Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 1: Solving Equations

Warm Up: Determine whether x = 8 is a solution to the equation

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34x −1 = −

12x + 9.

Modeling: Solving Equations by Inverse Operations

Directions: Find the value of x that solves each equation. In each case, first identify the operations that have occurred to x and then reverse them.

(a)

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x − 32

+ 7 = 23 (b)

What happened to x? What happened to x? Now reverse. Now reverse. Independent Task: What is the solution to the equation

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5(x − 3) + 2x = 4(x + 3)? Check your work. Group Task: Solve and check the following equations for x by applying inverse operations.

(a)

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7x −15 =1 (b)

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x + 24

= −2

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4(x +1) − 2 = −6

(c)

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−35x + 2 = 7 (d)

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5(x +1) + 46

= 4

Exit Slip: Solve and check the equation below for x by applying inverse operations.

Homework:

1. Solve each equation for x. Then, check to make sure the original equation ahs a true value for x.

(a)

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5(x +1) − 2x = 2(3+ x) (b)

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−32x + 2 = −4

2. Find the mistake in the work below and circle it. Then, to the side, show the appropriate work.

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9 − 6(x +1) = 2(x − 4) + 27

Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 2: Linear Word Problems Warm Up: The sum of a number and nine more than that number is 17. What is the number?

Hint: Create let statements and an equation to solve. Modeling:

(a) Twice the difference of a number and seven is three less than that number. What is the number?

(b) Dana and Abigail are comparing their ages. Dana is currently 36 years older than Abigail. Three years from now, Dana will be five times Abigail’s age at that point. How old is Abigail now?

Independent Task: Bill is seven times older than his niece, Cindy. In five years, Bill will be four times older than Cindy will be then. How old is Cindy now? Group Task:

1. Create an equation for the following: four times the sum of a number and six is equal to that number decreased by three.

2. Mia is eight years younger than Ryan. In two years, Ryan will be two less than twice Mia’s age. Find their current ages.

Exit Slip: Frank is 26 years older than Nina. In four years, Frank will be two more than five times Nina’s age. What are their current ages? Homework:

1. Six less than a number is equal to the quotient of that number and two. What is the number?

2. Jack is 20 years older than Diane. In two years, Jack will be three times Diane’s age. What is Jack’s current age?

3. A rectangular garden has a length that is six feet more than twice its width. It takes 120 feet of fencing to completely enclose the garden.

(a) Write an equation that could be used to find the width of the garden. Hint: How do you find the perimeter of a rectangle? Clearly define your variables. (b) Find the length of the garden algebraically.

Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 3: Consecutive Integers Warm Up: Three consecutive integers have the property that when you add the larger two, the result is nine less than three times the smallest of them.

(a) Fill out the table below. (b) Create an equation for the sentence above.

1st Integer 2nd Integer 3rd Integer

Modeling: Three consecutive even integers have the property that when the difference between the first and twice the second is found, the result is eight more than the third. Find the three consecutive even integers. Independent Task: In an opera theater, sections of seating consisting of three rows are being laid out. It is planned so each row will be two more seats than the one before it, and 90 people must be seated in each section. How many people will be in the third row? Group Task:

1. Find three consecutive odd integers such that the sum of the smaller two is three times the largest increased by seven.

2. The lengths of the sides of a triangle are consecutive even integers. What is the length of the longest side if the perimeter is 42 centimeters? Hint: draw a picture to represent the situation. Exit Slip: Find two consecutive integers such that ten more than twice the smaller is seven less than three times the larger. Homework:

1. Find two consecutive even integers such that their sum is equal to the difference of three times the larger and two times the smaller.

2. The sum of four consecutive integers is -18. What are the four integers?

3. Find three numbers that each differ by three, such that five times the largest integer is equal to three times the smallest increased by five times the middle.

Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 4: Unspecified Constants Warm Up: Solve each of the following equations for x. In (b), write your answer in terms of the unspecified constants, a, b, and c.

(a)

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5x + 3 = 33 (b) ax + b = c Modeling: Solving Equations with Unspecified Constants

(a)

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x + ab

− c = d Solve for x: (b)

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2(x − h) + k = 8 Solve for x:

Independent Task: The formula,

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C =59(F − 32), can be used to convert temperatures between

Fahrenheit and Celsius.

(a) Using the formula, convert 50° Fahrenheit to Celsius.

(b) Solve the formula for Fahrenheit, F, and then convert 50° Celsius into Fahrenheit.

Group Task:

1. Write the formula for the perimeter of a rectangle, P, in terms of its length, L, and width, W.

2. Rearrange the perimeter formula,

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P = 2L + 2W , so that it “solves” for L.

3. Determine the length of the rectangle when the perimeter is 20 feet and width is 4 feet.

Exit Slip: If the expression

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2xa

+ b = c is solved for x in terms of a, b, and c, then x =

(1)

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ac − ab2

(2)

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b + c2a

(3)

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ac − b2

(4)

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ab + c2

Homework: Solve the following equations for x.

(a)

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23x + y = z (b)

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a(x + b) − c = d (c)

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e(x + c)b

= 2

Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 5: Solving and Graphing Inequalities Warm Up: Determine the truth-value for each of the following statements.

(a)

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3x + 7 <1, when x = -2 (b)

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x 2 − 4 ≥ 0, when x = -2 Modeling: Given the linear inequality,

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8 − 2x >16, complete the following.

(a) Rewrite the expression as an equivalent expression using addition.

(b) Solve the inequality for x by applying inverse operations.

(c) Pick a number that is true based on your solution to (b) and show that it makes the original inequality true.

(d) Graph the solution to the inequality on the number line below.

Independent Task: Solve the inequality below using the properties of inequality and graph the final solution set on the number line provided.

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5x − 6 ≤ 24

Notes:  

Group Task:

1. Given a, b, c, and d are all positive, solve the following inequalities for x.

(a)

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ax + b ≥ cd (b)

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a(x + 2)b

> c

2. Two siblings, Edwin and Rhea, are both going skiing but choose different payment plans. • Edwin’s plan charges $45 for rentals and $5 per lift up the mountain. • Rhea’s plan was a bundle where her entire day cost $108.

(a) Set up an inequality that models the number of trips, n, up the mountain for which Edwin will pay more than Rhea. Then, solve the inequality for n.

(b) What is the greatest number of trips that Edwin can take up the mountain and still pay less than Rhea? Explain how you arrived at your answer.

Exit Slip: Solve the inequality below using the properties of inequality and graph the final solution set on the number line provided.

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2(5 − x) ≤12 Homework:

1. Determine the truth-value for each of the following statements.

(a)

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2x + 4 > 4x −1, when x = 1. (b)

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(x − 3)2 ≤ −3(x + 2) , when x = 3

2. Solve the inequalities below using the properties of inequality and graph the final solution set on the number line provided.

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6 + 4x <18

Unit 2 Review – Expressions, Equations, and Inequalities

Name: ___________________________ Algebra 1B Day 6: Compound Inequalities, Interval Notation, and Modeling Warm Up: Determine if x = 2 is part of the solution set to the following compound inequalities.

(a) x > -3 and x < 2 (b) x > -3 or x < 2 Modeling: Part A: On the number lines below, shade in all values of x that solve the compound inequality.

(a) x < 7 and x > -2 (b) x > 5 or x < -1 Part B: For each inequality, list one value that makes the compound true and one value that makes the compound false.

(a) x < 7 and x > -2 (b) x > 5 or x < -1 Part C: Express each inequality in interval notation. For part A, first rewrite the statement as a single inequality.

(a) x < 7 and x > -2 (b) x > 5 or x < -1 Independent Task: Graph the portions of the number lines described for each inequality and then write the inequalities in interval notation.

(a) -3 < x < 5 (b) x > 4

Notes:  

Group Task: Translate the following phrases into inequalities, and then find the solution by solving.

(a) When four times a number is decreased by three, the difference is less than 25.

(b) The sum of two consecutive integers is at least the difference between nine times the smaller and five times the larger.

(c) A fish tank has a maximum volume of 315 gallons of water. If a hose is filling the fish tank at a rate of 60 gallons every hour, how many hours can the hose be left on before the tank overflows?

Exit Slip: Cookies have to be in the oven between 8 and 12 minutes and brownies have to be in the oven between 9 and 14 minutes. Which of the following represent all times, t, when both are in the oven at the same time?

(a) 9 < t < 12 (b) 9 < t < 12 (c) 8 < t < 14 (d) 8 < t < 14 Homework:

1. Determine if each of the following values of x is in the solution set to the compound inequalities given below. Justify your answers by showing your calculations.

  (a)  x  =  -­‐1  for  

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3x + 7 > −11 or

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4 − 2x ≤18 (b) x = 2 for

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2(x +1)3

≤ 6 and

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−2(3 − 2x) < 2  

             2.  Aidan  wrote  the  interval  (-­‐5,  4]  and  claimed  it  was  equivalent  to  the  graph  below.  Explain  what  he  did  wrong  and  correct  his  mistake.              3.  Find  all  numbers  such  that  twice  the  sum  of  the  number  and  eight  is  at  most  four.  Solve  by  setting  up  and  solving  an  inequality.