Transcript M1L2 Remediation Notes

Module 1 Lesson 2
A Literal Equation is an equation with two or
more variables.
• You can "rewrite" a literal equation to
isolate any one of the variables using
inverse operations.
• When you rewrite literal equations, you
may have to divide by a variable or
variable expression.
Step 1
Locate the variable you are asked to solve
for in the equation.
Step 2
Identify the operations on this variable and
the order in which they are applied.
Step 3
Use inverse operations to undo operations
and isolate the variable.
A. Solve x + y = 15 for x.
x + y = 15
Since y is added to x, subtract y
–y –y
from both sides to undo the
x
= –y + 15
addition.
B. Solve pq = x for q.
pq = _x_
p p
Since q is multiplied by p, divide
both sides by p to undo the
multiplication.
Solve 5 – b = 2t for t.
5 – b = 2t
Locate t in the equation.
Since t is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
Solve for the indicated variable.
1.
for h
2. P = R – C for C
3. 2x + 7y = 14 for y
4.
for m
5.
for C
1. H = 3V
A
2. y = 14 – 2x
7
3. C = R – P
4. m = x(k – 6 )
5. C = Rt + S
The formula C = d gives the circumference of a circle C
in terms of diameter d. The circumference of a bowl is 18
inches. What is the bowl's diameter? Leave the symbol 
in your answer.
Locate d in the equation.
Since d is multiplied by , divide both
sides by  to undo the multiplication.
Now use this formula and the information given in the
problem.
The formula C = d gives the circumference of a circle C
in terms of diameter d. The circumference of a bowl is 18
inches. What is the bowl's diameter? Leave the symbol 
in your answer.
Now use this formula and the information given in the
problem.
The bowl's diameter is
inches.
Every Real Number is
either rational or
irrational.
We refer to these sets as
subsets of the real
numbers, meaning that all
elements in each subset
are also elements in the
set of real numbers.
Numbers
Examples
Natural Numbers
Whole Numbers
Integers
2,3,4,17
0,2,3,4,17
-5,-2,0,2,5
1 5
1
2
Rational Numbers
,  ,.4  ,0,.6 
2 1
5
3
Irrational Numbers
2,  ,  3
25 is a rational number because 25  5.
Example
Consider the following set of numbers.
1

3, 0, , .95,  , 8,
2

List the numbers in the set that are:
a. Natural Numbers
b. Whole Numbers
c. Integers
d. Rational Numbers
e. Irrational Numbers
f. Real numbers

16 

Example
Consider the following set of numbers.
1

3, 0, , .95,  , 8,
2


16 

List the numbers in the set that are:
a. Natural Numbers: √16 = 4, so that is the only Natural
Number
b. Whole Numbers: 0 , √16
c. Integers: -3, 0, √16
d. Rational Numbers: -3, 0, ½ , .95, √16
e. Irrational Numbers: √8
f. Real numbers: All of the numbers listed above!
Property
Addition
Multiplication
a+b=b+a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Identity
a+0=a
a*1=a
Inverse
a + (-a) = 0
a * 1/a = 1
Commutative
Associative
Distributive
a(b + c) = ab + ac
It doesn’t matter how you swap addition or
multiplication around…the answer will be the
same!
Rules:
Samples:
Commutative Property of Addition
Commutative Property of Addition
a+b = b+a
1+2 = 2+1
Commutative Property of
Multiplication
Commutative Property of Multiplication
ab = ba
(2x3) = (3x2)
It doesn’t matter how you group (associate)
addition or multiplication…the answer will be the
same!
Rules:
Samples:
Associative Property of Addition
Associative Property of Addition
(a+b)+c = a+(b+c)
(1+2)+3 = 1+(2+3)
Associative Property of
Multiplication
(ab)c = a(bc)
Associative Property of
Multiplication
(2x3)4 = 2(3x4)
What can you add to a number & get the same number back?
ZERO
What can you multiply a number by and get the number back?
ONE
Rules:
Identity Property of Addition
a+0 = a
Identity Property of Multiplication
a(1) = a
Samples:
Identity Property of Addition
3+0=3
Identity Property of Multiplication
2(1)=2
Think opposites!
The Inverse property uses the inverse operation to get to the identity!
Rules:
Inverse Property of Addition
a+(-a) = 0
Samples:
Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
a(1/a) = 1
Inverse Property of Multiplication
2(1/2)=1
You can distribute the coefficient through the parenthesis
with multiplication and remove the parenthesis.
Rule:
a(b+c) = ab+bc
Samples:
4(3+2)=4(3)+4(2)=12+8=20
• 2(x+3) = 2x + 6
• -(3+x) = -3 - x
1.
2.
3.
4.
5.
6.
7.
8.
9.
6+8=8+6
5 + (2 + 8) = (5 + 2) + 8
12 + 0 =12
5(2 + 9) = (5  2) + (5  9)
4  (8  2) = (4  8)  2
5/9  9/5 = 1
5  24 = 24  5
18 + -18 = 0
-34 1 = -34
1.
2.
3.
4.
5.
6.
7.
8.
9.
Commutative
Associative
Identity
Distributive
Associative
Inverse
Commutative
Inverse
Identity
All real numbers have closure.
The Closure property states the if a and b
are real numbers then:
• a + b is a real number
• ab is a real number.
So, if you add two rational numbers, your sum
will be rational. Also, if you add two irrational
numbers, that sum will be irrational.