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Chapter 1: Introduction to Algebra Notes
Name ______________
Date ______________
Introduction to Algebra
1.1 Variables: To simplify numerical expressions and evaluate algebraic expressions
Suppose you have an after school job at McDonald’s and you get paid $6.50 an hour. The amount of
money you earn depends on how many hours you work.
Number of hours
Money Earned
1
2
3
4
The money earned follows this pattern
Money earned =
=
The letter ____ stands for the hours shown on the table:1, 2, 3 or 4. Also, ____ can
stand for other hours not on the table. We call ____ a variable.
A variable is
The numbers are called _______ of the variable.
An expression that contains a variable such as 6.50  h , is called a
__________________.
An expression such as 6.50  4 , that names a particular number is called a
___________________ or _____________
Notation
 6.50  4 can also be written as 6.50  4 to represent multiplication
 Products of variables are usually written without the multiplication sign because it
looks too much like the letter x.
19  n can be written as ______
a  b can be written as ______
1
2  x can be written as ______

The symbol = means

4 + 2 = 6 is read
or
or

The symbol ≠ means

4 + 2 ≠ 5 is read


a
b
means
ab means

a + b means

a – b means
Order of Operations
P
E
M
D
A
S
1. 2 Grouping Symbols: To simplify expressions with and without grouping symbols.
There are three types of grouping systems
Parentheses
Brackets


Fraction Bar
When solving a problem involving grouping systems always evaluate inside the grouping symbol
firsts.
If an expression contains more than one grouping symbol, first simplify in the innermost grouping
system. Then work towards the outermost grouping symbol until the simplest expression is found.
Example 1 Simplify each expression
(13  11)
a) 9  (5  4)
b)
(6  2)
c) (12  11)  (2  11)
Substitution Principle- An expression may be replaced by another expression that has
the same value.
Example 2 Evaluate each expression if a = 1, b = 2, and c = 3
a) 6a
c) (5c) – 4
b) 9 – b
d) (a  b)  c
Example 3 Simplify the expression on each side of the ?. To make a true statement
replace the ? with the = or ≠ symbol.
b) (14 -3) -1 ? 14 - (3 – 1)
a) 6  3 ? 3 6
(8  2)
c)
?8–1
2
1.4, Translating Words and Sentences into Symbols: To translate phrases into
variable expressions
Write down all the words you can think of that mean each operation.
Addition
Subtraction
Multiplication
Addition
Division
Phrase
The sum of 8 and x
A number increased by 7
5 more than a number
Subtraction
The difference between a number and 4
A number decreased by 8
5 less than a number
6 minus a number
Multiplication
The product of 5 and a number
Translation
Seven times a number
One third of a number
Division
The quotient of a number and 8
A number divided by 10
CAUTION: Be careful with phrases involving subtraction.
The phrase “5 less than x” is translated to x - 5 not 5 – x
Example 1 Translate each phrase into a variable expression
a) 3 less than half of x
b) Half the difference between x and 3
Here are four formulas commonly used in Algebra.
A = lw
Area of a rectangle equals the length of the rectangle times the width of
the rectangle
P = 2l + 2w Perimeter of the rectangle equals two times the length plus two times the
width
D = rt
Distance traveled equals the rate times the time traveled
C = np
Cost equals the number of items times the price per item
Example 2 Find the area and perimeter of a rectangle with length 10 and width w.
Example 3 You travel (h + ½) hours at 80 km/h (kilometers per hour). How far do you
travel?
What words can you use to represent equals (=)?
Example 4 Twice the sum of a number and four is ten.
Example 5 When a number is multiplied by four the result is decreased by 6, the final
result is 10.
Guided Practice
Translate each phrase into a variable expression. Use n for the variable.
1. Eight times a number
2. The product of three and a number
3. Five more than a number
4. One fourth of a number
5. A number decreased by four
6. A number divided by five
7. Nine less than half a number
8. Nine more than twice a number
Complete each statement with a variable expression.
9. A rectangle has width 6 units and length x units. The area is ______ square
units.
10. You travel for (t – 2) hours at 75 km/h. You traveled ________ km.
11. Al earns (p + 3) dollars per hour. In 8 hours, he earns _______ dollars.
12. The Golden Gate Bridge was built n years ago. Three years from now it will be
standing ______ years.
13. Nine years from now Fenway Park will be g years old. It is now _____ years old.
14. Workers on an assembly like produce (x + 10) cars each day. In 5 days they
produce ______ cars.
Translate each sentence into an equation.
15. One third a number is seven.
16. Six less than a number is twelve.
17. Half of the sum of three and a number is four.
18. Four less than twice a number is nine.
19. Twice a number is 18 more than five times a number.
20. A number is 9 more than one third of itself.
21. Eleven less than twice n is seven more than n.
22. Ten times x is twice the sum of x and eight.
1. 5 Translating Problems into Equations: To translate word sentences into
equations
Sometimes you will need to use a formula in order to write an equation.
Example 1 Use the figure and the information below to write an equation involving x.
4
4
Perimeter = 14
x
Example 2 First choose a variable to represent the number described by the words in
the parentheses, then write an equation that represents the given information.
The distance traveled in 3 hours of driving was 240 km (hourly rate)
Steps to translate problems into equations
Step 1 Read the problem carefully
Step 2 Choose a variable to represent the unknowns
Choose a variable for the one unknown
Write an expression for the other unknown using the variable and one of the
facts
Step 3 Reread the problem and write an equation
Use the other fact from the problem to write an equation
Example 3 Translate the problem into an equation.
(1) Marta has twice as much money as Heidi
(2) Together they have $36
How much money does each have?
Example 4 Translate the problem into an equation.
(1) A wooden rod 60in long is sawed into two pieces
(2) One piece is 4 in longer that the other.
What are the lengths?
Guided Practice
Use the figure and the information below it to write and equation involving x.
1.
2.
8
3.
20
7
7
x
Perimeter = 18
x
x
8
Perimeter = 26
x
x
16
Perimeter = 60
First choose a variable to represent the number described by the words in the
parentheses, then write an equation that represents the given information.
4. A dozen eggs cost $1.19. (Cost of one egg)
5. Nine days ago a new radio station had been on the air for 13 days. (Station’s age
now)
6. A student solved all but the last four exercises in the homework assignment of 30
exercises. (Number of exercises solved)
7. A sixteen-year-old building is one fourth as old as a nearby bridge. (Bridges age
now)
8. A rectangular floor is tiled with 928 square tiles. The floor is 32 tiles long.
(Number of tiles in the width)
9. A season ticket good for 39 basketball games cost $1092. (Cost of one
admission with this ticket.
Translate each problem into an equation
10. (1) Lyn has twice as much money as Jo.
(2) Together they have $63.
How much does each have?
11. (1) State College has 620 students.
(2) There are 20 more women than men.
How many women are there?
12. (1) Brenda drove three times as far as Jim.
(2)Brenda drove 24 miles more than Jim.
How far did Jim drive?
13. (1) The Ravens won twice as many games as they lost>
(2) They played 96 games.
How many games did they win?
14. (1) Skip had eight fewer job interviews than Woody.
(2) Together they had 20 interviews.
How many interviews did each have?
15. (1) The number of items on two grocery list differs by 7.
(2)The total number of items is 33.
How many items are on each list?
16. The height of a tower is three times the height of a certain building. If the tower
is 50 m taller than the building, how tall is the tower?
17. The length of the rectangle is one unit more than its width. If the area is 30
square units, find the dimensions of the rectangle.
18. The triangle has two equal sides and a third side that is 15 cm long. It the
perimeter is 50 cm, how long is each side?
1.8, 1.9 Number Line and Absolute Value: To graph real numbers on a number line
and to compare real numbers and to use opposites and absolute values
Symbols
…
<
>
Vocabulary
Graph of a number
Integers
Coordinate of a number
Whole number
Positive integers
Real Number
Positive number
Opposite of a number
Negative integer
Absolute value of a number
Negative number
Example 1 Write a real number to represent the situation. Then write the opposite of
that situation and write a number to represent it.
a) Five wins
d) Four steps to the right
b) Up six floors
e) 7˚ above freezing (0˚ C)
c) 150m above sea level
f) A bank deposit of $50
Example 2 Translate each statement into symbols
a) Four is greater than negative two
b) Negative four is greater than negative six
c) Two is less than two and two tenths
d) Negative 12 is less than three
Example 3 List the letters of the points whose coordinates are given. Use the number
line below
A
B
a)-6, 3
Example 4 Simplify.
a) –(2 + 6)
b) –(7 - 3)
C D E
F G
H
I
J K L
M N O P Q
R
S
0
b) 2, -2, -3 ½
c) 0, -5, 2 ½
c)  0.2   1.8
d) 8   6
Example 5 Use one of the symbols >, < or = to make a statement true.
a) –(-3) ? -3
b) –(-4) ?  5
c)  8 ? 8