EVERYTHING YOU NEED TO ACE

MATH IN ONE BIG

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5⅞" x 8" • 512 pages

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978-0-7611-6096-0 • No. 16096

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2016

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EXAMPLE:

Similarly, because -3 is smaller than -2 and also smaller than -1, it is placed to the left of those numbers.

EXAMPLE:

Not only can we place integers on a number line, we can put fractions, decimals, and all other rational numbers on a number line, too:

RATIONAL NUMBERS AND THE NUMBER LINEAll rational numbers can be placed on a NUMBER LINE. A number line is a line that orders and compares numbers. Smaller numbers appear on the left and larger numbers on the right.

EXAMPLE:

Because 2 is larger than 1 and also larger than 0, it is placed to the right of those numbers.

2 3-3 -2 -1 0 1

321-3 -2 -1 0

-2.38 π 3-- 4

1-2 5

321-3 -2 -1 0

321-3 -2 -1 0

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EXAMPLE: - | - 1 6 | = - 1 6 (The absolute value of −16 is 16. Then we apply the negative symbol on the outside of the absolute value bars to get the answer −16.)

A number in front of the absolute value bars means multiplication (like when we use parentheses).

EXAMPLE: 2 | - 4 | (The absolute value of −4 is 4.)2• 4 = 8 (Once you have the value inside

the absolute value bars, you can solve normally.)

Absolute value bars are also grouping symbols, so you must complete the operation inside them first, then take the absolute value.

EXAMPLE: | 5 -3 | = | 2 | =2

Sometimes there are positive or negative symbols outside an absolute value bar. Think: inside, then outside-first take the absolute value, then apply the outside symbol.

EXAMPLE: - | 6 | = - 6(The absolute value of 6 is 6. Then we apply the negative symbol on the outside of the absolute value bars to get the answer −6.)

Multiplication can be shown in a few different ways—not just with x. All of these symbols mean multiply:

2 x 4 = 82 • 4 = 8(2 ) ( 4 ) = 82 ( 4 ) = 8

If you use VARIABLES, you can put variables next to each other or put a number next to a variable to indicate

multiplication, like so:ab = 83x = 1 5VARIABLE: a letter or symbol

is used in place of a quantity we don’t know yet

NOW THIS CHANGES EVERYTHING.

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EXAMPLE: You drive 150 miles in 3 hours. At this rate, how far would you travel in 7 hours?

150 miles- 3 hours = X miles-

7 hours

150•7 = 3 •x1050 = 3x (Divide both sides by 3 so you can get x alone.)350 = x

You’ll travel 350 miles in 7 hours.

Sometimes a proportion stays the same, even in different scenarios-for example, Tim runs 1-2

a mile, and then he drinks 1 cup of water. If Tim runs 1 mile, he needs 2 cups of water. If Tim runs 1.5 miles, he needs 3 cups of water (and so on). The proportion stays the same, and we multiply by the same number in each scenario (in this case, times 2). This is known as the CONSTANT OF PROPORTIONALITY or the CONSTANT OF VARIATION and is closely related to UNIT RATE (or UNIT PRICE).

EXAMPLE: A recipe requires 6 cups of water for 2 pitchers of fruit punch. The same recipe requires 15 cups of water for 5 pitchers of fruit punch. How many cups of water are required to make 1 pitcher of fruit punch?

We set up a proportion:

6 cups- 2 pitcher =

X cups- 1 pitcher or

15 cups- 5 pitcher = X cups-

1 pitcher

By solving for x in both cases, we find out that the answer is always: 3 cups.

We can also see unit rate by using a table. With the data from the table, we can set up a proportion:

EXAMPLE: Daphne often jogs laps at the track. The table below describes how much time she jogs, based on how many laps she finishes. How many minutes does Daphne jog per lap?

Total minutes jogging 28 42Total number of laps 4 6

28 minutes- 4 laps = X minutes-

1 laps or 42 minutes-

6 laps = X minutes- 1 laps

Solving for x, we find out that the answer is: 7 minutes.

Whenever you see “at this rate,”

set up a proportion!

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exaMple of gratuity: at the end of a meal, your server brings the final bill, which is $25. you want to leave a 15% gratuity. how much is the tip in dollars and how should you leave in total?

15% = 0.15

$25 x 0.15 = $3.75

the tip is $3.75.

$25 + 3.75 = $28.75

the total is $28.75.

exaMple of coMMiSSion: my sister got a summer job working at her favorite clothing store at the mall. her boss agrees to pay 12% commission on her total sales. at the end of her first week, her sales total is $3,500.

how much will she earn in commission?

12% = 0.12

$3,500 x 0.12 = $420.00

she earned $420.

Don’t forget: you can also solve these problems by setting up proportions, like this:

12- 100 =

x- 3500

100x = 42,000

x = $420

Again, the more your bill is, the more the gratuity or commission will be—they have a proportional

relationship.

uh-oh...

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EXAMPLE: Joey has $3,000. He deposits it in a bank that offers an annual interest rate of 4%. How long does he need to leave it in the bank in order to earn $600 in interest?

I = $600P = $3,000R = 4% (use .04)T = x

I = P x R x T

$600 = $3,000(.04)T

$600 = $120T

5 = T

So, Joey will earn $600 after 5 years.

EXAMPLE: In order to purchase your first used car, you need to borrow $11,000. Your bank agrees to loan you the money for 5 years if you pay 3.25% interest each year. How much interest will you have paid after the 5 years?

P = $11,000R = 3.25% = 0.0325T = 5 years

I = P x R x T

I = ($11,000) (0.0325) (5)

I = $1,787.50

You’ll have to pay $1,787.50 in interest alone!

With this in mind, what will be the total price of the car?

$11,000 + $1,787.50 = $12,787.50

The car will cost $12,787.50 in total.

(In this case, we know what the interest will be, but we don’t know the length of time. We use x to represent time and fill in all the other information we know.)

(Divide both sides by 120 to get T by itself.)

BANK

HAS IT BEEN 5 YEARS YET?

IT’S BEEN 2 HOURS.

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exaMple: linda and tim are racing around a track. their coach records their times below:

LindanuMber of lapS total MinuteS run

1 ?2 8 minutes

6 24 minutes

TimnuMber of lapS total MinuteS run

1 ?3 15 minutes

4 20 minutes

If each runner’s rates are proportional, how would their coach find out who runs faster? Their coach must complete the table and find out how much time it would take Tim to run 1 lap and how much time it would take Linda to run 1 lap, and then compare them. The coach can find out the missing times with proportions:

Linda:1 - x =

2 - 8

x = 4

So, it takes Linda 4 minutes to run one lap.

Tim:1 - x =

3 - 15

x = 5

So, it takes Tim 5 minutes to run one lap.Linda runs faster than Tim!

caution! We can only use tables if rates are

PROPORTIONAL! Otherwise, there is no ratio or proportion to extrapolate from.

woo-hoo!

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exaMple: 7 (x + 8) =

(x + 8) = 7 (x) + 7 (8) = 7x + 56

the DistributiVe ProPerty oF multiPliCation oVer subtraCtion looks like this a(b - c) = ab - ac. it says that subtracting two numbers inside parentheses, then multiplying that difference times a number outside the parentheses is equal to first multiplying the number outside the parentheses by each of the numbers inside the parentheses and then subtracting the two products.

exaMple: 9(5 - 3) = 9(5) - 9(3) (both expressions equal 18.)

exaMple: 6 (x - 8) =

(x - 8) = 6 (x) - 6 (8) = 6x - 48

Think about catapulting the number outside the

parentheses inside to simplify.

FaCtoring is the reverse of the distributive property. instead of getting rid of parentheses, factoring allows us to include parentheses (because sometimes it ’s simpler to work with an expression that has parentheses).

exaMple: 15y + 12 = 3(5y + 4)

Step 1: ask yourself, “What is the greatest common factor of both terms?” in the above case, the greatest common factor of 15y and 12 is 3. (15y = 3 • 5 • y) (12 = 3 • 4)

Step 2: Divide all terms by the greatest common factor and put the greatest common factor on the outside of the parentheses.

exaMple: 12a + 8 = 6(2a + 3)

the greatest common factor of 12a and 18 is 6. so, we divide all terms by 6 and put it outside of the parentheses.

You can always check your answer by using the Distributive propertY.

Your answer should match the expression you started with!