Download Warm-up Section 1.6/1.7: Simplifying Expressions

Remember to read the textbook before attempting to do your homework.
Warm-up
– Introduce yourself to your neighbors and answer the set of warm-up problems below. –
To change a percent to a decimal:
1. Drop the percent sign, %.
2. Divide by 100.
To change a decimal to a percent:
1. Multiply by 100.
2. Insert a percent sign, %
Example 1: Write as a decimal.
a. 3%
b.
4.5%
c.
6.75%
Example 2: Write as a percent.
a. 0.7
0.72
c.
0.0725
b.
Example 3: Below is the number of points a student received on different assessments. What letter
grade did the student earn? Please also write their assessment score as a percent. Use the standard
90% (A) - 80% (B) - 70% (C) scale. Ok to use a calculator, if desired.
a.
72.5 out of 100 points
b.
102 out of 120 points
Example 4: A student took a 2-part exam worth 120 points. Below is the number of points the student
received on the exam. What letter grade did the student earn? Please also include their score as a
percent. Use the standard 90% (A) - 80% (B) - 70% (C) scale. Ok to use a calculator, if desired.
Part 1: 25 out of 30 points
Part 2: 50 out of 90 points
KEY: 1a. 0.03
3a. A, 72.5%
1b. 0.045
3b. B, 85%
1c. 0.0675
2a. 70%
2b. 72%
2c. 7.25%
4. 62.5%, D (Hint: 75 points out of 120 points)
Remember to read the textbook before attempting to do your homework.
Section 1.6/1.7: Simplifying Expressions
Warm-up: Panchito needs your help. He has been assigned a set of homework problems. Write down
every single problem that he will have to do for that assignment. Note: eoo means “every other odd”
a. 7 – 11 odd, 16 – 20 ALL, 29 – 53 eoo
b. 29 – 53 eoo
Math 60 || Beginning Algebra || Cerritos College
– Pg 1 –
Chapter 1 Lecture Notes by Maria Torres
♥
Definitions:
A variable is a letter used to represent an unknown number, e.g. x, y, z, m, n
A number by itself is called a constant, e.g. 5, –2, 0, 12 , 0.3, −3 75 , –1.29, − 94
A term is a constant or a variable, or a product of constants and variables, e.g. 2, x, 2x3, –2xy, 23 x
An algebraic expression is a collection of numbers, variables, grouping symbols and operation symbols
such as +, –, ×, ÷. e.g. 3x – 6,
3 + 2[5(m – 1) + 6],
5xy – 3z2 + 9
The numerical coefficient (or coefficient) is the number in front of the variable,
e.g. Looking at 7bc, 7 is called the coefficient, and bc is called the variable part.
Answer: 2
What is the coefficient of the term − 12 x 3 ?
Answer: − 1
What is the coefficient of the term x?
Answer: 1
Why? x is the same as 1 • x, and 1 • x = 1x
What is the coefficient of the term –x?
Answer: –1
Why? –x is the same as –1 • x, and –1 • x = –1x
Moral of the story:
If there is not a number written next to a variable,
then the coefficient is understood to be 1 .
Moral of the story:
If there is only a negative sign, “ – ”, next to a variable,
then the coefficient is understood to be –1 .
What is the coefficient of the term 2x3?
2
Like terms are terms that are constants, or terms that contain the same variable(s) raised to the same
powers.
e.g. –1, 5, ¾, 0.25 ← all constants (numbers)
–x, 5x, 0.25x
← all have same variable part
2
2
12x , –4x
← all have same variable part
6x3y2z5, –7x3y2z5 ← all have same variable part
The cool thing about like terms is that we can combine them.
↓
Combining Like Terms:
To combine like terms, add or subtract their coefficients and keep the same variables with the same
exponents.
Example 1: Simplify. You may not use a calculator to answer.
a. 6m2 – m2 + 5m – 8m + 37
b. 3 – x2 – 9x – 11 – 6x2 + 9x
c. – (3a2 + 5a – 4)
c2 – (3a2 + 5a – 4) ← can be seen as . . .
means “what is the opposite of 3a2 + 5a – 4?”
Math 60 || Beginning Algebra || Cerritos College
– Pg 2 –
(3a2 + 5a – 4)
Chapter 1 Lecture Notes by Maria Torres
Example 2: Simplify. You may not use a calculator to answer.
a. −5 − 2  2 x − 4 ( x + 7 )  − 6
b.
7 9
2
− ( 2n + 1) − ( 4n − 3)
15 10
5
(end of Section 1.6/1.7 combo unit, so §1.6 and §1.7 are assigned for homework today – yay!)
Math 60 || Beginning Algebra || Cerritos College
– Pg 3 –
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Section 1.2: Translating Verbal Expressions into Variable Expressions
Key words that are code for . . .
Addition
add
Subtraction
subtract
Multiplication
multiply
sum
subtract from
product
plus
difference
times
more than
minus
increased by
less than
twice
code for “2 times”
of
with fractions
and percent
decreased by
less
Tip:
Division
divide
quotient
The word “and” is your friend! Why? Because that word separates the terms that you
are adding, subtracting, multiplying, or dividing.
Warning: Since subtraction is not commutative, you need to be very careful when writing
expressions involving subtraction. Whenever you see the word difference, you need to
subtract the terms in the order in which they are given.
Warning: Since division is not commutative, you need to be very careful when writing expressions
involving division. Whenever you see the word quotient, you need to divide the terms in
the order in which they are given.
Example 1: Write the algebraic expression described. Let x represent the unknown number.
a.
Five-ninths of the number
e.
Eighteen less than a number
b.
The quotient of eight and a number
f.
Four less than five times a number
c.
The sum of twice a number and seven.
g.
Twenty-six more than ten times a number
d.
The difference of twelve and fourteen
h.
The difference of one and twice a number
Math 60 || Beginning Algebra || Cerritos College
– Pg 4 –
Chapter 1 Lecture Notes by Maria Torres
Example 2: Write the algebraic expression described. Let x represent the unknown number.
a.
Twice the difference of six and a number.
b.
Three times the sum of a number and five.
c.
Six less than twice the difference between a number and seven.
d.
Eight more than four times the difference of negative five and three times a number.
e.
The sum of one-eighth of a number and one-twelfth of the number.
(Official end of Chapter 1. Please work on the additional homework problems assigned for §1.2)
Don’t forget to start working on today’s homework in a timely manner – that way you’ll be on track
to getting the grade that you are willing to earn.
You can check out the textbook for FREE through the Library’s Reserve System.
Instructor also has extra copies of the textbook for you to use during her office times.
Math 60 || Beginning Algebra || Cerritos College
– Pg 5 –
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Chapter 1 Bonus Material: Properties of Real Numbers
(Prealgebra review)
Sets of Numbers:
The set of natural numbers (aka counting numbers) is given by
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, … , 99, 100, 101, 102, 103, … , 2099, 2100, 2101, …}
The set of whole numbers include the set of natural numbers and 0.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, … , 99, 100, 101, 102, 103, … , 2099, 2100, 2101, …}
The set of integers includes the set of whole numbers and their opposites.
{–102, –101, –100, –99,…, –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7, … , 99, 100, 101, 102}
Properties of Real Numbers
If a, b, and c are real numbers, then we have:
Commutative Property of Addition:
Commutative Property of Multiplication:
a+b=b+a
a•b=b•a
e.g.
3+2=2+3
e.g.
3•2=2•3
(The order in which we add does not matter)
Associative Property of Addition:
a + (b + c) = (a + b) + c
e.g.
2 + (3 + 4) = (2 + 3) + 4
(The order in which we multiply does not matter)
Associative Property of Multiplication:
a • (b • c) = (a • b) • c
e.g.
2 • (3 • 4) = (2 • 3) • 4
The way in which the numbers are
grouped does not change their sum
The way in which the numbers are
grouped does not change their product
Multiplicative Identity:
Additive Identity:
Identity Property of Multiplication)
(aka Identity Property of Addition)
a • 1 = a, where 1 is the multiplicative identity
a + 0 = a, where 0 is the additive identity
e.g. 3 • 1 = 3
e.g.
3+0=3
1•3=3
0+3=3
Additive Inverse:
Multiplicative Inverse:
(aka Inverse Property of Addition)
(Inverse Property of Multiplication)
a + (–a) = 0, where –a is the additive inverse
a • 1a = 1, where 1a is the multiplicative inverse
e.g. 3 + (–3) = 0
e.g. 3 • 13 = 1
Note: “multiplicative inverse”
–4 + 4
= 0 Note: “additive inverse”
1
is code for “opposite”
-4 • ( - 4 ) = 1 is code for “reciprocal”
Distributive Property
(over addition):
a(b + c) = ab + ac
e.g.
2(x + 3) = 2 • x + 2 • 3
= 2x + 6
Distributive Property
(over subtraction):
a(b – c) = ab – ac
e.g.
2(x – 3) = 2 • x – 2 • 3
= 2x – 6
Wait! Are you looking for more fun? Then work on the attached sheet: Pre-Alg. 214
Math 60 || Beginning Algebra || Cerritos College
– Pg 6 –
Chapter 1 Lecture Notes by Maria Torres