Algebra I (SparkCharts)

NUMBER SYSTEMS The natural numbers are the numbers we count with: 1, 2. 3, 4, 5, 6,

, 27, 28, ..

The whole numbers are the numbers we count with and zero: 0, 1, 2, 3, 4, 5, 6, . The integers are the numbers we count with, their negatives. and zero: ... , -3, -2, -1, 0, I, 2, 3, . -The positive integers are the natural numbers. -The negotfve integers are the "minus" natural numbers: -1, -2. -3, -4, .

The rcmonal numbers are all numbers expressible as ~ fractions. The fractions may be proper (less than one; Ex: k) or improper (more than one; Ex: ft). Rational numbers can be positive (Ex: 5.125 =

rational: Ex: 4 =

y.

¥-) or negative (Ex:

- ~). All integers are

The real numbers can be represented as points on the number line. All rational numbers aTe real, but the real number line has many points that are "between" rational numbers and aTe called ilTotional. Ex:

0,

To,

v'3 -

Real numbers

.,

-!

1.25

2 3 etc. -1, -2, -3, ..

9, 0.12112111211112.

Irrationals

The imaginary numbers are square roots of negative numbers. They don't appear on the real number line and are written in terms of i = FI. Ex: J=49 is imaginary and equal to iV49 or 7i.

v'5~

The complex numbers are all possible sums of real and imaginary numbers; they are written as a + bi. where a and bare real and i = FI is imaginary. All reals are complex (with b = 0) and all imaginary numbers are complex (with a = 0). The Fundamental Theorem of Algebra says that every

Integers

~~

- - Whole Dwnbers Natural numbers

)'t2+3

etc.

Venn Diagram of Number Systems

polynomial of degree n has exactly n complex roots (counting multiple roots).

I

SETS A set Is ony collecllon-finite or infinite--:: (; SttS c 84 Sto c( ~

-a is a real number. a+ (-a) =(-a)+a=O Also, -(-a) = a. a

+ b is a real number.

1 is a real number. a-l=l-a=a 1 is the multiplkotive identity.

If a i- 0, ~ is a real number.

+ c) = a b + a c

+ c) . a = b· a + c· a

Inequality « and»

Property

Equality (=)

Reflexive

a=a

Symmetric

If a = b, then b = a. Ifa~bandb=c,

Transitive

INEQUALITY SYMBOLS Meaning

Other properties: Suppose a, b, and c are real numbers.

then a

= c.

t = a.

a b is a real number.

11 a < band b < c,

then a < c.

Example


greater than 1 > 0 and 56 > 4 i not equal to 0 i 3 and -1 i 1 ~ less than or equal to 1 ~ 1 and 1 ~ 2 2: greater than or equal to 1 2: 1 and 3 2: - 29 The shorp end aiways points toward the smoller number; the open end toword the larger.

Addition and subtraction

If a ~ b, then a+c=b+cand a-c=b-c.

Multiplicotion and division

b, then ac = be and ~ = ~ (if c i 0).

If a

~

11 a < b, then

a+ c < b+ c and

a-c< b-c.

If a < band c > 0,

then ac < be

and %< ~.

If a < band c < 0, then switch the

PROPERTIES OF EQUALITY AND INEQUALITY

u·~=~·a=l

Also,

(b

There are also two (derivative) properties having to do with zero.

Mullipllcotion by zero: a . 0 = 0 . a = O.

Zero product property: If ab = 0 then a = 0 or b = 0 (or both).

Sign

+ b) + c = a + (b + c) a· (b· c) = (a· b) . c

Assoclotive

a . (b

inequality: ac > be and ~ > ~.

Trichotomy: For any two rea) numbers a and b, exactly one of the following is true: a < b. a = b, or a > b.

LINEAR EQUATIONS IN ONE VARIABLE

A linear equation in one variable is an equation that, after simplifying and collecting like terms on each side. will look iike a7 + b = c or like IU" b ("J" + d. Each Side can Involve xs added to real numbers and muihplied by real numbers but not multiplied by other rS

Ex: 1(-~-3)+.c=9-(7-~) is a Iineor equation j" 9 = 3 and 7(X + 4) = 2 and Vi = 5 ore not linear

But

in one vW'iable will always have (a) exactly one real number solution, (b) rlO solutions, or (c) all real numbers as solutions.

FINDING A UNIQUE SOLUTION

J(-t -3) TT=9-(x-

real numbers are solutions.

Add 8x to both sides to get 5x + 8x - 18 = 77 or

13x-18=77.

Add 18 to both sides to get 13x = 77 + 18 or 13x = 95.

5. Divide both sides by the variable's coefficient. Stop if a = U.

Does ~ (2.~~

1. Get rid of fractions outside parentheses.

-

3) + ~ = 9 - (~- ~)? Yes! Hooray.

II~\I :C.ll,., II' j Ifill~I.] ~\ 14f'.'il'.' :JII Use the same procedure as for equalities, except flip the inequality when mliltiplying or dividing by a negative number. Ex: -x > 5 is equivalent to x < -5. -The inequality may have no solution if it reduces to an impossible statement. Ex: :£ + 1 > x + 9 reduces to 1 > 9. -The inequality may have all real numbers as solutions if it reduces to a statement that is always true. Ex: 5 - x 2: 3 - x reduces to 5 ~ 3 and has infintely many solutions. Solutions given the reduced Inequality and the condition:

Multiply through by the LCM of the denominators.

Ex: Multiplyby4 toget3(-t -3) +4x=36-4(x- ~).

2. Simplify using order of operations (PEMDAS).

DETERMINING IF A UNIQUE SOLUTION EXISTS

Use the distributive property and combine like terms on each side. Remember to distribute minus signs.

-

Multiply by 2 to get rid of fractions: 5x - 18 = 77 - 8x.

4. Move variable terms and constant terms to different sides.

Usually, move variables to the side that had the larger variable

Divide by Ij to get x = ~.

6. Check the solution by plugging into the original equation.

~).

Ex: Distribute the left-side parentheses: -~x - 9 + 4x = 36 - 4 (x - ~) .

Combine like terms on the left side: ~x

Any linear equation can be simplified into the form ax = b for some a and b. If a i O. then x = ~ (exactly one solution). If a = 0 but b i 0, then there is no solution. If a = b = 0, then all

coefficient to begin with. Equation should look like a::1: = b.

Linear eCJlUlh'ons

Ex:

Distribute the right-side parentheses; ~x - 9 = 36 - 4x + ~. Combine like terms on the right side: ~x - 9 = ¥ - 4x. 3. Repeat as necessary to get the form ax + b = ex + d.

9 = 36 - 4 (x

-

~)

a>O

-The original equation has no solution if, after legal

transformations, the new equation is false. Ex: 2 = 3 or 3x - 7 = 2 + 3x. -All real numbers are solutions to the original equation if, after legal transformations, the new equation is an identity. Ex: 2x = 3x -.r. or 1 = 1.

a b is true when b < 0 OR {b ~ 0 and a> b} OR {b ~ 0 and a < -b}. Ex: IIOI + II < 7x + 3 is eqUivalent to 7.e + 3 ~ 0 and -7x - 3 < lOx + 1 < 7x + 3. Thus the equations 7x + 3 ~ 0, -7x - 3 < lOx + I. and IOI + I < 7x + 3 must all hold. SolVing the equations. we see that

interval and every boundary point. The point x = 0 is often good to test. Ifs simplest to keep track of your information by graphing everything on the real number line.

Ex: IlOx + 11 < 7x + 3.

Solve the three equntions lOx + I = 7I + 3.

-lOx-l=7x+3. and 7x+3=0 to find

potentinl boundary points. The three points are.

not surprisingly. ~. and - ~. Testing the three boundary pOints and a point from each of the four intervals gives the solution - ~ $ x < ~.

At,

GRAPHING ON THE REAL.: NUMBER LINE

The real number line is a pictorial representation of the real numbers: every number corresponds to a point. Solutions to one· vorioble equations and (especiallyi inequalities may be graphed on the real number line. The idea is to shade in those ports of the line that represent solutions. Origin: A special point representing O. By

x ::; a: Shaded closed ray: everything to the left of and including a.

o

Open (ray or interval): Endpoints not

oa I

Closed (ray or interval): Endpoints included.

GRAPHING SIMPLE STATEMENTS x

= a:

Filled-in dot at a.

o I

a •

;

o I

a

•

I

a+b

both inequalities. Shade the portions that

I

would be shaded by both if graphed

independently.

I

b

Ix - al < b; I.e - "I ::; b:

The distance from a to x is less than (no more than) b; or x is closer than b to a. Plot the interval (open or closed) a - b < x < ,,+ b (or" - b::; x ::; a + b.)

'# a: Everything is shaded except for a.. around which there is an open circle.

o

'­

:

Filled-in circle if the endpoint is included,

a

I:

. I

a+b :

I

b •

I

a<x:O;b

Ix -

of the inequalities must be true.

Ex: Ix - 11 > 4 is really x < -3 OR x > 5.

Equivalently, it is{x, x < -3} U {x : x > 5}.

The graph is the union of the graphs of the

individual inequalities. Shade the portions

that would be shaded by either one (or both)

if graphed independently.

-Endpoints may disappear. Ex: x > 5 OR

6 just means that x > 5. The point 6 is no longer an endpoint.

x ~

b

b

a<x -3} n {x: x < 5}.

The graph is the intersection of the graphs of

Union: Inequalities joined by OR. At least one

Interval: A piece of the line; everything may not be included.

Absolute value is distance. Ix - al = b means

that lhe distance between a and x Is b.

Thraughout. b must be non-neg olive.

Ix - al = b: The distance from a to x is b. Plot

two points: x = a + b, and x = 0 - b.

b

x < a: Shaded open ray: everything to the left of (and not including) a. Open circle around a.

right of a given point. The endpoint may or may not be included. between two endpoints. which mayor

OTHER COMPOUNO INEQUALITIES

oI a¢

represent negative numbers, and points to the

Ray: A half-line; everything to the left or the

I

x > '" Shaded open ray: everything to the right of (and not including) a. An open circle around the point a represents the not· included endpoint.

convention, points to the left of the origin

right of the origin represent positive numbers.

a

I

..

GRAPHING ABSOLUTE VALUE STATEMENTS

< b lopen IntervolJ

al

-4

0

2

I I I , I I I

Ix - al > b; Ix - al ~ b:

Ix -

The distance from a to x is more than (no less than) b; or x is further than b away from a. Plot the double rays (open or closed) x < a - band x> ,,+ b (or x ::; a - b and x ~ a + b).

a-b 0 a I

I

I

b

b

Ix -

41 :0; 2 AND x

.

on

i' 2.5

i

IF YOU CAN DETERMINE ALL POTENTIAL ENDPOINTS ... Plot the points and test all the Intervals one by one by plugging a sample point into the equation. Also test all endpoints to determine if they're included. see exam pie in Soiving Inequalities with Absolule Volue, above.

a+b

I

6

I [ I •

al ~ b

THE CARTESIAN.' PLANE

The Cartesian lor coordinatel plane is a method for giving a nome to each point In the plane on the basis of how for it is from two special perpendicular lines. called axes.

y-axHI

Quadrants: The four regions of the plane cut

by the two axes. By convention, they are numbered counterclock"wise starting with the upper right (see the diagram at right).

Sign (±) of the x- and y-coordinates in the

+

four quadrants: QuadrantIl

TERMINOLOGY OF THE CARTESIAN PLANE x-axis: Usually, the horizontal axis of the coordinate plane. Positive distances are measured to the right; negative. to the left. y-axis: Usually, the vertical axis of the coordinate plane. Positive distances are measured up; negative. down. Origin: {O, 0), the point of intersection of the

ordered pair of coordinates enclosed in

parentheses. The first coordinate is measured along the x-a.xis; the second, along the y-axis. Ex: The point (I. 2) is 1 unit to the right and 2 units up from the

III

Quadrant I

x

• (a,b)

Point: A location on a plane identified by an origin (0,0)

Quadrant ill

.r-~

Quadrant 1V

+

IV

+

I

y---+---+-­

LINES IN THE CARTESIAN PLANE

origin. Occasionally (rarely). the first

coordinate is called the abscissa; the second, the ordinate. Cortes;an planewtth Quadrants I. II. 1ii,1V: point (a, b).

x-axis and the y-axis. CONTINUED ON OTHER SIDE

THE CARTESIAN PLANE (CONTINUED) a measure of how fast the line moves "up" for every bit that it moves "over" Oeft or right). If (a, b) and (c. d) are two points on the line, then the slope is change in y d- b change in x = C - fl,' -Any pair of points on a straight line will give the same slope value. -Horizontollines have slope O. -The slope of a vertical line is undefined; it is "infinitely large," -Lines that go "up right" and "down left" (ending in I and III) have positive slope. -Lines that go "up left" and "down right" (ending in II and IV) have negative slope. -Parallel lines have the same slope. -The slopes of perpendicular lines are negative reciprocals of each other: if two lines of slope ffil and 7112 are perpendicular, then mtm2 = -1 andm2 = -~.

Rough direction of lines with slope m:

1::::

lL

01

t:::

II

FINDING THE EQUATION OF A LINE

l=

Any line in the Cartesian plane represents some linear relationship between x and y values. The relationship always can be expressed as Ax + By = C for some rea] numbers A, B, C. The coordinates of every point on the line will satisfy the equation. m