Download Unit 1 – Linear Functions

Unit 1A – Linear Functions
Lesson 1-8: Properties of Real Numbers
The properties of real numbers allow you to write equivalent expressions and simplify expressions. The
summary below reviews properties of real numbers that apply to both addition and multiplication.
PROPERTIES OF REAL NUMBERS
For every real number a, b, and c,
CLOSURE PROPERTY OF REAL NUMBERS
__________________________________________
COMMUTATIVE PROPERTY OF ADDITION
__________________________________________
COMMUTATIVE PROPERTY OF MULTIPLICATION
__________________________________________
ASSOCIATIVE PROPERTY OF ADDITION
__________________________________________
ASSOCIATIVE PROPERTY OF MULTIPLICATION
__________________________________________
IDENTITY PROPERTY OF ADDITION
__________________________________________
IDENTITY PROPERTY OF MULTIPLICATION
__________________________________________
INVERSE PROPERTY OF ADDITION
__________________________________________
INVERSE PROPERTY OF MULTIPLICATION
__________________________________________

The following summary reviews some additional properties of real numbers.
PROPERTIES OF REAL NUMBERS
For every real number a, b, and c,
DISTRIBUTIVE PROPERTY
__________________________________________
MULTIPLICATION PROPERTY OF ZERO
__________________________________________
MULTIPLICATION PROPERTY OF -1
__________________________________________
Examples:
1. Name the property that each equation illustrates. Explain.
a. 1m  m
b. (3  4)  5  3  (4  5)
c. 3(8  0)  (3  8)0
d. 2  0  2
e. 9  7  7  9
f. (d  4)  3  d  (4  3)
g. t  0  t
h.  q  1q
i. np  pn
j. p  q  q  p
You can also use the properties to reorganize the order of numbers in sums or products so that you can
calculate more easily.
2. A fast-food restaurant offered a special on their Biggie Burger to entice customers at lunch time.
The 99 ¢ special ran Monday through Friday. The number of Biggie Burgers sold each day is
recorded in the table below. What was the average amount received from the sale of Biggie
Burgers during each day?
M
246
T
303
W
182
T
341
F
378
3. At the supermarket, you buy a package of cheese for $2.50, a loaf of bread for $2.15, a cucumber
for $.65, and some tomatoes for $3.50. Find the total cost of the groceries.
_____________________________________________ is the process of reasoning logically from given
facts to a conclusion. Using deductive reasoning, you justify each step in simplifying an expression with
reasons such as properties,
d. 2(3t  1)  2 definitions, or rules.
3. Simplify each expression. Justify each step.
a.  4b  9  b
b. 7 z  5(3  z )
c. 5a  6  a
Homework Assignment: PH ALG 1 page 56 odds 1-39, 40-52
Unit 1 A– Linear Functions
Lesson 2.1: Solving One- Step Equations
Lesson 2-2: Solving Two-Step Equations
Lesson 2-3: Solving Multi-Step Equations
Mr. Hallam needed to buy some stamps. He bought some 25 cent stamps and three times as
many 29-cent stamps. He paid a total of $22.40. How many of each type of stamp did she buy?
Definition
A mathematical statement that can be either true or false depending on what values (replacements) are
used is called an _______________________
Definition
The _______________________ to an open sentence are the values (replacements) that make the
sentence true
Definition
An _____________________ states that two mathematical expressions are equal

Real numbers have certain properties that we can use when we solve equations or open sentences.
Some of those properties are listed below.
For all real numbers
___________________________________
Reflexive Property of Equality
Symmetric Property of Equality
if a = b, then
___________________________________
Transitive Property of Equality
if a = b and b = c, then___________________________________
Addition Property of Equality
if a = b, then
___________________________________
Subtraction Property of Equality
if a = b, then
___________________________________
To find a solution, you can use properties of equality to form equivalent equations. Equivalent
equations are equations that have the same solution (or solutions).

INVERSE OPERATIONS

REVERSE ORDER OF OPERATIONS
Single Step Equations
Examples:
1. Solve each equation. Check your answer.
a. m  10  2
c.  9  b  5

d. x  3  8
Multiplication and division are inverse operations. When you multiply or divide to solve equations,
you use the following properties.
For all real numbers
Multiplication Property of Equality
if a = b, then
__________________________________
Division Property of Equality
with c  0 , if a = b, then________________________________
2. Solve each equation. Check your answer.
n
a
5
a.
b.   20
6
5

SKILL:
o In the next example, the coefficient of the variable is a fraction. You can use
RECIPROCAL to solve the equation.
3. Solve each equation. Check your answer.
3
x9
a.
4

1
b.  m  8
4
SKILL:
o When you solve equation involving MULTIPLICATION, DIVIDE each side of the
equation b y the same number.
4. Solve each equation. Check your answer.
a. 4c  96
b. 3a  12
c. 20  2x
d.  8  5 y
Two-Step Equations
Summary
SOLVING TWO-STEP EQUATIONS
Step 1
Step 2
Use the Addition or Subtraction Property of Equality
to get the term with a variable alone on one side of
the equation.
Use the Multiplication or Division Property of
Equality to write an equivalent equation in which
the variable has a coefficient of 1.
Examples:
5. Solve each equation. Check your answer.
m
a. 10   2
4

b. 7  2 y  3
SKILL:
o You can write two-step equations to MODEL real-world situations. Some real-world
situations require whole-number answers. So check that your answer is reasonable in the
given situation.
o
6. You order iris bulbs from a catalog. Iris bulbs cost $.90 each. The shipping charge is $2.50. If you
have $18.50 to spend, how many iris bulbs can you order?

SKILL:
o In some equations, the variable may have a NEGATIVE SIGN in front of it, such as
 x  3 . To help you solve these equations, recall that x  1 x and  x  1 x . You can
solve for x by multiplying or dividing by -1.
7. Solve each equation. Check your answer.
a.  x  7  12
b.  b  6  11
Multi-Step Equations
SUMMARY
Steps for Solving a Multi-Step Equation
Step 1 Clear the equation of fractions and decimals.
Step 2 Use the Distributive Property to remove
Parentheses on each side.
Step 3 Combine like terms on each side.
Step 4 Undo addition or subtraction.
Step 5 Undo multiplication or division
Examples:
8. Solve each equation. Check your answer.
a. 2c  c  12  78

b. 3x  4x  6  2
SKILL:
o You can MODEL real-world situations using multi-step equations.
9. A gardener is planning a rectangular garden area in a community garden. His garden will
be next to an existing 12-ft fence. The gardener has a total of 44 ft of fencing to build the other three
sides of his garden. How long will the garden be if the width is 12 ft?

SKILL:
o In the equation  2(b  4)  12 , the parentheses indicate multiplication. Use the
DISTRIBUTIVE PROPERTY to multiply each term within the parentheses by -2. Then
use the properties of equality to solve the equation.
11. Solve each equation. Check your answer.
a.  2(b  4)  12
b. 3(k  8)  21

SKILL:
2x x
  7 by adding the fractions or by clearing the
3 2
equation of fractions. To CLEAR the FRACTIONS, you multiply each side of the equation
by a common multiple of the denominators.
o You can solve an equation like
4. Solve each equation.
2x x
 7
a.
3 2

b.
m m 5
 
4 2 8
SKILL:
o You can CLEAR an equation of DECIMALS by multiplying by a power of 10. In the
equation 0.5a  8.75  13.25 , the greatest number of digits to the right of a decimal point is
2. To clear the equation of decimals, multiply each side of the equation by 10 2 , or 100.
5. Solve each equation.
a. 0.5a  8.75  13.25
b. 0.025x  22.95  23.65
Keep the steps in the summary below in mind as you solve equations that have variables on one side of
the equation.
SUMMARY
Steps for Solving a Multi-Step Equation
Step 1 Clear the equation of fractions and decimals.
Step 2 Use the Distributive Property to remove
Parentheses on each side.
Step 3 Combine like terms on each side.
Step 4 Undo addition or subtraction.
Step 5 Undo multiplication or division
Homework Assignment: PH ALG 1 page 78 odds 55-79, page 84 odds 41-69, page 92 odds 39-49,
57,61.
Unit 1A – Linear Functions
Lesson 1-3:
Lesson 3-2:
Lesson 3-3:
Lesson 3-4:
Properties of Real Numbers
Solving Inequalities Using Addition and Subtraction
Solving Inequalities Using Multiplication and Division
Solving Multi-Step Inequalities
Algebra deals with operations and relations among numbers, including real numbers and imaginary
numbers. Listed below are some of the subsets of the real numbers. Imaginary numbers will be
introduced in Unit 2.
Subsets of Real Numbers
Symbol
Examples
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
1. Name the set(s) of numbers to which each number belongs.
a. 

17
31
b. 23
c. 0
d. 4.581
e. -12
f.
g. -4.67
h. 6
5
12
Suppose 110 students are going on a field trip. Each bus can hold 40 students. To find the number
of buses needed. What number set best describes this situation (reasonable)?
2. Which set of numbers is most reasonable for each situation?
a. the number of students who will go on the class trip
b. the height of the door frame in your classroom
c. the cost of a scooter
Graph
3.
4.

All of the properties that we learned about equations and solving equations hold true for inequalities
EXCEPT ?
Investigation:

Investigate the properties of inequalities.
Addition
Subtraction
Multiplication
Division
2
1
0
-1
-2

If you _______________________ or __________________ by a _______________________ then
you flip(reverse) the inequality
Examples
 Solve each inequality. Graph and check your solution.
x
b 1
 1

3.
a.
b.
2
4 2
a. 
2
n2
3
b. 
k
 1
4
a.  5z  25
b.  2t  8
a. 7  6a  19
b.  3x  4  14
a. 2(t  2)  3t  1
b. 4 p  2( p  7)  8
a. 6x  15  4z  11
b. 3b  12  27  2b
4. The student council votes to buy food for a local food bank. A case of 12 jars of spaghetti sauce
costs $13.75. What is the greatest number of cases of sauce the student council can buy if they
use at most $216 for this project?
5. The school band needs a banner to carry in a parade. The banner committee decides that the
length of the banner should be 18 feet. What are the possible widths of the banner if they can
use no more than 48 feet of trim?
Homework Assignment: PH ALG 1 pg155 odds 1-39,52.
Unit 1 A– Linear Functions
Lesson 3-5: Compound Inequalities

Two inequalities that are joined by the word and or the word or form a ____________________.
You can write the compound inequality x  5 and x  7 as  5  x  7 .
The graph above shows that a solution of ______________ is in the overlap of the solutions of the
inequality x  5 and the inequality x  7 .
You can read  5  x  7 as:
“x is greater than or equal to -5 and less than or equal to 7.”
“x is between -5 and 7, inclusive.”
Examples:
1. Write a compound inequality that represents each situation. Graph the solutions.
a. all real numbers that are at least -2 and at most 4
b. Today’s temperatures will be above 32F , but not as high as 40F .
A solution of a compound inequality joined by and is any number that makes both inequalities _____.
One way you can solve a compound inequality is by writing _______ inequalities.
Example
2. Solve each inequality. Graph your solution.
a.  4  r  5  1
b.  6  3x  15
You could also solve an inequality like  4  r  5  1 by working on all three parts of the inequality at
the same time. You work to get the variable alone between the inequality symbols.
Example
3. The acidity of the water in a swimming pool is considered normal if the average of three pH readings
is between 7.2 and 7.8, inclusive. The first two readings for a swimming pool are 7.4 and 7.9. What
possible values for the third reading p will make the average pH normal?
4. Your test grades in science so far are 83 and 87. What possible grades can you make on your next
test to have an average between 85 and 90, inclusive?

A solution of a compound inequality joined by _____ is any number that makes either inequality
true.
Example
Write a compound inequality that represents each situation. Graph the solution.
5. all real numbers that are less than -3 or greater than 7
Solve the compound inequality. Graph your solution.
6. 4v  3  5 or  2v  7  1
7.  2x  7  3 or 3x  4  5
Homework: PH Alg 1 pg 163 evens 2-42
Unit 1A – Linear Functions
Lesson 3-6: Absolute Value Equations and Inequalities

Recall that the absolute value of a number is its distance from zero on a number line. Since
absolute value represents distance, it can never be negative.
x 3
The two solutions of the equation x  3 are -3 and 3.
You can use the properties of equality to solve an absolute value equation.
Examples:
1. Solve each equation. Check your solution.
a. x  5  11
c. 3 n  15
b. t  2  1
d. 4  3 w  2
e. Is there is a solution of 2 n  15 ? Explain.

Some absolute value equations such as 2 p  5  11 have variable expressions within the absolute
value symbols. The expression inside the absolute value symbols can be either positive or
negative.
RULE
Solving Absolute Value Equations
To solve an equation in the form A  b , where A represents a variable expression
and b > 0, solve _______ and ________
Examples
2. Solve each equation. Check your solution.
a. 2 p  5  11
b. c  2  6
You can write absolute value inequalities as compound inequalities.
n 1  3
n 1  3
RULE
Solving Absolute Value Inequalities



To solve an inequality in the form A  b , where A is a variable expression and b > 0, solve
__________________
To solve an inequality in the form A  b , where A is a variable expression and b > 0, solve
__________________
Similar rules are true for A  b or A  b .
3. Solve and graph the solution.
a. v  3  4

b. |y – 5| ≤ 2
To maintain quality, a manufacturer sets limits for how much an item can vary from its
specifications. You can use an _______________ equation to model a quality-control situation.
4. The ideal diameter of a piston for one type of car engine is 90,000 mm. The actual diameter can vary
from the ideal by at most 0.008 mm. Find the range of acceptable diameters for the piston.
Homework: PH Alg 1 pg.169 evens 2-54, 58.
Function Sort
Unit 1 A
Students work in groups and each group is given all of the cards and must match
the correct graph, table and function. They should be discussing how they know they
have the correct matches.
Function
Table
Graph
7
16
13
36
9
10
3
6
21
23
20
27
12
X
-9
-6
-3
0
3
6
9
Y
-21
-15
-9
-3
3
9
15
26
X
-9
-6
-3
0
3
6
9
Y
15
9
3
-3
-9
-15
-21
35
X
-9
-6
-3
0
3
6
9
Y
159
69
15
-3
15
69
159
34
X
-9
-6
-3
0
3
6
9
Y
-165
-75
-21
-3
-21
-75
-165
24
X
-9
-6
-3
0
3
6
9
Y
1455
429
51
-3
-57
-435
-1461
32
X
-9
-6
-3
0
3
6
9
Y
-1461
-435
-57
-3
51
429
1455
4
X
-9
-6
-3
0
3
6
9
Y
-2.998…
-2.984…
-2.875
-2
5
61
509
29
X
-9
-6
-3
0
3
6
9
14
X
-9
-6
-3
0
3
6
9
Y
162
72
18
0
18
72
162
19
X
-9
-6
-3
0
3
6
9
18
X
-9
-6
-3
0
3
6
9
Y
2
2
2
2
2
2
2
Y
509
61
5
-2
-2.875
-2.984…
-2.998…
Y
Not defined
Not defined
Not defined
0
3.4641
4.899
6
15
X
-9
-6
-3
0
3
6
9
Y
Not defined
Not defined
Not defined
0
-3.4641
-4.899
-6
7
16
36
9
3
6
23
10
13
21
27
20
5
y = 2x – 3
22
y = -2x – 3
11
y = 2x2 – 3
17
y = -2x2 – 3
30
y = -2x3 – 3
31
y = 2x3 – 3
1
y = 2x – 3
2
y = 2-x – 3
25
y = 2x2
28
y = 2x0.5
33
y=2
8
y = -2x0.5
Name:________________________________________________Date________Hour________
Unit 1 A
Function Sort Homework
For each problem below, you will be supplied with an equation, table or graph. Use the given
information to find the other information. For example, if you are given an equation, find the
table and graph.
1. y = x + 2
2.
What is the equation? _______________________________________________
3.
What is the equation? _______________________________________________
Algebra I
Families of Functions Posters
Name: ______________________
Date: ______________________
Hour: ______________________
The purpose of this assignment is to help you understand characteristics of certain
kinds of functions, how to classify functions, and how to use the graphing calculator.
Directions
You will do this assignment two times, each time with a different symbolic rule.
1.
Get a slip of paper that has a symbolic rule on it. On a poster-sized sheet, record
your name and the symbolic rule.
2.
On the poster, write the calculation procedure in words.
3.
Create a table of values for inputs from –10 to 10 counting by ones. Use the
graphing calculator and enter the rule under “Y=.” Use the calculator to make
this table. Record it on the poster-sized sheet of paper.
4.
On graph paper, create a graph of your rule. Label your scale and your axes.
Tape or staple this graph to your poster.
5.
Predict what would happen if you made your graph for x-values beyond –10 and
10. Use dotted lines to show what you predict will happen.
6.
Make a list of anything you notice about the rule, the table, and the graph on the
poster sized sheet of paper. Include what you know about tables and graphs.
7.
Please get a second function and complete 1-6 on a new sheet of poster paper.
When finished, you will be presenting BOTH of your functions to the class.
An example of a poster may look like this:
Your Name
y = 2x2
-10
-9
-8
-7
-6
.
.
.
10
What you
noticed about
the table and
graph and
rule . . .