Algebra Honors ( Period 4 & 7)

Inequalities Involving Absolute Value 5-5

The inequality │x│< 3 means that the distance between x and 0 is less than 3

Graph:

so x > -3 and x <  3

The set builder notation or solution set is {x │ -3 < x < 3}

When solving absolute value inequalities there are two cases to consider:

Case 1 The expression inside the absolute value symbols is nonnegative

Case 2  The expression inside the absolute value symbol is negative

The solution is the intersection of these two cases.

When the absolute value inequality is less than… I think of “less thAND” … it is the intersection of two parts. It is the “YO”  we talked about in class

│m + 2 │ < 11

Rewrite  │m + 2 │ < 11       for both the above cases

m + 2 < 11                   and            -(m+2)  < 11

m  < ; 9                   this one  becomes m + 2 >; -11

m < ; 9     and     m >  -13

or think m + 2 < 11   AND  -11 < m + 2

The solution set is { m │ -13 < m <; 9}

│y -1│ <  -2    WAIT… this can NEVER be TRUE. There is no solution The solution set is the empty set.  {  }  or

When the absolute value inequality is a greater than… I think of  “greatOR”  it is the union
of two parts..  It is the “DORKY DANCER we talked about in class… going one way and then the other.
│x│ > 3 means that the distance between x and 0 is greater than 3…

Graph:

so x <  -3 OR  x >; 3 The solution set is { x │ x < -3 or x >  3}

We must consider two cases always
Case 1 The expression inside the absolute value symbols is nonnegative
Case 2 The expression inside the absolute value symbols is negative

Solve      │3n + 6│≥ 12

Case 1
3n + 6 is non negative

3n + 6 ≥ 12              n ≥ 2        That was easy

Case 2
3n + 6 is  negative

The book sets it up  -(3n + 6) ≥ 12

That would mean    3n + 6 ≤ -12

n  ≤ -6     so

 n  ≥2   OR   n  ≤ -6

The solutions set is { n │ n  ≥2   OR   n  ≤ -6}

Graph   

Algebra Honors ( Periods 4 & 7)

Multiplication Properties of Exponents 7-1

Monomial: A number, variable, or product of a number and variables with NON-NEGATIVE INTEGER EXPONENTS.

Negative integer exponents mean that a variable is in the denominator. Numbers can be in the denominator.

Integer exponents mean that you can’t have variables with square roots or other roots (these would be fractional exponents)

Therefore, monomials are ONE TERM and terms are separated by addition or subtraction NOT multiplication or division.

A constant is a monomial that is a real number (it can be negative, positive, a fraction/decimal or in a radical sign)

Linear expressions have each variable to the 1 power. Nonlinear expressions have either one variable to a power of 2 or more OR multiple variables attached together.

The base of an exponent is the number or variable at the bottom.

The exponent is the little superscript number at the top.

This is called exponential form or a power of the base raised to the specific exponent.

Expanded form is when you show the repeated multiplication.

Standard or simplified form is the number answer (simplified for a variable is the same as the exponential)

3⁴ is exponential form or we say it’s 3 raised to the power of 4

3●3●3●3 is expanded form

81 is standard form

ODD/EVEN RULES WITH POWERS AND NEGATIVE:

An odd number of negatives = negative

An even number of negatives = positive

So

An odd power with a negative integer in a ( ) = negative

An even power with a negative integer in a ( ) = positive

BUT

An EVEN power with a negative power WITHOUT ( ) = NEGATIVE…there’s only ONE negative here and that’s odd…I call this negative the ZAPPER because it zaps the answer at the very end of the simplifying

An odd power without ( ) = still negative…there’s only ONE negative here and that’s odd

EXAMPLES:

EVALUATING WITH POWERS:

If you are given a variable to a power, you simply plug and chug the value given for the variable.

USUALLY YOU SHOULD PLACE THE VARIABLE IN ( ) WHEN PLUGGING IN:

a² – b⁴ if a = 2 and b =3

(2) ² – (3) ⁴ = 4 - 81 = -77

VS this case where you’re doing an operation INSIDE the (  ) FIRST, then doing the power:

(a – b) ⁴ = (2 – 3) ⁴ = (-1)⁴ = 1

Algebra ( Period 1)

Multiplication Properties of Exponents 7-1

Monomial: A number, variable, or product of a number and variables with NON-NEGATIVE INTEGER EXPONENTS.

Negative integer exponents mean that a variable is in the denominator. Numbers can be in the denominator.

Integer exponents mean that you can’t have variables with square roots or other roots (these would be fractional exponents)

Therefore, monomials are ONE TERM and terms are separated by addition or subtraction NOT multiplication or division.

A constant is a monomial that is a real number (it can be negative, positive, a fraction/decimal or in a radical sign)

Linear expressions have each variable to the 1 power. Nonlinear expressions have either one variable to a power of 2 or more OR multiple variables attached together.

The base of an exponent is the number or variable at the bottom.

The exponent is the little superscript number at the top.

This is called exponential form or a power of the base raised to the specific exponent.

Expanded form is when you show the repeated multiplication.

Standard or simplified form is the number answer (simplified for a variable is the same as the exponential)

3⁴ is exponential form or we say it’s 3 raised to the power of 4

3●3●3●3 is expanded form

81 is standard form

ODD/EVEN RULES WITH POWERS AND NEGATIVE:

An odd number of negatives = negative

An even number of negatives = positive

So

An odd power with a negative integer in a ( ) = negative

An even power with a negative integer in a ( ) = positive

BUT

An EVEN power with a negative power WITHOUT ( ) = NEGATIVE…there’s only ONE negative here and that’s odd…I call this negative the ZAPPER because it zaps the answer at the very end of the simplifying

An odd power without ( ) = still negative…there’s only ONE negative here and that’s odd

EXAMPLES:

EVALUATING WITH POWERS:

If you are given a variable to a power, you simply plug and chug the value given for the variable.

USUALLY YOU SHOULD PLACE THE VARIABLE IN ( ) WHEN PLUGGING IN:

a² – b⁴ if a = 2 and b =3

(2) ² – (3) ⁴ = 4 - 81 = -77

VS this case where you’re doing an operation INSIDE the (  ) FIRST, then doing the power:

(a – b) ⁴ = (2 – 3) ⁴ = (-1)⁴ = 1

Math 6H ( Period 5)

Equivalent Ratios in the Real World 1-6

There are 2 ways to determine if 2 ratios are equivalent:

1. Restate them both as unit rates. If they’re the same unit rates, they’re equivalent.

2. Use equivalent fractions if finding the unit rate is difficult (let’s say you need to go several decimal places in the division and don’t have a calculator)

Example using both methods:

You know that one store will charge you $2 for 10 photo prints and another has a sign offering 30 prints for $6.

Are these 2 stores charging EQUIVALENT AMOUNTS?

First way: Unit Rates

The unit rates are the same:

You can either find the prints per dollar or the unit price per print.

Both will show that the stores are equivalent.

Prints per dollar:

10 prints/$2 = 5 prints/$1 and 30 prints/$6 = 5 prints/$1 as well

Unit price:

$2/10 prints = $.20/print and $6/30 prints = $.20/print as well

Second way: Equivalent Fractions

The equivalent fraction approach asks you to see if you can find a common factor that the numerator and denominator were both either multiplied by or divided by to get to each other:

Can I get from 10/2 to 30/6 by multiplied both the numerator and denominator by the same number?

Yes I can!

I can multiply by 3 (scale forward by 3)

When I do this, it’s helpful to show a curved arrow with the x 3 on each arrow:

So by using equivalent fractions, the 2 stores are offering the same deal.

Algebra (Period 1)

 LITERAL EQUATIONS AND DIMENSIONAL ANALYSIS 2-8 

We’ve already talked about FORMulas being equations with KNOWN RELATIONSHIPS

The Common Core lingo for a formula is LITERAL EQUATION

Many times, you’ll need to solve for one of the variables.

Example: d = rt
Sometimes, you want to solve for distance, but other times you need the time or the rate.
You can manipulate the variables by balancing the equation until you solve for the wanted variable.
In the formula, d = rt, to solve for rate, divide both sides by t
To solve for time, divide both sides by r
Don't think of this as 3 different equations!
Just learn the main one and use that to solve for what you need!

Example: Solve for l:
P = 2l + 2w
Subtract 2w for both sides: P - 2w = 2l
Divide by 2 on each side:
P - 2w = l
2

When the variable is in the denominator, it's usually easiest to use cross products:
Solve for x:

3z = 4
x      y

Cross products:
3yz = 4x

Divide both sides by 4:
x = 3yz
       4

If you have a fraction, you can simply use the multiplicative inverse:
Solve for b:
3ab = 7c

4          

Either use cross products or multiply each side by 4/3a:
b = 7c (4/3a) = 28c/3a

Why can't you use cross products for the following?
Solve for f:
d = 2a + 2b

      c f  

Algebra ( Period 1)

 ABSOLUTE VALUE EQUATIONS 2-5

 When you plug into an expression with absolute value, the absolute value signs function as parentheses in Order of Operations.

So make sure you simplify INSIDE before turning that POSITIVE!

Generally, you solve these the same way you solve regular equations.
 Make sure you balance equally on both sides!

 Follow the steps of a 2-step equation.
 1. Add the opposite (you can subtract as well)
 2. Multiply by the reciprocal (you can divide as well)

 THE DIFFERENCE?
 YOU HAVE 2 POSSIBLE ANSWERS! (+ and -)

 EXAMPLE: 2 IxI + 1 = 15
 2 IxI  + 1 - 1 = 15 - 1
 2 IxI  = 14
 1/2 ( 2 IxI  ) = 1/2 (14)
  IxI  = 7
 x = {-7, 7}

 REMEMBER, IF YOU AFTER YOU GET THE ABSOLUTE VALUE ALONE ON ONE SIDE, YOU FIND THAT THE CONSTANT ON THE OTHER SIDE IS NEGATIVE, 

THE ANSWER IS THE NULL SET!

 EXAMPLE: 2 IxI + 16 = 15
 2 IxI + 1 - 1 = 15 - 16
 2 IxI= -1
 1/2 ( 2 IxI ) = 1/2 (-1)
  IxI = -1/2
 NOT POSSIBLE! So the answer is the null set

Algebra ( Period 1)

SOLVING EQUATIONS WITH VARIABLE ON BOTH SIDES 2-4 

COLLECTING TERMS FIRST:
Sometimes, you will have to COLLECT LIKE TERMS ON THE SAME SIDE OF THE EQUATION before balancing:

8y + 12 – (-2y) = -6

10y + 12 = -6

10y = -18

y = -18/10 =  -9/5

TWO STEPS WITH DISTRIBUTIVE PROPERTY
Usually, you want to do DISTRIBUTE FIRST!
UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED
OUT OF BOTH SIDES PERFECTLY!!!!

EXAMPLE:
5y - 2(2y + 8) = 16
5y - 4y - 16 = 16 [distribute]
y - 16 = 16 [collect like terms]
y = 32 [solve by adding 16 to both sides]

EXAMPLE WHEN YOU DON'T NEED TO DISTRIBUTE FIRST:
-3(4 + 3x) = -9
4 + 3x = 3 [Don't distribute! Divide by -3. The -3 goes into both sides perfectly!)
3x = -1 [Subtract 4 from both sides]
x = -1/3 [Divide both sides by 3]

REVIEWED: IDENTITY OR NO POSSIBLE SOLUTION EQUATIONS:

An identity equation is where ANY NUMBER can be substituted for the VARIABLE, the equation will be TRUE. What will happen is that while you’re balancing the equations, you will ultimately end up with the SAME EXACT EXPRESSION ON EACH SIDE of the equation.

 

You can keep going, but as soon as you have the same thing on both sides, you know you have an IDENTITY

A no possible solution equation is one where no matter what number you substitute into the equation, the equation will be FALSE. What will happen is that while you’re balancing the equations, you will ultimately end up with one number will equal a DIFFERENT number (which can never be true).

 

NOTICE SOMETHING ELSE ABOUT SOLVING EQUATIONS IN GENERAL:

WHENEVER YOU HAVE THE SAME EXACT TERM WITH THE SAME SIGN ON DIFFERENT SIDES OF THE EQUATION, YOU CAN SIMPLY CROSS THEM OUT BECAUSE  WHEN YOU USE THE ADDITIVE INVERSE PROPERTY ON BOTH SIDES TO BALANCE, BOTH TERMS WILL DROP OUT!

VARIABLES ON BOTH SIDES:

Simplify each side of the equation first.
Then use the ADDITIVE INVERSE PROPERTY to move variables to the other side of the equation so that all variables are on the same side.
Usually, we try to move the smaller coefficient to the larger because sometimes that avoids negative coefficients,

BUT that is not always the case, and you may move to whatever side you choose!
EXAMPLE:
3y - 10 - y = -10y + 12
2y - 10 = -10y + 12
+10y +10y
12y - 10 = 12
+ 10 +10
12y = 22
12     12
y = 11/6

FINDING THE VALUE OF AN UNKNOWN SO THAT 2 PERIMETERS OR AREAS ARE THE SAME:

This type of problem is a perfect example of using the Distributive Property with variable on both sides of an equation.

Example 4 on p. 99

You have 2 rectangles whose areas are the same. One rectangle has sides of x and 10 cm. and the other rectangle has sides of x + 3 and 6 cm.

Set up the following equation and solve:

10x = 6(x + 3)

10x = 6x + 18

4x = 18

x = 18/4 = 9/2 cm

CHECK THAT THE RECTANGLES DO HAVE THE SAME AREA IF x = 9/2 OR 4.5 cm

First rectangle: Sides are 4.5 and 10 so A = (4.5)(10) = 45 cm2

Second rectangle: Sides are 4.5 + 3 or 7.5 cm and 6 cm. so A = (7.5)(6) = 45 cm2

You have 2 rectangles whose perimeters are the same. One rectangle has sides of x and 6 cm. and the other rectangle has sides of 2x + 2 and x cm.

Set up the following equation and solve:

2x + 2(6) = 2(2x + 2) + 2x

2x + 12 = 4x + 4 + 2x

2x + 12 = 6x + 4

12 = 4x + 4

8 = 4x

x = 2

CHECK THAT THE RECTANGLES DO HAVE THE SAME PERIMETER IF x = 2 cm

First rectangle: Sides are 2 and 6 so P = 2(2) + 2(6) = 16 cm

Second rectangle: Sides are 2 and 2(2) + 2 = 6 so P = 2(2) + 2(6) = 16

Algebra ( Period 1)

SOLVING MULTI-STEP EQUATIONS 2-3

You’re doing Order of Operations working BACKWARDS

TWO STEP EQUATIONS
1. Use the ADDITION/SUBTRACTION PROPERTIES OF EQUALITY first
(get rid of addition or subtraction)
2. Use the MULTIPLICATION/DIVISION PROPERTIES OF EQUALITY second
(get rid of multiplication/division)

When the coefficient is a variable, use the multiplicative inverse property and multiply by the reciprocal.

CONSECUTIVE INTEGER PROBLEMS:

Consecutive integers are integers that are one after another like 1, 2, 3, etc,

So if n is the first consecutive integer, the next one would be n + 1 and the 2nd one would be n + 2

Consecutive EVEN integers are 2 apart beginning with an EVEN integer like 2, 4, 6 etc.

So if n is the first consecutive EVEN integer, the next one would be n + 2 and the 2nd one would be n + 4

Consecutive ODD integers are also 2 apart beginning with an ODD integer like 1, 3, 5 etc.

So if n is the first consecutive ODD integer, the next one would be n + 2 and the 2nd one would be n + 4

You can write equations with consecutive integers.

For example, the sum of 3 consecutive ODD integers is -51, find all 3 integers:

n + (n + 2) + (n + 4) = -51

3n + 6 = -51

3n = -57

n = -19

n + 2 = -19 + 2 = -17

n + 4 = -19 + 4 = -15

CHECK TO THE ORIGINAL WORD PROBLEM:

-19 + -17 + -15 = -51

Algebra Honors ( periods 4 & 7)

Percent of Change 2-7

Percent of change is the ratio of the change over the original amount.

It can be an increase or a decrease.

Sometimes you know the percent of increase or decrease and you want to find either the original amount or the new amount.

Simply plug in the given information and solve for the missing item.

REAL LIFE APPLICATIONS OF PERCENTS OF CHANGE:

Percent of Decrease: Sales Tax or Discount

Sometimes a store will not tell you the percent off merchandise is…Instead, they’ll tell you the amount off.

You can find the discount % by looking at it as a percent of change.

Example:

A laptop is $100 off of the original price of $700. What is the discount percent?

The amount off is the change.

100/700 ≈ .143 or 14.3%

Percent of Increase: Markups

To make a profit, stores must mark up what they manufacture or buy to their customers.

That markup is an increase.

Example:

A company makes something that costs them $500 to produce. They mark it up $200 and sell it. What is their markup percent?

The $200 is the increase.

200/500 = .4 or 40% markup 

5-1 Solving Inequalities by Adding or Subtracting

Graphing an inequality - open dot is < or >
Closed dot mean less than or EQUAL or greater than or EQUAL
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
Different from equations: Inequalities have many answers (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution

Graphing an equation's solution is easy
1) Say you found out that y = 5, you would just put a dot on 5 on the number line
2) But now you have the y ≥ 5
You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions
3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!

TRANSLATING WORDS:
Some key words to know:

AT LEAST means greater than or equal

NO LESS THAN also means greater than or equal

AT MOST means less than or equal

NO MORE THAN also means less than or equal

I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes,
but I'm hoping for even less!

Solving Inequalities with adding and subtracting
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get:
y > 5
Your answer is infinite!

Any real number bigger than 5 will work!

Always finish with the variable on the left.

If you don’t, you may misunderstand the answer and graph it in the opposite position.

5 > y is not the same as y > 5!

5 > y means that y < 5!

Check with whatever solution is easiest in the solution set!

Set builder notation:

Get familiar with the following notation:

{x I x ≥ 5} which is read: “x SUCH THAT x is greater than or equal to 5.

Algebra ( Period 1)

 SOLVING ONE-STEP EQUATIONS 2-2

REVIEW: EQUATION BALANCING PROPERTIES OF EQUALITY:

There are 4 of these.

Whatever YOU DO TO BALANCE an equation, 

that operation is the property of equality that was used.

 If you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally.

If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally.

If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3.

If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3.

SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY

CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?

Since we say we never subtract and we really never divide, it's those 2.

GOING BACK TO OUR PREVIOUS EXAMPLES:

If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.

If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY. 

(We always multiply by the MULTIPLICATIVE INVERSE).

REVIEW OF SIMPLE EQUATIONS!
GOAL? Determine the value of the variable
HOW? Isolate the variable (get it alone on one side of equation)
WHAT DO I DO? Use inverse (opposite) operations to "get rid" of everything on the side with the variable
WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED?
Always focus on the variable(s) first!

IDENTITY PROPERTIES AND INVERSE PROPERTIES 

are also used to justify solving equations!

When you have a one-step equation such as x + 5 = 12, you ADD -5 (or just subtract 5) from each side equally. The reason you chose -5 is that it was the ADDITIVE INVERSE of 5.  The reason the +5 then "disappears" is due to the IDENTITY PROPERTY OF ADDITION. Since +5 + (-5) = 0, it's not necessary to bring down the 0 in the equation.

JUSTIFYING A SIMPLE ONE-STEP:

                                                   x + 5 = 12     GIVEN

                                                      - 5   -5     Subtraction Prop =

                                                        0            Additive Inverse Prop

                                                   x       =  7     Identity of Addition

FORMAL CHECK:
1. Rewrite original equation
2. Substitute your solution and question mark over the equal sign
3. Do the math and check it!

EXAMPLE FROM ABOVE:

                                      1.  Rewrite:                                  x + 5 = 12

                                                                                                    ?  

                                      2. Substitute your solution:        7 + 5 = 12

                                      3.  Do the math!                               12 = 12 √

Quick review of a couple of specific types of one-steps:

Do you remember from 7th grade how you balance an equation that has a fractional coefficient?

Multiply by the reciprocal (our BFF, the multiplicative inverse property ;)

Another special type of one-step equation are those where the VARIABLE IS NEGATIVE.

Remember: You’re solving for the POSITIVE VARIABLE.

There are a couple of ways to do this.

DID YOU KNOW THAT YOU CAN MOVE A NEGATIVE SIGN

IN 3 DIFFERENT PLACES ON ANY FRACTION????

 

So if you see a negative sign on a variable in a fraction, just MOVE IT to the number!

 

If you don’t move the negative sign first, BE CAREFUL because you’ll need to either multiply or divide by -1 at the very end to find POSITIVE y:

If there’s a negative on a variable and it’s not part of a fraction, you can multiply or divide both sides by

-1 AS I JUST SHOWED ABOVE

or you can just reason out the answer:

Algebra ( Period 1)

CHAPTER 2-1: STRATEGIES FOR TRANSLATING WORDS TO ALGEBRAIC EQUATIONS

Algebraic expressions just are the ones that have variables

Numeric expressions have only numbers

Equations must have an = sign while expressions do not

STRATEGY #1:TRANSLATE WORD BY WORD

You did this in Chapter 1!

Always try this first.

Just be careful of less THAN and subtracted FROM because these are switched from the order that you read/say them:

A number less THAN 12 is 12 – n but if you say a number less 12, this would be n – 12

12 subtracted FROM a number is n – 12, but 12 subtract a number would be 12 – n

The only other translation to be careful of is when you multiply a SUM or DIFFERENCE by a number or variable:

12 times the SUM of a number and 5 is 12(n + 5), but the sum of 12 times a number and 5 would  be 12n + 5

12 times the DIFFERENCE of a number and 5 is 12(n – 5), but the difference of 12 times a number and 5 would be 12n – 5

If you have 2 or more unknowns, use different variables:

The difference of a number and ANOTHER number would be x - y

STRATEGY #2: DRAWING A PICTURE

(When in doubt, draw it out! ;)
I have 5 times the number of quarters as I have dimes.
I translate to: 5Q = D
I check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG! I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20
D = 4
Does this make sense? YES! I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes
to make that pile the same height as the number of quarters!

STRATEGY #3: MAKE A T-CHART
To translate known relationships to algebra (known as dimensional analysis), it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with 3 lines of numbers and look for the relationship between the 2 columns.
Then, you use that mathematical relationship with a variable.

EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:
d days number of hours
1                   24
2                   48
3                   72
Now look at the relationship between the left column and the right column.
You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d
d days number of hours
1                    24
2                   48
3                   72
d                  24d

EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:
h hours number of days
24             1
48             2
72             3
(Why did I start with 24 and not 1 hour this time?)
Now look at the relationship between the left column and the right column.
You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h
h hours number of days
24           1
48           2
72           3
h        h/24

WHEN IT’S AN EQUATION AND NOT AN EXPRESSION….

You use the same strategies, but you’ll have an = sign and then you can solve for the unknown variable if there is only 1 variable:

10 less than the product of 5 and a number is 25:

5n – 10 = 25

n = 7

FORMulas

Equations that represent KNOWN RELATIONSHIPS are called formulas because there is a specific format that must be used that never changes.

For example, d = rt is a FORMula…it’s also an equation, but it has a more specific name because this is a KNOWN RELATIONSHIP in the real world.

You can translate words for formulas:

Distance is the product of the rate of speed and the time traveled.

TRANSLATING ALGEBRAIC EQUATIONS TO WORDS:

Going back the other way, you’ll have choices in the words you can use to represent the same equation.

2n = 40

You can say:

2 times a number is 40

Double a number is 40

Twice a number is 40

The product of a number and 2 is 40

The product of 2 and a number is 40 (multiplication is commutative!)

TRANSLATING GIVEN INFORMATION INTO YOUR OWN WORD PROBLEM…YOU’RE WRITING A STORY!

I’ve included example 5 from your book below (p. 77)

Maxine’s time every time she drove = t

Tia’s time every time she drove = t + 4

Given: 2t + (t + 4) = 28

Write a word problem or story

Maxine and Tia took a trip together and took turns driving.

Maxine took 2 turns driving and Tia only 1, but when Tia drove, she drove 4 more hours than Maxine drove on each of her turns.

The trip took 28 hours.

How long did each of them drive?

Algebra Honors ( Periods 4 & 7)

Ratios & Proportions 2-6
A ratio is a comparison of two things.

Comparisons:

Say you are a dog walker and you want the ratio of large dogs to small dogs to remain at 8 small dogs to 2 large dogs for your business.

3 ways to write a ratio:

8 small dogs to 2 large dogs or
8 small dogs : 2 large dogs or
8 small dogs /2 large dogs

 You can also reverse the order and put the large dogs first.

Just as with fractions, since a ratio functions like a fraction, you ALWAYS SIMPLIFY the ratio:

4 small dogs to 1 large dog or
4 small dogs : 1 large dog or
4 small dogs /21 large dog

Proportions:

A proportion is 2 EQUAL ratios.

You can use CROSS PRODUCTS or SIMPLIFYING to determine if two ratios are equivalent.

 Means:

The means in a proportion are the two middle terms if written with a : or the denominator of the 1st term and the numerator of the 2nd term.

Extremes:

The extremes in a proportion are the two outside terms if written with a : or the numerator of the 1st term and the denominator of the 2nd term.

CROSS PRODUCTS PROPERTY:

The product of the MEANS is always equal to the product of the EXTREMES

Rate:

A ratio with 2 DIFFERENT units of measure like miles per gallon

Unit Rate:

A rate with a denominator of 1 unit that is found by dividing the numerator by the denominator of a rate

Scale Rate:

A rate that is used to make a model bigger or smaller of an actual sized item that is usually too big to draw or use…Example: a building sketch or a map.

Remember: It’s all about the labels! After setting up the proportion, you have your choice of 3 methods:

1)  equivalent fraction method (doesn’t always work- the numbers must be compatible)

3/5 = y/15   y = 9 because 5(3) = 15 must multiply 3(3) to get numerator in second fraction

2)  balancing equation method  (always works) multiply by the multiplicative inverse

3/5 = y/15  multiply both sides by 15 ( the multiplicative inverse of 1/15

3)  cross products method (this is the only time that name is accurate) Multiply the “corners” making an X. Same example but this time you would set up the cross product equation or

 5y = 3(15). Don’t be too quick to multiply 3(15).  Divide by 5 first. It may simplify.   so   y = 9

Algebra Honors ( Periods 4 & 7)

Interpreting Graphs of Functions 1-8

There are several key features of different functions that help you identify what type of function it is and also interpret how it’s going to act.
 LINEAR OR NONLINEAR:

If a graph has a curve, it’s nonlinear. If it’s a straight line, it’s linear.
You can see this easily when it’s graphed.
On the graphing calculator, you’ll discover that if the x power is 1, it’s a line (linear)
When we change the x power to 2 or 3 or higher, it’s nonlinear.

INTERCEPTS:

These are points where the graph intersects the x or y axis.
x-intercept: where the graph intersects the x axis…the coordinate would be of the form (x, 0)
y-intercept: where the graph intersects the y axis…the coordinate would be of the form (0, y)

If the graph goes through the ORIGIN, both intercepts would be (0, 0)
A horizontal line would not have an x-intercept UNLESS the line is the x axis (the y value would always be 0 or y = 0)
A vertical line would not have an y-intercept UNLESS the line is the y axis (the x value would always be 0 or x = 0)

IS IT POSSIBLE FOR A GRAPH TO HAVE MORE THAN ONE X OR Y INTERCEPT???
If it’s a line (linear), NO. A line can’t come back around again.
However, if a graph has a curve (nonlinear), YES it can…it can intersect say the x axis and then curve around and intersect the x axis again.

MOVING THE Y-INTERCEPTS UP OR DOWN:
Adding a POSITIVE constant at the end of a function moves the graph UP and adding a NEGATIVE constant moves it DOWN.
y = x goes through the origin           y = x + 2 moves it up 2            y = x – 3 moves it down 3

 SLOPE:
When the coefficient of x is POSITIVE, it looks like you’re going up the mountain.
When the coefficient of x is +1, the slope going up is a 45 degree angle.
As the coefficient of x gets greater than 1, the steepness of the line INCREASES.
As the coefficient goes into the range between 0 and 1 (a fraction or decimal), the slope starts to level out.
When the coefficient is negative the line switches direction and looks like you’re going down the mountain.

 SYMMETRY:
Just as you learned in geometry, line symmetry means that one half of a graph looks like
the other half along some vertical line.
We’ll see that y = x² is symmetrical along the y axis.
If we move the graph to the right so it’s all in the first quadrant and look at it as the trajectory of a ball, the symmetry could be interpreted as it took the same amount of time for the ball to rise up in the air as it did to come down.

POSITIVE AND NEGATIVE PARTS OF A GRAPH: This is pretty obvious!
A function is positive where the graph is ABOVE the x axis…the RANGE is positive above the x axis.
A function is negative where the graph is BELOW the x axis…the RANGE is negative below the x axis.

INCREASING AND DECREASING PARTS OF A GRAPH:
When the graph is going UP, the function is INCREASING.
When the graph is going DOWN, the function is DECREASING.
REMEMBER WE’RE LOOKING AT THE GRAPH FROM LEFT TO RIGHT!

EXTREMA:
Extrema comes from the word extreme so we’re talking about extreme values of a function…either high range values or low range values (y values)
There are two kinds of extrema: minimums and maximums
A minimum means that there are no other y values (range values) lower anywhere in the function
A maximum means that there are no other y values (range values) higher anywhere in the function

A RELATIVE minimum means there are no other y values lower NEARBY (but there may be lower points in another region of the function)
A RELATIVE maximum means there are no other y values higher NEARBY (but there may be higher points in another region of the function)

 END BEHAVIOR:

Every graph has an “end” on both sides of the domain values (x values)
As x gets smaller towards negative infinity (meaning you’re going to the left on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

As x gets larger towards positive infinity (meaning you’re going to the right on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

Generally, we summarize end behavior by comparing what x (the domain) is doing to what y (the range or function value) is doing at the same time:

 As x decreases—>y also decreases OR  y increases

As x increasesà y also increases OR y decreases

DOMAIN AND RANGE ON A GRAPH:
You already know that the x values are the domain and y values are the range.
On a graph, we look at all the possible x values to determine if the domain is all real numbers or if it’s limited in some way.
We do the same thing with the range.

For example: f(x) = x²
This is a U shaped graph that only goes up from the origin so the range is limited to y ≥ 0
The domain would be all real numbers because you can square any number and, looking at the graph, you can see that eventually the graph will go to both negative and positive infinity to the left and to the right.

REAL WORLD INTERPRETATIONS OF GRAPHS:

Sales of a company:
By looking at a graph of sales over time, you can analyze how the company is doing.
The increasing parts of the graph mean that the company is growing while the reverse is also true.
If you see a flat part of the graph, that part would show the company is staying the same.

Between an increasing and decreasing part of sales would be a relative max to sales…meaning for some reason the company is in decline.

Between a decreasing and increasing part of sales would be a relative min to sales…meaning for some reason the company is doing well again.

The end behavior over time TO THE RIGHT would predict the success of the company in the future. (to the left would be the actual history of sales)

Algebra (Period 1)

Interpreting Graphs of Functions 1-8

There are several key features of different functions that help you identify what type of function it is and also interpret how it’s going to act.
 LINEAR OR NONLINEAR:

If a graph has a curve, it’s nonlinear. If it’s a straight line, it’s linear.
You can see this easily when it’s graphed.
On the graphing calculator, you’ll discover that if the x power is 1, it’s a line (linear)
When we change the x power to 2 or 3 or higher, it’s nonlinear.

INTERCEPTS:

These are points where the graph intersects the x or y axis.
x-intercept: where the graph intersects the x axis…the coordinate would be of the form (x, 0)
y-intercept: where the graph intersects the y axis…the coordinate would be of the form (0, y)

If the graph goes through the ORIGIN, both intercepts would be (0, 0)
A horizontal line would not have an x-intercept UNLESS the line is the x axis (the y value would always be 0 or y = 0)
A vertical line would not have an y-intercept UNLESS the line is the y axis (the x value would always be 0 or x = 0)

IS IT POSSIBLE FOR A GRAPH TO HAVE MORE THAN ONE X OR Y INTERCEPT???
If it’s a line (linear), NO. A line can’t come back around again.
However, if a graph has a curve (nonlinear), YES it can…it can intersect say the x axis and then curve around and intersect the x axis again.

MOVING THE Y-INTERCEPTS UP OR DOWN:
Adding a POSITIVE constant at the end of a function moves the graph UP and adding a NEGATIVE constant moves it DOWN.
y = x goes through the origin           y = x + 2 moves it up 2            y = x – 3 moves it down 3

 SLOPE:
When the coefficient of x is POSITIVE, it looks like you’re going up the mountain.
When the coefficient of x is +1, the slope going up is a 45 degree angle.
As the coefficient of x gets greater than 1, the steepness of the line INCREASES.
As the coefficient goes into the range between 0 and 1 (a fraction or decimal), the slope starts to level out.
When the coefficient is negative the line switches direction and looks like you’re going down the mountain.

 SYMMETRY:
Just as you learned in geometry, line symmetry means that one half of a graph looks like
the other half along some vertical line.
We’ll see that y = x² is symmetrical along the y axis.
If we move the graph to the right so it’s all in the first quadrant and look at it as the trajectory of a ball, the symmetry could be interpreted as it took the same amount of time for the ball to rise up in the air as it did to come down.

POSITIVE AND NEGATIVE PARTS OF A GRAPH: This is pretty obvious!
A function is positive where the graph is ABOVE the x axis…the RANGE is positive above the x axis.
A function is negative where the graph is BELOW the x axis…the RANGE is negative below the x axis.

INCREASING AND DECREASING PARTS OF A GRAPH:
When the graph is going UP, the function is INCREASING.
When the graph is going DOWN, the function is DECREASING.
REMEMBER WE’RE LOOKING AT THE GRAPH FROM LEFT TO RIGHT!

EXTREMA:
Extrema comes from the word extreme so we’re talking about extreme values of a function…either high range values or low range values (y values)
There are two kinds of extrema: minimums and maximums
A minimum means that there are no other y values (range values) lower anywhere in the function
A maximum means that there are no other y values (range values) higher anywhere in the function

A RELATIVE minimum means there are no other y values lower NEARBY (but there may be lower points in another region of the function)
A RELATIVE maximum means there are no other y values higher NEARBY (but there may be higher points in another region of the function)

 END BEHAVIOR:

Every graph has an “end” on both sides of the domain values (x values)
As x gets smaller towards negative infinity (meaning you’re going to the left on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

As x gets larger towards positive infinity (meaning you’re going to the right on the x axis), we look at what the function values are doing (the y or range values)…Is the function also going to negative infinity (down)?....Is it going to positive infinity (up)?

Generally, we summarize end behavior by comparing what x (the domain) is doing to what y (the range or function value) is doing at the same time:

 As x decreases—>y also decreases OR  y increases

As x increases à y also increases OR y decreases

DOMAIN AND RANGE ON A GRAPH:
You already know that the x values are the domain and y values are the range.
On a graph, we look at all the possible x values to determine if the domain is all real numbers or if it’s limited in some way.
We do the same thing with the range.

For example: f(x) = x²
This is a U shaped graph that only goes up from the origin so the range is limited to y ≥ 0
The domain would be all real numbers because you can square any number and, looking at the graph, you can see that eventually the graph will go to both negative and positive infinity to the left and to the right.

REAL WORLD INTERPRETATIONS OF GRAPHS:

Sales of a company:
By looking at a graph of sales over time, you can analyze how the company is doing.
The increasing parts of the graph mean that the company is growing while the reverse is also true.
If you see a flat part of the graph, that part would show the company is staying the same.

Between an increasing and decreasing part of sales would be a relative max to sales…meaning for some reason the company is in decline.

Between a decreasing and increasing part of sales would be a relative min to sales…meaning for some reason the company is doing well again.

The end behavior over time TO THE RIGHT would predict the success of the company in the future. (to the left would be the actual history of sales)