Math 6 Honors ( Periods 1, 2, & 3)

Solving Other Equations & Inequalities 2-5



Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!

Our math example was
the sum of three and a number b, times even
(3 + b)7

and
the sum of three and a number b times seven
3 + 7b

Our Language Arts example... is one of my favorites.

Where does the comma belong in the following:
A woman without her man is nothing.

I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!

We solve by "undoing" the operations in each equation.

We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!


GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


Here are the steps and justifications (reasons)

1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)

2. What is the operation the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?

that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:

" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone (isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

Algebra Honors (Period 6 & 7)

Multiplying Polynomials by Monomials 4-5
This is just the distributive property
x(x + 3) = x² + 3x

-2x(4x² - 3x + 5)

-8x³ + 6 x² -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.

n(2-5n) + 5(n² -2 ) = 0
2n - 5n² + 5n² - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}

1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}

Math 6 Honors ( Periods 1, 2, & 3)

Solving Equations & Inequalities 2-4


To solve equations you need to ISOLATE the variable on one side of the mathematical sentence.
isolate--> means to get the variable alone on one side of the equal sign or the inequality sign.


Properties of Equality:
Property of Equality allows us to add or subtract the same number from BOTH sides of the equation.

Addition Property of Equality abbreviated as +prop=
Subtraction Property of Equality ( written as -prop= )



Identity Property of Addition ( our textbook calls it the Addition Property of Zero)
We abbreviated it as ID(+) a + 0 = a

Remember to JUSTIFY with the properties

We completed a yellow form-- see tonight's homework assignment if you need to print it out. That sheet gets glued into our spiral notebook


w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46


What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=

What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition.

How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)

p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186

What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=

What property allows us to write p instead of p + 0
The Identity property of Addition.
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!

What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation

b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1

h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14

Our textbook gives the answer as "greater than 14" but I want you to put it in math symbols. That is, please answer with
h > 14.