Transcript X - Mrs. Astudillo`s Class

About Me
 Graduated from Texas A&M University
 Born and Raised in San Antonio
 6th year teaching at Pearce
 Teach Algebra II and Pre AP Algebra I
 Coach Volleyball and Track
 Grading
About the Class
 Test/ Projects/ After Party = 50%
 Quizzes = 30%
 Homework and Participation = 20%
 Test
 Given at least a weeks notice
 Quiz
 Given at least 3 day notice
 Homework
 Graded by completion or partial completion
 Answers are put on board by students/teacher, gone over
as a group
 Late work will follow Pearce Policy
About the Class
 After Party




Take home test
2-4 a Six Weeks
Will all accumulate to equal one Test grade
Easy way to get a 100 as a Test Grade
 Projects
 At least one a Semester
 First Semester: Linear Project
 Second Semester: Gym Project, Final Project
 Mornings
Tutoring
 Monday, Tuesday, Thursdays, and Fridays
 8:15am- 8:55am
 Afterschool
 Monday or Wednesday with Mr. Tiam
 4:10- 5:00 in room B207
Class Online
EdLine
 Class Announcements
 Class Calendars
 Power Point Notes
 Video Lessons
 Homework Assignments
Grades
 https://portal.risd.org
 Or you can get there from www.risd.org
Supplies
Binder/ Folder
Pencils
Erasers
Colored pen/ pencil
Grid Paper Spiral Notebook
Class gift you pick
Box of Tissues
Hand Sanitizer
Pencils
Calculator and Book
TI-83
 Bring every day
 Buy your own or borrow one from the school
 Bring 4 AAA batteries
Book
 One will be issued to you to take home
 A class set will be available for you to borrow in school
 Online access through EdLine
To Contact Me…
 Email: [email protected]
(Have your parents email by the end of the week)
Addition
Plus
All Together
More Than
Greater Than
Total
Sum
Increase
Subtraction
Difference
Subtract
Less Than
Decrease
Lost
Multiplication
Times
Per
Each
Product
Multiples
Multiply
Distribute
Division
Quotient
Per
Each
Fractions
Divide
Factors
Exponents
Squared
Raised to
Cubed
Exponentially
To the power of …..
Parenthesis
The sum of ……
The difference of ……
The quotient of ……
The product of ……
Quantity of …….
VOCABULARY
Variable:
a letter that represents an unknown value
i.e. If 5 = 4x, then x is variable
Operation:
an action or procedure which produces a new value from one or more input
values
i.e. addition, subtraction, multiplication, division
Constant:
a non-varying value; does not change
i.e. If 5 = 4x, then 5 and 4 is constant. 5 will always equal 5
Coefficient:
A number multiplied by a variable
i.e.
If 5 = 4x, then 4 is the coefficient
VOCABULARY
Expression:
a mathematical phrase that contains operations, numbers, and/or
variables. It does not have an equal sign
i.e.
4 + 20 – 7
5(2x+3y)
Verbal Expression:
a mathematical expression that is represented in using words
i.e.
Four plus twenty minus seven.
Five times the sum of twice a number and three times another number.
Numerical Expression:
An expression involving only numbers and operations
i.e.
4 + 20 – 7
Algebraic Expression:
An expression containing at least one variable
i.e.
5(2x+3y)
VOCABULARY
Equation:
a mathematical statement that states that two expressions are
equivalent. It must have an equal sign.
i.e.
34x + 3 = 6x
Equation or Expression?
4c  5  6
67  9
4(3  7 a )
6
9 25 
3
Write an algebraic expression for each verbal expression given
below.
1) The product of the difference of a number and six, and eight
2) The sum of x and y, multiplied by the quotient of 3 and 4
3) Choose the correct verbal expression for the algebraic
expression.
3
x
5
a) the quotient of x squared and 5
b) the quotient of x cubed and 5
c) the product of x squared and 5
d) the product of x cubed and 5
Write a verbal expression for each algebraic expression given
below.
4)
5)
1
n4
2
6)
3 x 2  5y
4m  5n
Write an algebraic expression or an equation for each verbal expression.
7. The measure of an angle is (5x)° . What is the measure of that angle’s complement?
Recall:
COMPLEMENTARY ANGLES:
Two angles that add up to 90
degrees
ANGLES:
8. A square has aSUPPLEMENTARY
side length of s. What is it’s
perimeter?
Two angles that add up to 180
degrees
Write an algebraic expression or an equation for each verbal expression.
9. Three plus the quotient of 7 and a number subtracted from two times the same number
10. Write an expression that best represents the amount of money in a bag of quarters, where q is
the number of quarters.
11. Lara wants to buy a Rock Band game that is on sale for 35% off the regular price. The regular
price of the game is p dollars. Which expression represents the sale price of the game?
a) 0.65p
b) p  0.35 p
c) p  35p
d) 0.35p
e) 0.35p  p
Evaluate the following expressions for a 
12. Vocabulary
2
bc  4a
3
, b  2.4 , and c  6 .
4
13. 0.5c  a
a
EVALUATE:
Recall:
to find the value of an
ORDER of OPERATIONS:
algebraic expression by
Parenthesis
substituting a number for each
variable and simplifying byExponents
using the order of operations
Multiple
Divide
Add
Subtract
Warm Up #1
Write the algebraic expression for each verbal expression.
1. Three less than twice a number.
2n – 3
2. The quotient of 3 and a number subtracted from 8.
8 – (3/x)
3. The sum of a number and five times two.
2(m + 5)
VOCABULARY
Term:
a single number/variable, or the product of several numbers and/or
variables separated from another term by a + or - sign in an overall
expression.
i.e. in 3 + 4x + 5yzw , the 3, 4x, and 5yzw are all terms.
Like Terms
4x and 7x
Not Like Terms
-2x and 9x2
-3a2b3 and 3a2b3
5xy and 7x2y
ac and ac
3abc and 3ab
If possible, simplify each expression by
combining like terms:
1. 4x + x – 3
2. 13x2 + 4 + 10x2
3. 12x2 + 6x4
For all real numbers, a, b and c,
a(b + c)=ab + ac, and (b + c)a = ba + ca
 You multiply a single term over a close set of terms
 To simplify the expression 2(x + 4), you must distribute the 2 to
EVERYTHING inside the parentheses.
2( x + 4) = 2*x + 2*4 =2x+8
Simplify.
4. 8(y + 2x) + 7y
Evaluate the expression if x = 1 and y =
4

1
3
Simplify.
5. 6(a – b) – a + 3b
Simplify.
1
6. 18 – (2x – 4xy) – (x + y)
2
Write an expression for the following problem, then simplify.
7. The length of a rectangle is 8x – 4 meters long and the width is 4x
meters long.
8x – 4
4x
a) Write an expression to represent the
perimeter of the rectangle in simplest form.
b) Evaluate the perimeter if x = 3.
Write an expression for the following problem, then simplify.
8. Find the perimeter of the square in terms of r, the radius of the
circle.
r
Write an expression for the following problem, then simplify.
9. Find the perimeter of the image, in simplified form, in terms of x.
2x
x2  3
2x 2  5x  1
3x  4
Write an expression for the following problem, then simplify.
10. Two angles are complementary. The larger angle
measures 6 more than three times the smaller angle.
Write an equation that can be used to find out the
measures of both angles, then simplify.
Write an expression for the following problem, then simplify.
11. The triangle has angles that measure x°, 3x+10°, and
2(x + 2) °. Write an equation to represent the sum of
the angles in the triangle.
Recall:
ANGLES OF A TRIANGLE:
all three angles in any triangle
have a sum of 180 degrees
Warm Up #2
1 .By hand simplify the following.
1
1 3
1
x  x
2
6 4
3
– (1/4)x + (1/2)
2. What is the difference between complementary and
supplementary angles?
Complementary = 90 degrees
Supplementary =180 degrees
3. How many degrees are in triangle? circle?
Triangle = 180 degrees
Circle = 360 degrees
If a = b, then a can be substituted for b
in any equation
 Every time an a appears in an equation or expression, you can
replace a with b without changing the value of the expression or
equation
Let a  2
If 3a  4, then 3 * 2  4
For all real numbers, a, b and c,
a(b + c)=ab + ac, and (b + c)a = ba + ca
 You multiply a single term over a close set of terms
 To simplify the expression 2(x + 4), you must distribute the 2 to
EVERYTHING inside the parentheses.
2( x + 4) = 2x + 8
For all real numbers, a, b and c, if a = b
then a+ c =b + c.
 If two expressions are equal to each other then you can add the same
number to both expression without change their equivalence.
3x-5 = 14
3x-5+5=4+5
For all real numbers, a, b and c, if a = b
then a – c =b – c.
 If two expressions are equal to each other then you can subtract the
same number to both expression without change their equivalence.
2x+5 = 17
2x+5-5 = 17-5
For all real numbers, a, b and c, if a = b
then ac =bc.
 If two expressions are equal to each other then you can multiply the
same number to both expression without change their equivalence.
x
=12
2
x
2 * =12 * 2
2
For all real numbers, a, b and c, if a = b
a b

then c c
 If two expressions are equal to each other then you can divide the
same number to both expression without change their equivalence.
3x  15
3x 15

3
3
Name the property that is illustrated in each equation.
1.
14  6t  14  4  14
4.
7t 21

7
7
2.
3.
4
5  x  20  5
5
5.
2(5  x)
2(5) + 2(x)
6.
18x  9  9  36  9
2(5x  12)  23
5x(2)  12(2)  23
Simplify the equation. Justify each step by naming the property
being used.
7. Equation:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
2(7  3t)  4
14  6t  4
14  6t  14  4  14
6t  18
6t 18

6
6
t  3
Algebra Tile Legend
= x or the
variable
= -x or the
opposite
of the
variable
= 1 unit or a
constant
= -1 unit or a
constant
Solve the following equations for the variable given. Show your
process using Algebra Tiles and Justify your answer by identify
the property you used.
8.
x+6 = 2
Solve the following equations for the variable given. Show your
process using Algebra Tiles and Justify your answer by identify
the property you used.
9.
4x =12
Solve for the Variable.
10.
10
2
x
11.
4.5x  27
VOCABULARY
Reciprocal:
1
for a real number a  0 , the reciprocal of is . The product of reciprocals
a
is 1.
i.e. given 5 its reciprocal is
Solve for x
5
x  10
9
1
5
Define a variable, write an equation, solve and check
your answer. Write you answer in a complete
sentence.
13. This year Hays High School had 578 sophomores enrolled.
This is 89 less than the number enrolled last year. Write and
solve an equation to find the number of sophomores
enrolled at Hays High School last year.
Define a variable, write an equation, solve and check
your answer. Write you answer in a complete
sentence.
14. Bob Waters sells boats. He gets to keep one-eighth of his
sales as a commission. How much must he sell in order to
earn $10,000 in commission?
Warm Up #3
1. What operation does the word quotient indicate?
Divide
2. What is the coefficient of the term 3xy2?
3
3. Simplify the following: 3(x + 1) – 2(x – 2)
3x+3 – 2x+4
x+7
BOOLEAN
ALGEBRA
1 = TRUE
0 = FALSE
Solve for the Variable. Use your calculator to
check your answer.
1.
6  4  2x
2.
4  7x  3
Solve for the Variable. Use your calculator to
check your answer.
3.
n
22
7
4.
x7
 2
4
5. The measures of the angles of a triangle are x  , (x  5),(2x  3) .
Solve for x and then find the measure of each angle.
6. Ashlyn’s scores on her last 4 Algebra tests were 82, 86, 91 and 96. What
does Ashlyn need to make on her fifth test if she needs to make a 90
average?
VOCABULARY
Consecutive Integers:
are integers that follow each other in order. They have a difference of 1
between every two numbers.
If n is an integer, then n, n+1, and n+2 will be consecutive integers
Even consecutive integers:
are even integers that follow each other. They have a difference of 2
between every two numbers.
If n is an integer, then 2n, 2n+2, and 2n+4 will be even consecutive
integers.
Odd consecutive integers
Odd consecutive integers are odd integers that follow each other. They
have a difference of 2 between every two numbers.
If n is an integer, then 2n+1, 2n+3, and 2n+5 will be odd consecutive
integers.
7. The sum of two consecutive integers is 47. Find the integers.
8. The sum of two consecutive odd integers is 36. Find the integers.
Homework
page 96
#2-18 even
48-52 even
Warm Up #4
1. What is the Multiplication Property?
When you multiply both sides of an equation by the same
term.
2. Solve for the variable:
m = –16
3. Simplify:
4m
10 
2
2 3 1
27
  
7 5 2
70
Solve for the Variable.
1.
2( x  3)   4
Solve for the Variable.
2.
7  4x (2 x)
Solve for the Variable.
3.
2(x  3)
 12
5
Solve for the Variable.
4.
1
4  (3n  9) 6n
3
Solve for the Variable.
5.
4n  1
1

3
2
Draw a picture for each, then solve.
6. The length of a rectangle is three times the width. The
perimeter is 96 cm, what is the area of the rectangle?
Set up an equation, then solve.
7. An angles is 30 degrees more than twice its compliment.
Find the measure of both angles.
Draw a picture for each, then solve.
8. <A is supplemental to <B . If <A measures 4(4x+5)° and
measures <B 2(x+8)°, find the measure of largest angle.
Warm Up #5
1. List 2 terms that are alike and 2 terms that are not
alike?
~varies~
2. What is the coefficient of –xyz ?
–1
3. Write an equation that is represented for the
following algebra tiles.
Algebra Tile Legend
=x
= -x
=1
–
= -1
2x – 6 = – 3x + 4
Solve the following problems.
1.
4  2n  5  n
Solve the following problems.
2.
12h  8  3h  46
Solve the following problems.
3.
6x  4
2
 x 6
Solve the following problems.
4.
2( x  3)  5  3( x  1)
5.
Solve the following problems.
1
2(n  3)  ( 6n  32)
2
Solve the following problems.
6. You have two consecutive odd numbers. Twice the greater
number is 13 less than three times the lesser number. Find
the integers.
Solve the following problems.
7. A house-painting company charges $376 plus $12 per hour.
Another painting company charges $280 plus $15 per hour.
a) How long is a job for which both companies will charge the
same amount?
b) What will that cost be?
Warm Up #6
1. What is the difference between 2x² and (2x)² ?
(2x)² = 4x²
2. Is the following true or false?
True
1
1
1
1

   ( )
2
2
2
2
3. Put the following in order from least to greatest.
1 6
1 2
, , .3, ,
3 7
6 9
1 2
1 6
, ,.3, ,
6 9
3 7
VOCABULARY
All Real Numbers/ Infinite Solutions:
When the variables cancel out and there is a true statement.
No Solution:
When the variables cancel out and there is a false statement
Solve the following problems.
1.
3x  8  2( x  4)  x
2.
Solve the following problems.
1
2 x  3( x  1)  3 x  (4  4 x)
2
3.
Solve the following problems.
1
1
(14 x  6)  4  (24 x  24)  ( x  1)
2
3
4.
Solve the following problems.
1
1
(6x  18)  (9x  27)
2
3
Warm Up #7
1. What is the difference between infinite solutions,
no solutions, and one solutions?
Infinite- any number with make the equation true
No Solution- there is no real number that will solve the
solution
One Solution- there is only one answer that can solve the
equation.
 Paper notes/activity
Warm Up #8
1. Show two different way to start solving the
following equation.
3(x – 1) = 15
3x – 3 = 15
3(x – 1) =15
3
3
Write a let statement and equation for the following
situations and then solve as directed.
1. Allison’s cell phone bill for last month was $110. This
includes a $60 monthly fee plus the cost of 500 texts. What is
the cost of sending one text message that month?
Write a let statement and equation for the following
situations and then solve as directed.
2. Jessica goes to the State Fair of Texas. The cost of admission is $14.00. The cost
for each food coupon is $0.50.
a) Write let statements and an equation to represent Jessica’s total cost at the fair.
b) If Jessica buys 40 food coupons, how much does she spend total at the Fair?
c) If Jessica spends $26.50, how many food coupons did she buy?
d)Jessica has $31.00 to spend at the fair, how many food coupons may she buy?
Write a let statement and equation for the following
situations and then solve as directed.
3. Bulk Up Fitness Club has an enrollment fee of $225 plus $35 per month.
a) Write let statements and an equation that represents the total cost c for an individual
working out at this fitness club for m months
b) What is the cost of a one-year membership?
c) WWF contestant, Hulk Man, spent $1030 working out at Bulk Up. How many months
did he work out?
Write a let statement and equation for the following
situations and then solve as directed.
4. Jenna’s PAP Algebra grade after the first 6-weeks was 70%. Jenna wanted to
improve her grade, so she began to study at least one extra hour each day. As Jenna
tracked the number of extra hours that she studied, she realized her average went up by
2% for every 5 hours of extra study time.
a) Write an equation that represents Jenna’s total average as function of h hours.
B) What will Jenna’s average be after 300 minutes of extra study time?
c) If Jenna’s average is now 86%, how many extra hours did Jenna study?
Warm Up #9
Solve for the variable.
1.
3x  1
4
4
x =5
.
2.
m
 3
7
m = -21
JUST HW
Warm Up #10
Solve for the variable.
1.
32  30  2( x  4)
X= -5
2.
4
x  2  14
7
x = 21
VOCABULARY
Ratio- a comparison of two numbers by division.
i.e. a : b
a
b
when  b  0 
Proportion- states that two ratios are equivalent.
i.e. a c
b

d
when  b  0, d  0 
Cross Products-are equal to each other
i.e. a c
b

d
then ad  bc
Solve the following proportions for the variable.
1.
2.
5
8

2n 3n  24
m  9 m  10

5
11
Solve the following using ratios
3. The ratio of the angles of a triangle is 2:3:4. Find the
measure of the largest angle.
4. The length and width of a rectangle are in the ratio 3:2.
The perimeter of this rectangle is 73 cm. Find the length
and width of the rectangle.
VOCABULARY
Rate- a ratio of two quantities with different units
i.e. 34miles
2 gallons
Unit Rate- is when the second quantity is a unit of 1.
i.e.
34miles 17miles

2 gallons 1gallons
The unit rate is 17 miles per gallon.
VOCABULARY
Direct Variation - is a linear relationship between two variables, x and y,
that can be written in the form , where k is a nonzero constant
y  kx; k  0
*Proportions are a type of direct variation.
*In direct variation as one quantity increases (decreases) so does the
other quantity.
*In the form y = kx nothing is being added or subtracted to the kx
Identify if each of the following is an example of direct
variation. If so, identify the constant of variation.
5.
y  4x
6.
2x  y  10
7.
3x  5y  0
Answer the following questions.
8. Y varies directly with x, and y is 84 when x is 16.
a) What is the constant of variation?
b) Write a direct variation equation to represent this situation.
9. If y is directly proportional to x and y = 8 when x = 10, what is the value
of y when x = 5?
Answer the following questions.
10. Look at the table below. If y varies directly with x, what is the constant
of variation?
11. The amount of chlorine, c, needed for a swimming pool varies directly
with the amount of water, w, needed to fill the pool. If 16 units of
chlorine are needed for every 1250 gallons of water, write an equation
that represents this situation.
Warm Up #11
Set up and solve a proportion for each problem.
1. In Mrs. Willis’s first period Algebra I class, the ratio of boys to girls is 3:4.
If there are 28 students in the class, how many girls are there in the
class?
3boys:4girls= 7 total
x girls = 28 total  16 girls
Direct Variation – A Chili Situation
Inverse Variation – A Chilly Situation
•A Linear Relationship
between two variables; x and y
• If x increases,
then y Increases
•If x decreases,
then y decreases
•
y  xy
y
k
x
What is k?
•Constant of
proportionality
•A Relationship
between two variables;
x and y
• If x increases,
then y decreases
•If x decreases,
then y increases
k
•y  k k  xy
y x
x
1. The number of hours, h, it takes for a block of ice to melt varies inversely
as the temperature, t. If it takes 2 hours for a square inch of ice to melt at 65º,
find the constant of proportionality, k.
2. Do the tables below demonstrate a relationship of inverse variation? Explain why or
why not.
Table A
Table B
Table C
x
y
x
y
x
y
1
30
1
20
2
6
2
15
2
10
3
9
3
10
4
5
6
18
Product Rule of Inverse Variation
The product of an x value and it’s corresponding y is equal to the
product of any other x and y value in the same situation.
x1 y1  x2 y2
x1 y1
3.The number of chairs on a ski lift is inversely proportional to the distance
between them. A lift has 70 chairs when they are spaced 24 m apart. If 80
evenly-spaced chairs are used on the lift, how much space will be left
between them?
4. Which of the following equations shows a relationship in which y is
inversely proportional to x?
A
B
C
D
E
I only
II and III only
I, II and III
II only
I and II only
5. The time it takes to fly from LA to New York varies inversely as the speed
of the plane. If the trip takes 6 hours at 900 km/hr, how long would it take at
800 km/hr?
6. The weight of an object on the moon varies directly as its weight on Earth.
With all of his gear on, Neil Armstrong weighted 360 pounds on Earth. When
he became the first person to walk on the moon, on July 20, 1969, he
weighed 60 pounds in moon weight. If Tara weighs 60 pounds on Earth, how
much will she weigh on the moon?
7. The length of a violin string varies inversely as the frequency of its
vibrations. A violin string 10 inches long vibrates as a frequency of 512
cycles per second. Find the frequency of an 8 inch long string.
8. In kick boxing, it is found that the force, f, needed to break a board, varies
inversely with the length, l, of the board. If it takes 5 lbs of pressure to break
a board 2 feet long, how many pounds of pressure will it take to break a
board that is 6 feet long?
9. On a balanced seesaw, weight varies inversely as the distance from the
fulcrum to the weight. The diagram shows a balanced lever. If the child
weighs 80 pounds, approximately how far away from the fulcrum should she
sit in order to balance the seesaw?
Warm Up #12
Set up and solve a proportion for each problem.
1. What is the formula for direct variation?
y = kx
2. What is the formula for inverse variation?
y=k
x
3. What does k represent?
Constant of proportionality
Percenta ratio that compares a number to 100
i.e.
25
25% 
100
The Percent Proportion-
part (is)
%

whole (of) 100
Set up and use the Percent Proportion to find
the missing value.
1. What is 40% of 75?
2. 75% of what number is 33?
3. 30 is what percent of 50?
4. What percent of 200 is 5?
VOCABULARY
Dimensional Analysis – method for comparing the dimensions in a
problem to find relationships between the quantities.
Answer the following questions.
5. How many seconds are in a day?
6. You're throwing a pizza party for 15 and figure each person might eat 4
slices. You call up the pizza place and learn that each pizza will cost
you $14.78 and will be cut into 12 slices. How much is the pizza going
to cost you?
homework
 Page 117 #4-9 all
 Page 118#20-24 all
 Page 119 #44-50 even
Percent Changeis increase are decrease given as a percent of the original
amount.
amount of increase or decrease
percent change 
*100%
original amount
Percent increase- a growth in amount
Percent decrease- a reduction in amount
Find the percent of change. Is it an increase or decrease?
1. From 25 to 49
2. From 50 to 45
Discount- an amount by which an original price is reduced
total  original  discount
Markup- an amount by which a wholesale cost is increased
total  original  markup
Commissionis money paid to a person or company for making a sale.
total pay  base salary  commission
3. Admission to a museum is $8. Students receive a 15%
discount. How much do students pay for admission?
4. Laura purchased a daily planner for $32. The wholesale cost
was $25. What was the percent markup?
5. Jeff receives a 7% commission for selling a house. If his
commission was $13,475, what was the selling price of the
house?
6. A car salesman earns a 3.5% commission on every car he
sells. What is his commission for selling a car for $25,000?
7. A 15% tip is automatically put on the bill at a restaurant
when there are more than 6 people in the party. If the eight
members of the Bernardo family eat dinner together at the
restaurant and their bill totals $156.00 before the tip is
added, what will the Bernardo family’s total bill be after the
tip?
homework
 Page 135 #2-8 even
 Page 141 #24-40 even
Warm Up #11
Set up and solve a proportion for each problem.
1
2
1. Marlene can walk her dog, Bruno, for miles
in 40 minutes.
2
Determine how far she would walk Bruno in one hour.
3.75 miles
2. If Marlene increases her rate to 3 miles in 40 minutes. At this new rate,
how far will she walk in 2 hours?
9 miles
What is the difference between the two?
5x  10
5x  10
VOCABULARY
Equations:
• Shows two expression that are equal
• Has an equal sign
• Have a limited number of solutions
• 5x = 10
Inequalities:
• Shows some type of relationship between two expressions
• Have an inequality sign
• Have an unlimited number of solutions
• 5x < 10
Graphing Inequalities:
If the endpoint IS a solution….
then a closed circle is used to represent that number
If the endpoint IS NOT a solution….
then an open circle is used to represent that number
Interval Notation:
If the endpoint IS a solution….
then a bracket is used to represent that number
If the endpoint IS NOT a solution….
then a parenthesis is used to represent that number
Sign
Less than
Graph
Internal
Notation
( )
Less than and
equal to
<
>
≤
Greater than
and equal to
≥
[ ]
Greater than
( )
[ ]
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
1.
x2
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
2.
x  1
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
3.
8  x 3
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
4.
4  3  x
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
5.
x  4  2x
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
6.
2x  4  x  3
Graph:
Interval Notation:
Identify the inequality that goes with each of the following
 Less than
 Maximum
 Greater Than
 More Than
 Minimum
 No more than
 At most
 No less than
 At least
Define a variable, write an inequality, and solve each problem.
7. Thirty-six is at least the sum of 20 and a number.
Interval Notation:
Graph:
Define a variable, write an inequality, and solve each problem.
8. The difference of a number and 4 is at most -5.
Interval Notation:
Graph:
Define a variable, write an inequality, and solve each problem.
9.Sami can spend at most $30. She has already spent $14. Write,
solve, and graph an inequality to show how much more she can
spend.
Interval Notation:
Graph:
Define a variable, write an inequality, and solve each problem.
10. Mrs. Lawrence wants to buy an antique bracelet at an auction. She
is willing to bid no more than $550. So far, the highest bid is $475.
Write and solve an inequality to determine the amount Mrs.
Lawrence can add to the bid.
Interval Notation:
Graph:
Define a variable, write an inequality, and solve each problem.
11. Josh wants to try to break the school bench press record of 282
pounds. He currently can bench press 250 pounds. Write and solve
an inequality to determine how many more pounds Josh needs to lift
to break the school record.
Interval Notation:
Graph:
HOMEWORK
Bookwork
Page 177-179
#8-28 even
#36-39 all
#43-45all
Warm Up
Write an inequality for each statement.
X >0
1. X is a positive number
X≥0
2. X is a nonnegative number
X<0
3. X is a negative number
-∞< x<∞
4. X is a real number
or all real numbers
1.16 Solving MultiStep
Inequalities
Plug in values for x.
2x  10
2x  10
What do you notice?
What is different about the two inequalities?
Can you come up with the rule?
RULE:
When solving for a variable in an
inequality and you multiple or
divide by a negative, you MUST
flip the inequality!
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
1.
2  3x  7
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation.
2.
2 x  5  5x  4
Graph:
Interval Notation:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
3.
Interval Notation:
6m  (2m  3)  13
Graph:
Solve and graph the following inequalities. Then write the
solutions to the inequalities in interval notation
4.
Graph:
2x  7
9
5
Interval Notation:
Define a variable, write an inequality, and solve each problem.
5. Four times a number is less than the sum of two times the
same number and six. Find the number.
Interval Notation:
Graph:
Define a variable, write an inequality, and solve each problem.
6. Three more than one-third of a number is no more than nine.
Interval Notation:
Graph:
HOMEWORK
Bookwork
Page 191-193
#18-26 even
#38-48 even
#50-54all
1.17 Compound
Inequalities
Define a variable, write an inequality, and solve each problem.
6. Three more than one-third of a number is no more than nine.
Interval Notation:
Graph: