Merced College; Don Power

INTERMEDIATE ALGEBRA - CH 1, LECTURE

1.1 Real Number System

Definitions/Relationships:

Real numbers

Within real numbers: Rational vs Irrational

Rational: decimal form terminates or repeats (demo)

Within rational: Integers vs Non-integer fractions

Within integers: Negative Integers vs Whole numbers

Within whole numbers: Natural numbers vs Zero

Ex 1: Identify which categories apply for pi, -sqrt(4), sqrt(5)

Ex 2: Plot points on a real number line

Ex 3: List the odd integers between -3π and sqrt(7)

Ex 4:

Rational vs irrational: For rationals, decimal representation either terminates or repeats

Class: Turn each form into a fraction: 0.18, 0.18181818...

You may not see the repeat on the calculator:

Ex 4/41 is .09756097561 on TI-85. Rounding issue. How to write?

4/43 is .093023255814. The problem here is that the repetend is 21 digits long.

Ordering real numbers:

Insert the appropriate inequality symbol < or > between two numbers

Key: the smaller numbers are to the left on the number line

Ex: Place the correct inequality symbol between -2π and -sqrt(38) [use calculator]

Examples in book are all rational

How to compare fractions with different denominators, say 3/5 and 4/7

LCD technique

Distance:

Rule: if a ≤ b, then dist(a,b)=b-a (Key: large – small)

Ex4 If they’re both the same sign, easy (just subtract abs values)

Trickiest: if they’re opposite signs, you get a double negative

What is the dist between 3 and -6?

Not in book: If you don't know which will be larger, d(a,b) = |b-a| = |a-b|

Concept of opposites: two numbers the same distance from 0 on a number line

We write the opposite of a number a as –a

So the opposite of –a is –(-a), which is the same as a itself.

Notice that a + (-a) = 0, or the sum of a number and its opposite is 0

Therefore, we also call opposites “additive inverses”

Absolute Value: Then that distance from 0 is the absolute value of both numbers

Note: distance is always positive

Formal def of abs value:

|a| = a, if a > 0 or a = 0; or |a| = -a, if a < 0.

Ex5-6 Finding absolute values, simplifying abs value expressions

Notice -(-a) is not the same as -|-a|

Contrast abs(-4) with -abs(4)

1.2 Operations with Real Numbers

Addition, Subtraction, Multiplication, Division

Sign Rules: Ex: 7 operation -14; -9 operation -6

Fraction x and / Review #33, 38

Fraction + and –

Like denoms 3/8 + 1/8 Reduce

Unlike denoms: Supply the missing factor(s) from the LCD

For 2 fractions:

If no common factors in denoms, 2/7 + 5/9

The LCD is the product of the two denoms, so…

The missing factor is just the other LCD

If there is a common factor in denoms, 13/16 – 7/20

The LCD is the product of the common factor with both uncommon factors, so…

The missing factor is the uncommon factor from the alternate fraction

Supplemental: Review of Addition/Subtraction with Fractions

Prime factorization technique to find LCD's

Copy each base, then, for each base, copy the highest exponent that appears.

Study tip (text, pg 13): a/b + c/d = (ad+bc) / (bd)

Works best if there are no common factors fro the two denoms

Variation (not in text): If e is a common factor,

a/be + c/de = (ad+bc) / (bde)

To implement this easy/new technique for any 2 fractions:

Multiply each fraction (top and bottom) by the reduced denominator from the other fraction.

i.e., by the "uncommon" factor from the other fraction

Ex: 5/12 + 7/15 = 53/60 [12 and 15 reduce to 4 and 5; multiply by 5/5 and 4/4 respectively]

Ex: 37/840 - 21/560 = 11/1680

Reduction of the 840 and 560 results in 3 and 2: these are the reduced denominators

This technique also works for fractions of algebraic expressions.

Mixed numbers

X and / Convert to improper fractions first

+ and - Better: Get a common denom for the fraction part

Add/subt both the integer part and the fraction part

If necessary

Borrow before subtracting: 212 1/5 – 78 2/3

Subt 1 from integer part

Add denom to num to get the new num for the fraction part

Carry after adding: 47 5/6 + 81 5/9

Add 1 to integer part

Subt denom from num to get the new num for the fraction part

Write repeated mult in exp form and vice versa; evaluate exponential expressions

(-4)² vs -4² Also |–4|² and –|4|²

Note: applying the sign is a multiplication step (mult by –1)

Know order of ops

P: Ops inside grouping symbols. Fraction bar means (__) / (___)

So you can't reduce 3's in (3+x)/(3+y). Try (3+5) / (3+1)

E: Exponents

MD: Multiplication and division, in order from left to right

Equivalently: Build fraction with all X's in num and all /'s in denom

Note: a neg sign is the equiv of mult by –1

AS: Addition and subtraction, in order from left to right.

Equivalently: Add all +'s, add all –'s, finish with one subraction

Note style: important to write entire problem at each step.

Regular example: Review #49

Fraction example: Ex #126

Evaluate expressions using a calculator and order of operations

Working with negatives and parentheses

Ex #139b

Applied problems:

#133 Find the unknown fractional part of a circle graph

#138 Create a table of yearly gains and losses

#143 Find area of a triangle

#150 vs #151 For false results, you can use a counterexample (-1)¹¹ = -1, not pos.

1.3 Properties of Real Numbers

Commutative: a+b = b+a, and a*b = b*a

Not for subtraction: 2-5 ¹5-2. But 2 + (-5) = -5 + 2. Why?

Treat as addition, so that the Commutative Law applies;

In practice, when you move the number, move the sign along with it.

Not for division: 4 / 2 ≠ 2 / 4. But 4 * 1/2 = 1/2 * 4

Treat as division, so that the Commutative Law applies

Associative: (a+b)+c = a+(b+c) and (a*b)*c = a*(b*c)

Again, not for subtraction, but we can view subtraction as addition of a negative number.

Challenge: Insert parentheses to combine the last two terms first in the expression 5-4-3.

Solution: 5-4-3 ¹ 5-(4-3) or 5(-4-3), but 5+(-4-3) Why? Law is for addition

Distributive - Use to multiply or to factor.

Geometric example at beginning of chapter to show a(b+c) = ab+ac

Ex: 12(7x/4 - 2y/3) Technique: reduce, then mult; App: clearing fractions.

Combining like terms - Actually an app of distributive law

Ex: Compare sol technique for 2x+3x = 6 and 2x + ax = 6

Identity: Addition of 0 or mult by 1 leaves any real number unchanged.

Inverse:

Addition of the "opposite" results in the additive identity, i.e. 0.

Opposites - Numbers that sum to 0; i.e. numbers with opposite signs.

Multiplication by the "reciprocal" results in the multiplicative identity, i.e. 1.

Reciprocals - Numbers that multiply to 1, e.g. –4/5 and –5/4

0 has no reciprocal: division by 0 is undefined

If you forget which of 0/6 or 6/0 is valid, try checking (how would you check 6/3=2 ?)

Ex: Find reciprocal and opposite of 2, 3/2, -4, -1/5, -x

Properties of equality

Addition property of equality Does this also work for subtraction?

Multiplication property of equality Does this also work for division? Except when …

Cancellation property of addition

Cancellation property of mult ac = bc and c ≠ 0 implies a = b

Not valid if cancelling 0: equivalent to division by 0: 2x = 3x doesn’t mean 2 = 3.

Sample problems:

Identify the property (commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, identity property, inverse property, distributive property) for each example:

a. If 6x = 6, then x must be 1.

b. 2(y+x) = 2(x+y)

c.

d. 2(x+4) = 2·x + 2·4

e. 2+(3+y) = (2+3)+y

f. z(3+4) = (3+4)z

g. To solve 3x-2=7, add 2 to both sides of the equation, then divide both sides by 3.

1.4 Algebraic Expressions

Definitions:

Terms vs factors

Terms, and coefficients, and constants: Be able to identify/distinguish

Apply properties (from 1.3) to algebraic expressions

Identify the property (#19-24)

Or… Rewrite an expression using a given property: (#25 vs #26)

Clearing parentheses and combining like terms

Def: Like terms have the same variable factors (including variables and exponents) Ex: #46

I suggest underlining like terms, if needed, instead of introducing extra parentheses.

Ex: Subtract (-4x²+5x-3) - (2x²+x-7)

Use order of operations: Ex: –3t[5–(7–t)]+8t–3

More examples: #59-82

Evaluate algebraic expressions by substituting values for the expressions

Ex: Evaluate 4x³-3x²-7 if x = -2

Applications:

#102: Write and simplify an expression for area of the figure

#107-8: Bar chart and formula, with substitutions; use of indexing for the year.

#109: Find area of a roof by calculating and adding its sections: area of trapezoid

1.5 Constructing Algebraic Expressions

This is a lesson that will help you do word problems:

Translating between verbal phrases and algebraic expressions

Constructing algebraic expressions that represent quantities described verbally:

Many of these involve rates or known “easy” formulas: substitute and solve:

Distance = rate * time #48 vs #50

Total value of coins = Value per coin * nr coins + ... #44, 46

Tax = Amount * tax-rate #54

Area = length * width #74

Sum = one nr. + second nr.

Contrast: product

Sum of consecutive numbers #60

Sum of consecutive odd integers, first of which is 2n+1 #61

Total cost = Fixed cost + variable cost #56, 58

McKEAGUE:

1.1 Linear and Quadratic Equations in One Variable

Def: Solution set [why a set?], equivalent equations [e.g. steps in an equation solution]

Properties of equality

Addition property

Mult property

Symmetric property (you can switch sides) [not in text]

Transitive property (a=b and b=c implies a=c) [not in text]

Permits simplification of each side independently; permits substitutions

1.2 Formulas

Given a formula, make substitutions and solve for the remaining letter

Height of an object in free-fall (Text's Ex 4): Note general formula is h = -16t²+v₀t+h₀

Example: If an object is thrown upward at 48 fps from top of 160 ft bldg (v₀=48, s₀=160),

1. What equation results (this will be given to you in this text)? Use h = -16t² + 48t +160.

2. Find t if h = 96 ft? Interpret the solution of t = 4 or -1.

3. When will the object hit the ground? Hint: What is the height if it is on the ground?

Revenue

Formula R = xp

Meaning: Revenue = number of items sold (x) times theprice of each item (p)

Problem similar to Ex: 5:

If the price per item and the number of items sold are related by the equation x = 1500 - 20p,

what must you set the price at to get a revenue of $22,000?

X83 x = 1800-100p; What should the price be to get a revenue of $7000?

Solving for a given letter:

Text: formulas from geometry [know these formulas on pg 116]

Also, p 117, surface area of a cylinder: sides 2prh, top and bottom each pr².

Formulas for number sequences

Ex for the following: Let "a" be the sequence a = {1, 4, 7, 10, 13, 16, ...}

Review from R8: arith or ...? What is common difference?

Given a sequence, we'll relate the position in the sequence to the value.

Use subscript to note the position; use a letter with a subscript for the value in the corresp position.

So in our example, a₁ = 1, a₂ = 4 ,a₃ = 7, ...

Task for this lesson: Find the first 5 terms of a seq, given the "general term" a_n = ....

(the subscript n indicates the formula works for every position in the sequence, for n=1, 2, 3, ...)

In this case, a_n = 3n-2

Ex: Find first 5 terms in seq a_n = 23 - 4n

Increasing vs decreasing sequences: define informally, identify for our two examples above.

1.3 Applications

Blueprint on page 126: Extra comments:

Step 1: Not mentally: list on paper. Plus, draw and label a picture, or build a table

If too complicated, pose and solve a related, simpler problem to get some ideas.

Step 2: Label a picture, or let x = the item we know the least about

Step 3: What known formula(s) might be involved?

Examples from text: most can be solved with either one or two vars

Ex 1: Geom (perimeter of rectangle) #7

Ex 2: % increase prob #18

Logic: old amount + increase = new amount

where the increase/decrease is always a percent of the old amount

Know vocab for angles on pg 129: 360 degrees, theta, right/straight, acute, obtuse angles

Complementary, supplementary

Examples

Ex 3: Complementary angles

Ex 4: Investments/charges at two interest rates

Know facts from geometry: equilateral, isosceles triangles; Pythag thm

Ex 5: Angles in an isosceles triangle

Ex 6: Sides in a right triangle

Ex 7: Building tables

Return to:  Merced College; Don Power               Updated 08/25/08 by Don Power