Regents Review #5

Set NotationUniversal Set = {10,11,12,13,14,15,16,17,18,19,20,21}

A = {14, 21}

B = {12,15,18,21}

A and B are subsets of the Universal Set

A B A intersection B “In Common”

{21}

A B A union B “Combination of both Sets”

{12,14,15,18,21}

B’ or Bc The Complement of B “Not B”

{10,11,13,14,16,17,19,20}

Interval Notation

[ means to include ( means do not include < or >For Example: [12,16) means…. All Real Numbers 12 through 16, including 12 but not 16

Ex: Which interval notation represents the following inequality? -4 < x 6

(1)[-4,6] (2) (-4,6) (3) [-4,6) (4) (-4,6]

(4) (-4,6]

or

Properties

1) Commutative (addition)

2) Commutative (multiplication)

3) Associative (addition)

4) Associative (multiplication)

5) Distributive

A + B = B + A

(A)(B) = (B)(A)

(A + B) + C = A + (B + C)

(A x B) x C = A x (B x C)

A(B + C) = AB + ACor

A(B – C) = AB – AC

Properties

6) Identity (addition)

7) Identity (multiplication)

8) Inverse (addition) (additive inverse)

9) Inverse (multiplication) (multiplicative inverse)

A + 0 = A

(A)(1) = A

A + (-A) = 0

A x (1/A) = 1

Properties

10) Closure

Given a set of numbers, if an operation is performed and the result is part of that set, then the set of numbers is closed under that operation.

Example: Is the set of whole numbers closed under addition?

Yes, any whole number added to another whole number will always result in a whole number

Example: Is the set of whole numbers closed under subtraction?

No, if the difference is taken between 3 and 4, the result is an integer3 – 4 = -1, therefore the set is not closed under subtraction

PropertiesName The Property

1)

2) (q)(y + z) = (y + z)(q)

3) a x 1 = a

4) (9 + 8) + 11 = 9 + (8 + 11)

5) r + 0 = r

6) m(p + q) = mp + mq

7) c + (-c) = 0

8) For which operations are the set of integers closed?

11

x

x Inverse Property of Multiplication

Commutative Property of Multiplication

Identity Property of Multiplication

Associative Property of Addition

Identity Property of Addition

Distributive Property

Inverse Property of Addition

Add, Sub, Mult

Ratios, Rates, ProportionsA paper copy machine takes 4 minutes and 20 seconds to duplicate 520 pages. Find the rate at which the machine duplicates in pages per second.

4 minutes = 4(60) = 240 seconds240 seconds + 20 seconds = 260 seconds

2sec260

pages520 2 pages per second

2 pgs/sec

Ratios, Rates, Proportions

Two inches of rain fell in an hour and a half. At this rate, how many inches of rain, to the nearest hundredth, will fall in 10 minutes?

min90

inches2

hour1.5

inches2assametheis

Set up a proportion to find the number of inches it will rain in 10 minutes. ....222222.

2090

min10min90

2

x

x

xin

0.22 inches of rain will fall in ten minutes

Ratios, Rates, ProportionsMike takes 3 minutes to read an article of 312 words. How many minutes will it take him to read an article of 884 words at the same rate?

Set up a Proportion

words884words312

min3 x

312x = 2652 x = 8.5

8 ½ minutes8 minutes and 30 seconds

Ratios, Rates, ProportionsThe measures of two supplementary angles are in the ratio of 4:11. What is the measure of the larger angle?

Let 4x = the measure of the smaller angle

Let 11x = the measure of the larger angle

4x + 11x = 180

15x = 180

x = 12

The smaller angle = 48 degrees

The larger angle = 132 degrees

The larger angle measures 132○

Ratios, Rates, ProportionsConverting Units of MeasurementYou can convert units of measurement by setting up a proportion

EX: Convert 20 feet to yards

1 yard = 3 feet

ft

ydx

ft

yd

203

1

3x = 20 x = 6.66666….

6 2/3 yards

Ratios, Rates, ProportionsConvert 25 yards to inches

1 yard = 3 feet1 foot = 12 inches

inx

ft

in

ft 75

12

1

75 = x

75 feet

ftx

yd

ft

yd 25

3

1

900 = x

900 inches

ProportionsAnita’s wages vary directly as the number of hours she works. If she can earn $29.80 for working 4 hours, how much will she earn when she works 30 hours?

This problem can be solved using a proportion

hours

x

hours 304

80.29$

4x = 894

x = 223.5 Anita will earn $223.50

PercentsRonald bought a DVD that cost $18.99. His total payment including sales tax was $20.51. Find the sales tax rate to the whole percent?

Tax = % (Tax Rate) x Original Price Tax = 20.51 – 18.99Tax = $1.52

1.52 = N% (18.99)

N% = 1.52/18.99

N% = 0.080042…. 8% tax rate

PercentsThe accompanying circle graph shows expenditures for state and local governments.

1. If the total spending is $50,000, how much money was spent on highways?

2. Approximately how many times the amount of spending on highways is spent on education?

3. Approximately what fraction of the total expenditures are spent on highways and public welfare combined?

1. N = .07(50,000) N = $3,500

2. N = .34(50,000) N = $17,00017,000/3500 = 4.857….About 5 times as much

3. 7% + 14% = 21% 21/100 is about 20/100 About is spent on highways and pw 5

1

TrigonometryMake sure calculator is in degrees! Go to Mode and highlight degree (ENTER)

Finding a SideUse a Proportion Sine, Cosine or Tangent

Finding an AngleUse the inverse Trig Functions (2nd Key on Calc)Sin-1 ,Cos-1 or Tan-1

...186.20

)16(cos21211

16cos

)(21

)(16cos

x

x

x

hyp

adjx

20 units

A ladder leans against a building. The foot of the ladder is 6 feet from the building. The ladder reaches a height of 14 feet on the building. To the nearest foot, find the measure of the angle that the ladder makes with the ground.

...8014.66

)6/14(tan

)(6

)(14tan

1

x

x

adj

oppx

67◦

StatisticsQuantitative Data is data that is numerical, meaning it can be counted or measuredEx: height of a flagpole, weight of a backpack

Qualitative Data is data that is not numerical, meaning it can be observedEx: type of toppings on a pizza, favorite ice cream flavor

Univariate refers to single variable dataEx: the number of pets each student owns

Bivariate refers to two variable dataEx: A person’s shoe size compared to their height

StatisticsCorrelation vs. CausationA correlation is a relationship between two variablesEx:

Not every correlation shows causation (one variable causes the other)Which graph above shows causation?

Money Spent on potted plants

Number of potted plants purchased

Number of swimsuits sold

Number of winter coats sold

The first graph because the more money you spend on plotted plants, the more plants you will have

The second graph shows a correlation but the number of swimsuits sold does not cause the number of winter coats sold. The temperature outside and seasons cause the sales of the items.

StatisticsA population is a group that you want information about.A sample is part of a population that is used to make estimates about the populationIn a random sample, each member has an equal chance of being selected, and the sample is representative of the entire populationA biased sample favors one or more parts of the population over othersEx: You want to conduct a survey to find out what type of music people listen toDetermine which of these scenarios is biasedA.You ask every fifth person leaving a Taylor Swift concert about the type of music they listen toB.You ask every fifth person leaving the local mall about the type of music they listen to

THE REGENTS IS WEDNESDAY, JUNE 12th

STUDY! STUDY! STUDY!

END OF REGENTS REVIEWSNow it’s your turn to review on your own! Using the information presented today and your review packet, complete the practice problems in the packet.