(-2)*(-2)³ with absolute value

6-|2-3*4|+(-5)²/5²

Simplify expressions with fraction bars

Read fraction as (____) / (____); You must finish the num and denom before reducing the fraction

Ex: ; ;

Ex: (num of 0):

Ex: (denom of 0):

2.1 Intro to Variables

Obj: Identify variables, constants, coefficients [and exponents] from and expr like 3v+2

Obj: Evaluate an expression like 4u-3 for a given value of the variable

Obj: Write properties of operations using variables: Do X28,30 in class

Obj: Understand the use of exponents with variables:

Rewrite x^4 w/o expo

Eval expr like -3x^3*y^4 when x = … and y = …

2.2 Simplifying Expressions

Like terms: identify; combine

Actual: add coeff, copy vars and exponents

Why? do by distrib law, in reverse

Example: #38

Simplify 4(-3a^3) etc

Use distrib for -9(3p-2)

Use distrib and combine like terms -9(3p-2)+4p+5 etc

Return to: Merced College; Don Power Updated 10/05/06 by Don Power