Transcript Unit 1 PowerPoint

Bell Work:
Be ready to hand in your signed course syllabus, and have your
notebook out, open, and ready for notes!!!
Unit 1
EQUATIONS AND INEQUALITIES
Unit Essential Question
How do we translate real world applications of equations and inequalities into
mathematical expressions?
Lesson 1.2
EVALUATE AND SIMPLIFY ALGEBRAIC EXPRESSIONS
Lesson Essential Question (LEQ)
How do we use the properties of real numbers to help us simplify algebraic
expressions?
Evaluating Powers
Examples:
1) Evaluate: 43 and 54
2) Evaluate: −43 and −54
3) Evaluate: (−4)3 and (−5)4
PAY ATTENTION TO PARENTHESIS!!!
Evaluating Expressions
Examples:
4) Evaluate 5𝑥 − 3 + 2𝑥 2 if x = 3.
5) Evaluate −3𝑦 3 + 2𝑦 2 − 5𝑦 + 4 if y = -2
Make sure you evaluate the powers first!!!
Combining Like Terms and Constants
Examples:
6) Simplify: 8𝑥 − 9 − 12𝑥 + 7 − 𝑥
7) Simplify: 4ℎ3 − 2ℎ + 4 + 3ℎ3 + 3ℎ
Remember: You can only combine terms if they have the same variable(s) raised
to the same exponent!!!
Review Examples:
8) Evaluate −34 𝑎𝑛𝑑 (−2)4
9) Evaluate 𝑏 3 − 2𝑏 2 − 8𝑏 + 10 𝑖𝑓 𝑏 = −4
10) Simplify 8𝑥 − 4 − 9𝑦 + 10 − 𝑦 + 3𝑥 + 4
11) Simplify
3
4
8𝑥 2
− 32𝑥 − 8 −
7
(20𝑥
5
− 15 + 5𝑥 2 )
Homework:
Pages 13 – 15 #’s 3-31 odds, 35, 41, 42, 50, 57, and 59
Bell Work:
1) Simplify: 9𝑥 − 3 + 𝑥 3 − 9𝑥 − 4 − 4𝑥 3 + 2𝑥 2
2) Evaluate the simplified expression from number 1 if x = - 4
Lesson 1.3
SOLVING LINEAR EQUATIONS
Lesson Essential Question (LEQ):
How can we represent word problems as linear equations?
THE SUPER EASY ONES!
Solve the following equations:
Examples:
1) 𝑥 + 9 = −3
2) 5𝑥 = −130
3) 18 =
𝑥
2
THE KINDA EASY ONES!
Solve the following equations:
Examples:
4) −2𝑥 + 12 = 32
5)
𝑥
3
6)
3𝑥−2
5
− 18 = −21
=5
MEDIUM LEVEL EQUATIONS
Solve the following equations:
Examples:
7) 5𝑥 − 8 + 4𝑥 − 4 = 21
8) −3 − 2 4𝑥 − 5 + 𝑥 = 35
9) 4 1 − 2𝑥 − 3𝑥 − 9 = 46
Totally Sweet Equations:
Solve the following equations:
Examples:
10) 3𝑥 − 9 = 2(5𝑥 − 22)
11)
1
𝑥
5
1
2
2
3
− = 𝑥+3
12) −9 + 4
1
𝑥
2
+
15
4
= −3 2x − 5 + 8x − 9
Remember!!!
It is possible for an equation to have:
NO SOLUTION, if the equation ends as a false statement.
Or
ALL REAL NUMBERS as the solution, if the equation ends as a true statement.
Extra Practice Problems…
Solve the following equations.
13)
4𝑥−2
3
14)
2 3𝑥−1
15
=
2𝑥−4
4
=
1+2𝑥
5
Homework:
Pages 21 – 23 #’s 11 – 25, 33 – 53, 63 – 67 odds only, then 68 – 71 all.
Bell Work:
Solve the equations below.
1) 5𝑥 − 2 3𝑥 − 5 = 2𝑥 + 3 3𝑥 − 5 − 15
2)
2 4𝑥−5
2
3)
1
𝑥
2
= −2(5 − 2𝑥)
− 15 =
1
𝑥
5
+9
Bell Work Continued!!!
4) Levi and Emily sell luxury cars for a living. Levi has a base salary
of $48,000 and earns 5% commission. Emily has a base salary of
$64,000 and earns 3% commission. At what point will the two be
earning the same amount?
5) It takes Trish 10 minutes to wash a car and it takes Jacob 12
minutes to wash a car. How long will it take the two of them to wash
22 cars if they work together???
Lesson 1.4
REARRANGING FORMULAS AND EQUATIONS
Lesson Essential Question:
Why is it necessary to rearrange a specific formula to solve for a different variable?
Important…
Why would we want to rearrange formulas or equations?
Examples:
1) Solve 𝐴 = 𝑃𝑟𝑡 for t.
2) Solve 𝐴 =
1
𝑏ℎ
2
3) Solve 𝐴 =
1
4
for h.
𝑎 + 𝑏 𝑐 for b.
Examples:
4) Solve 4𝑥 − 2𝑦 = 18 for x.
5) Solve
2
𝑥
3
+ 6𝑦 = 30 for y.
6) Solve
2
𝑥
3
3
− 𝑦
7
= −2 for x.
Examples:
7) Solve 4𝑎𝑏 − 3𝑎 = 12 for a.
8) Solve 5𝑥𝑦 + 10𝑦 = 20 for y.
9) Solve 𝑎𝑏 − 𝑏 = 3 for b.
10) Solve 2𝑥𝑦 = 3𝑥 + 4𝑦 for x.
Homework:
Pages 30-32 #’s 6, 7 – 25 odds, 28 – 32 all, 35, 37, and 39
Bell Work:
1) Solve.
2
𝑥
3
1
5
− =
2) Solve for x.
1
𝑥
1
𝑥
2
−
1
4
= 3 + 2𝑦
3) Solve for w. 𝑃 = 2𝑙𝑤 + 2ℎ𝑙 + 2ℎ𝑤
Lesson 1.5
PROBLEM SOLVING STRATEGIES
Lesson Essential Question:
What are some of the strategies used to solve word problems?
Examples:
1) A car used 16 gallons of gasoline and traveled a total distance of 460 miles.
The car’s fuel efficiency is 30 mpg highway and 25 mpg city. How many
gallons of gasoline were used on the highway?
2) You are hanging four championship banners on the gym wall. The banners
are 8 feet wide and the wall is 62 feet long. There needs to be an equal
amount of space between the ends of the walls and the banners. How far
apart should the banners be spaced?
Group Work:
Pages 37 – 39 #’s 3 – 15 odds, 16 – 26 all, 29 – 31, 33
Bell Work:
1) Solve for x.
2
3
18 − 9𝑥 =
3
𝑥
2) Solve for x. 9 − = 6𝑦
3) Page 39 # 32
3
(10𝑥
5
− 40)
Bell Work:
1) The sum of three numbers is 126. The second number is
four less than three times the first. The third number is five
more than the second. Find the three numbers.
2) Write the formula for the area of a square in terms of its
perimeter.
Examples:
3) Luke unloads 40 boxes from a truck that contains 8lb
boxes of green beans and 5lb boxes of asparagus. If the
total weight of the boxes he unloaded is 266lb, how many of
each box did he unload?
Example:
4) The combined age of Mr. Kelsey, Mr. Kelsey’s father, and Mr.
Kelsey’s grandfather is 176 years. His grandfather is three times as
old Mr. Kelsey, and his father is 26 years older than Mr. Kelsey. Find
each persons age.
Example:
5) You are laying tiles for a new bathroom floor. The tiles are 10
inches long, and the room is 11.375 feet long. You plan on laying 12
tiles, with equal space between each tile, but no space between the
tiles next to the wall. What will be the space between each tile in
terms of inches?
Example:
6) Kylan runs 24.5 miles in 3 hours and 30 minutes. He sprints for
part of the race at 9 miles per hour and jogs part of the race at 2
miles per hour. How much time did he spend jogging?
How far did he travel while sprinting?
Example:
7) The combined weight of four wrestlers is 530 pounds. The
second wrestler weighs 40lb more than the first. The third weighs
22 less than the second. The fourth weighs 8 less than the first. Find
the weights of the four wrestlers.
Example:
8) Write the formula for the volume of a sphere in terms of its
circumference.
9) Prove that the new formula is correct by find the volume of a
sphere with a radius of 5cm using the new and original formula for
volume.
10) Solve for x.
1
𝑥
1
𝑦
− =𝑧
Example:
11) Write the formula for the area of a cube in terms of its surface area.
Quiz Tomorrow:
The quiz will be on:
Evaluating Expressions
Combining Like Terms and Constants
Solving Equations
Rearranging Equations
Equation Word Problems
Bell Work:
1) What are the four main inequality symbols?
2) Why do we graph solutions for inequalities on a number line?
3) What is interval notation?
Lesson 1.6
SOLVING LINEAR INEQUALITIES
Lesson Essential Question:
What are inequalities, how do we solve them, and how do we
represent the solutions?
Solving Linear Inequalities
Examples on board…
Homework:
Pages 45 – 47 #’s 25 – 51 odds 52, 53, 56, 60
Bell Work:
Solve the inequality, graph the solution on a number line, and write
the interval notation.
1) −22 < 4𝑥 − 2 5 − 𝑥 < 20
Lesson 1.7
ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
Lesson Essential Question:
What do the absolute value symbols do to an equation and inequality?
Absolute Value Equations
If 𝑎𝑥 + 𝑏 = 𝑐, then we can rewrite this as two separate equations:
1) 𝑎𝑥 + 𝑏 = 𝑐
2) 𝑎𝑥 + 𝑏 = −𝑐
Because absolute value shows the distance from zero on the number
line, we have to remember that there are two sides of the number
line, so it is possible to have two solutions for absolute value
equations.
Extraneous Solutions/No Solutions
It is possible that one of the solutions you found does not work
when substituted back into the original equation. These are called
extraneous solutions. It is imperative that you check your work to
insure that the answers you found work.
It is also possible that none of the solutions work. This will occur if
the equation is set equal a negative number.
Homework:
Page 55 #’s 3 – 41 odds
Bell Work:
Pop Quiz: Grab a small piece of paper from the front of the room and begin.
1) Solve the inequality and write the interval notation. 5𝑥 − 2 3𝑥 − 5 < 20
2) Solve the absolute value equation. 6𝑥 + 3𝑥 − 9 − 9 = 27
Bell Work:
Solve the following equations:
1
2
1) 5 − 𝑥 = 25
2) 3𝑥 − 15 = 2𝑥 + 10
3) 4𝑥 − 25 = −25
Absolute Value Inequalities:
BLUE TABLE ON PAGE 53!!!
You should memorize this!
Examples:
Homework:
Page 56 #’s 43 – 65 odds
Bell Work:
1) You plan to buy 18 picture frames for your friends. 8x10 photo frames cost
$10 each and 6x8 photo frames cost $7.50 each. If you want to spend at least
$150 but no more than $175 for the 18 picture frames, write and solve an
inequality to show the different possible combinations that you can purchase.
2) Emily wants to hang four pictures on her bedroom wall that is 12 feet and 1
inch long. Each picture that she hangs is 10 inches wide, the space between
each picture needs to be the same, and the distance between the end pictures
and the wall needs to be twice as much as between the pictures. What will be
the space between each picture? What is the space between the end pictures
and the walls?
Test on Thursday
You need to know:
Evaluating Expressions and Combining Like Terms
Solving Equations (watch out for ARN/NS)
Solving Inequalities (Interval Notation as well)
Word Problems
Rearranging Equations/Formulas
Absolute Value Equations/Inequalities
Examples:
Solve each absolute value equation:
1) 5𝑥 − 30 = 25
2) 2𝑥 − 19 − 17 = 20
3) −3𝑥 + 𝑥 − 10 + 14 = 0
Examples:
Solve the absolute value inequalities. Write the final inequalities in interval
notation.
4) 6𝑥 − 15 > 9
5) −40 +
1
𝑥
2
− 4 ≤ −35
6) −4 5 − 2𝑥 ≻ −28
Homework:
This will be collected tomorrow!
Pages 55 – 56 #’s 30 – 40 evens, 54 – 62 evens
Bell Work:
1) A single cup has a height of 6.5 inches. When another identical cup is stacked
on top of it, the total height of the two cups is 9 inches. Write an equation to
show the height of n cups. Use this equation to show the height of 25 cups in
terms of feet and inches.
2) Solve. −3𝑥 − 2𝑥 − 10 − 45 = 0
3) Solve for x.
10
𝑥
2
𝑦
− = 20𝑧
4) A piece of string 20 feet long is cut into 4 pieces. The first two pieces are the
same length, the third piece is 8 inches longer than the first two, and the fourth
piece is twice the length of the third. Find the length of the four pieces.
Bell Work:
1) Solve for y. 8𝑥
5
−
6𝑦
=𝑧
2) Solve. −4 3𝑥 − 18 > −48
3) Solve.
3
−
4
16 − 8𝑥 = 2(3𝑥 − 6)
Word Problems:
4) Jane is driving from Pittsburgh to Watsontown. She drives the 300 miles in 5 hours.
She was driving 75 mph on the highway and 25 mph in the city. How much time did she
spend on the highway? What was the total distance she traveled on the highway?
5) The sum of three numbers is 400. The second number is half the first number, and
the third number is 25 more than twice the second. Find the three numbers.
6) The height of 4 chairs stacked on each other is 4 feet 8 inches. The height of 9 chairs
stacked is 8 feet 10 inches. Write an equation to show the height of n chairs. Use this
equation to find the height of 15 chairs in terms of feet and inches.
Word Problems:
7) Five championship banners are being hung in the gymnasium. Each banner is
6 feet wide, and the space between each banner will be the same. The space
between the end banners and the wall needs to be 8 feet more than the space
between each banner. If the wall is 88 feet long, what will be the space between
each banner?
8) Solve for a.
6𝑏
𝑎
3
4
− = 9𝑐
1
2
9) Solve. 15 − 2 7 − 𝑥 = −1
Review Problems:
10) Solve. 3 − 2𝑥 = 4𝑥 − 15
11) Solve.
2
−
3
9𝑥 − 21 = −2(4 − 𝑥)
12) Solve.
1
−
2
4𝑥 − 2 − 8 < −3