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