Unit 1

Solving Equations, Inequalities & Absolute Value

Standard 1a Solve one-step equations Standard 1b Solve multi-step equations

Standard 1c Solve equations with distributive property Standard 1d Explain the steps to solve an equation Standard 2 Write linear equations Standard 3 Solve literal equations (solve for y)

Standard 4a Solve linear inequalities

Standard 4b Solve compound linear inequalities Standard 5a Write linear inequalities from a graph Standard 5b Write linear inequalities from a word problem

Standard 6 Solve absolute value equations

1 . 1 Solv ing One & Two Step Equations NOTES A-REI 1 : Expla in each step in solv ing a simple equation as fol lowing from the equal ity of numbers asserted at the previous step, start ing from the assumption that the orig inal equation has a solut ion. Construct a v iable argument to just ify a solut ion method.

1 . 1 1 .2 1 .3 1 .4 1 .5

Recognize Addition

Recognize Subtraction

Recognize Multiplication

Inverse Operations

One Solution

Only one number will work as a solution.

4𝑥 = 2𝑥 + 6

No Solution

No possible number works as a solution.

2𝑥 + 10 = 2𝑥 + 20

Inf in ite Solutions

All numbers work as a solution.

−3𝑥 − 4 = −3𝑥 − 6+ 2

1 . 1 Solv ing One & Two Step Equations Assignment

Solve each equation. Explain your steps in words.

Work Explain

1. y – 7 = 8

2. w + 14 = –8

3. –22 = 11v

4. – !! = –13

5. !!𝑞 = –4

6. 4𝑛 + 8 = 12

7. 3𝑐 − 4 = 5

8. !!+ 3 = 11

9. 12 = 7−𝑚 10. !!− 5 = 3

11. 15𝑥 − 8 = 14𝑥 + 13 12. 9𝑛 − 7 = 5𝑛 + 5

13. 4𝑧 − 15 = 4𝑧 + 11 14. −7𝑎 + 9 = 3𝑎 + 49

15. Find the perimeter of the square.

16. A video game store has sold 3/5 of their X-Box games, but 10 were returned. Now the store has 62 X-Box games. How many were there originally? 17. Len read ¾ of a graphic novel over the weekend. Monday, he read 22 more pages. If he has read 220 pages, how many pages does the book have?

1 .2 Solv ing Equations with Distr ibut ive Property NOTES

The Distributive Property

3( b + f + s)

-2 (x -3 + y)

Like Terms SAME variable & SAME exponent

3x +5x = 8x 2x –x +2y = x + 2y

3x + 4xy -5x + 3xy

12(x+3) -5x

Simplify the expression: 𝑏! + 13𝑏 + 13+ 2𝑏! 7𝑎 + 4− 6𝑎! − 2𝑎 2 3𝑥 − 𝑦 + 5𝑥 Solve for the variable: 5𝑥 + 5 = 3 5𝑥 − 4 − 10𝑥 3 2𝑏 − 1 − 7 = 6𝑏 − 10 6 𝑦 − 5 = 2(10+ 3𝑦) Word Problems Consecutive Integer The sum of three consecutive integers is 372. What are the integers? A Surface Pro 3 is $799. You have saved $300 and you get paid $15 an hour to mow your neighbor’s lawn. How many hours will you have to work so you have enough money to buy the Surface Pro 3? Show your linear model used to solve the problem.

1 .2 Solv ing Equations with Distr ibut ive Property ASSIGNMENT

Solve the equation. Explain your steps.

Work Explain

1. 2𝑤 + 𝑤 = −6

2. 12− 4𝑤 + 3 = −1

3. 5𝑑 − 3+ 2𝑑 = 4𝑑 + 9

4. 6.8𝑡 − 10− 3.2𝑡 = 3𝑡 − 1

5.  14𝑑 − 43 = 6𝑑 − 13− 7𝑑

6.  3 x+ 5 = 23

7. 14− 4𝑧 = 2(17− 𝑧) 8. −4 2w− 5 = 3w− 13

9. 11w+ 2 w+ 1 = 5w− 6 10. 10+ 2 2a+ 1 = 7a− 3

11. Find the perimeter of the square. 12. A company that replicates DVDs spends $1500 per day in building overhead plus $.80 per DVD in supplies & labor. If the DVDs sell for $1.59 per disk, how many DVDs must the company sells each day before it makes a profit? 13 . Fran is making bracelets. Each bead to make the bracelet costs $.89 and the fishing line she uses to string the beads costs $2.99. She wants to know how many beads she should put on her bracelet so the bracelets cost $9.22. 14. The ages of three brothers are consecutive integers with the sum of 96. How old are the brothers? 15 . Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car Keisha wants to buy costs at least $5440. How many hours does Keisha need babysit to buy the car?

1 .3 Literal Equations NOTES Why do we solve literal equations?

• An equation has to be solved for y to graph in slope-intercept form.

𝑦 = 3𝑥 + 4 instead of −3𝑥 + 𝑦 = 4

• Substitution in a table is easier if you solve for the variable first, then substitute. If you use 𝐷 = 4.5𝑡 then you have to solve for t every time. By using an equation already solved for t : !

!.!= 𝑡, then all you do is

substitute. One-Step

Solve for r:  𝐷 = 𝑟𝑡 Solve for h: 𝑉 = 𝑙𝑤ℎ Solve for a: 𝑃 = 𝑎 + 𝑏 + 𝑐 Two-Step

Solve for b: 𝐴 = !!

! Solve for w: 𝑃 = 2𝑙 + 2𝑤 Solve for y:  3𝑥 − 4𝑦 = 12

Multi-Step

Solve for R: 𝐼 = !! Solve for F:  𝐶 = !

!(𝐹 − 32)

D t 1 2 3

A-CED 4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

1 .3 Literal Equations ASSIGNMENT

Write the equation in function form. Explain your steps.

1.  𝑦 + 10𝑥 = 3 2. 3𝑦 = 9𝑥 + 27

3. 15𝑥 − 5𝑦 = 20 4.  8𝑥 + 𝑦 − 4 = 0

5. 2𝑥 + 5𝑦 = 20 6. Solve  𝑦 = 𝑚𝑥 + 𝑏  for  𝑚.

7. Solve  𝐹 = 𝑚𝑎  for  𝑚. 8. Solve  𝑎𝑥 − 𝑏𝑦 = 𝑐  for  𝑥.

9. Solve  𝐶 = 2𝜋𝑟  for  𝑟. 10. Solve  𝐼 = !!  for  𝑅

11. You are decorating a clean soup can to make a pencil holder. You are going to glue yarn around the top and bottom of the can. The total amount y of yarn (in inches) you need is given by the equation y = 4πr, where r is the radius of the can. a. Solve the equation for r

b. What is the radius of the can if you need 37.68 inches of yarn? Use 3.14 for π.

Solve the equation: 12. !

!= −4 13. 6(3𝑎 + 1− 30) = 3(2𝑎 − 4) 14. !

!𝑞 = −15

15. Find the value of x so that the figures have the same perimeter.

1 .4 Linear Inequal it ies NOTES

Solving Inequalities Just like solving an equation EXCEPT when you multiply or divide by a negative FLIP the inequality sign. −2x+ 3 ≥ 13 Graphing Inequalities

Compound and

You can have a new Xbox & a new game

Compound Or

You can have a new Xbox or a new game

or

Solve −23 < 5𝑥 + 2 < 17

Solve 7𝑋 − 2 > −2  𝑜𝑟  4𝑥 − 3 ≤ −25

GRAPH

GRAPH

A-REI 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

1 .4 Linear Inequal it ies ASSIGNMENT Solve & Graph each inequality. Show you work when solving. 1. 17 > 8+ 𝑠 2. −2− 𝑔 ≤ 18 3. !

!< 11 4. 27 ≤ −3𝑛

5. 2+ 6𝑟 < −40 6. 6+ !

!≥ −10

7. 53 ≥ −61+ 5 8. 14>-5h+6h 9. 7𝑞 − 175 < 2(6− 5𝑞) 10. 2𝑑 < 10  𝑜𝑟  𝑑 + 5 > 16

11. 42 < 6𝑞 < 66 12. −12 ≤ 𝑎 + 12 ≤ −4 13. 7𝑐 < 7  𝑜𝑟  𝑐 + 3 > 7 15. !

!≤ 7  𝑜𝑟   !

!≥ 5

16. Solve for y: 2x-3y = 12 17. Solve for x: -3(x+5) -2 = 4(x-1) 18. Solve for m: 2𝑛 = !!

!

19. Benito has $6 to spend. A sundae costs $3.25 plus $0.65 per topping. How many toppings can he order? 20. How many bracelets can Caitlin buy for herself and her friends if she wants to spend no more than $22?

1 .5 Absolute Value Equations NOTES

In 2007, a telephone poll was conducted to determine the reading habits of people in the U.S. People in this survey were allowed to select more than one type of book. The survey had a margin of error of ±3%. This means that the results could be three points higher or lower. So, the percent of people who read religious material could be as high as 69% or as low as 63%.

How would you represent this margin of error in the percent of people who chose religious books with an absolute value equation?

The distance between 63 & 66 on a number line is the same as the distance between 63 & 66.

𝑓 + 5 = 17

Positive Negative 𝑏 − 1 = −3 2𝑥 − 1 = 7

The average January temperature in a northern Canadian city is 1℉. The actual January temperature for that city may be about 5℉ warmer or colder. Write and solve an equation to find the maxiumum and minmum temperatures.

A-REI 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Can the absolute

value of a number

ever be negative?

1 .5 Absolute Value Equations ASSIGNMENT Solve the equation: 1. 𝑛 + 7 = 5 2. 4𝑛 − 1 = −6 3. 2𝑡 − 4 = 8 4. 8𝑤 + 5 = 21 5. !

!𝑎 − 3 = 9 6. 2 −2𝑦 + 6 = 6

7. The circle graph at the right shows the results of a survey that asked, “How likely is it that you will be rich some day?” If the margin of error is ±4%, what is the range of the percent of teens who say it is very likely that they will be rich? 8. The 4x400 relay is a race where 4 runners take turns running 400 meters, or one lap around the track.

If a runner runs the first leg in 52 seconds, plus or minus 2 seconds, write an equation to find the fastest and slowest times.

If the runners on the second and third legs run their laps in 53 seconds plus or minus 1 second, write an equation to find the fastest and slowest times.

Suppose the runner of the fourth leg is the fastest on the team. If he runs an average of 50.5 seconds plus or minus 1.5 seconds, what are the team’s fastest and slowest times?

9. Which equation represents the second step of the solution process?

10. Solve for x: −3 𝑥 + 5 ≤ 4𝑥 + 2 11. Solve for y: 5x – 2y=16 12. Find the value of x so that the figures have the same perimeter.

13. Solve for v: !!!!

!"= !

! 14. Solve for r: !

!𝑟 = 18

15. At The Family Farm, you can pick your fruits and vegetables. The cost of a bag of potatoes is $1.50 less than ½ of the price of apples. Write and solve an equation to find the cost of potatoes.