# Inequalities. and Graphing Inequalities 4.3 Solving Inequalities Using. What would you have?

Save this PDF as:

Size: px
Start display at page:

Download "Inequalities. and Graphing Inequalities 4.3 Solving Inequalities Using. What would you have?"

## Transcription

1 Inequalities.1 Writing ii in and Graphing Inequalities. Solving Inequalities Using Addition or Subtraction. Solving Inequalities Using Multiplication or Division. Solving Two-Step Inequalities If you reached into your water bowl and found more than \$0... And then reached into your cat food bowl and found more than \$0... What would you have? Dear Precious Pet World: Your ad says Up to 75% off on selected items. I select Yummy Tummy Bacon-Flavored Dog Biscuits.

2 What You Learned Before Move farther back. We still have an inequality. Example 1 Graph x. Use a closed circle because is a solution. Shade the number line on the side where you found the solution Test a number to the left of. x 0 is not a solution. Test a number to the right of. x 6 is a solution. Graph the inequality. 1. x 1. x < 5. x 0. x > 1 Example Compare 1 and Graph. 6 1 Graph is to the left of 1. So, 5 6 < 1. Copy and complete the statement using < or >

3 .1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle each number that makes the statement true, and then answer the questions. a. You are in at least 5 of the photos What do you notice about the numbers that you circled? Is the number 5 included? Why or why not? Write four other numbers that make the statement true. b. The temperature is less than degrees Fahrenheit What do you notice about the numbers that you circled? Can the temperature be exactly degrees Fahrenheit? Explain. Write four other numbers that make the statement true. c. More than students from our school are in the chess tournament What do you notice about the numbers that you circled? Inequalities In this lesson, you will write and graph inequalities. use substitution to check whether a number is a solution of an inequality. Is the number included? Why or why not? Write four other numbers that make the statement true. d. The balance in a yearbook fund is no more than \$ What do you notice about the numbers that you circled? Is the number 5 included? Why or why not? Write four other numbers that make the statement true. 1 Chapter Inequalities

4 ACTIVITY: Understanding Inequality Symbols Work with a partner. a. Consider the statement x is a number such that x > 1.5. Can the number be exactly 1.5? Explain. Make a number line. Shade the part of the number line that shows the numbers that make the statement true. Write four other numbers that are not integers that make the statement true. b. Consider the statement x is a number such that x 5. Can the number be exactly 5? Explain. Make a number line. Shade the part of the number line that shows the numbers that make the statement true. Write four other numbers that are not integers that make the statement true. ACTIVITY: Writing and Graphing Inequalities Math Practice Check Progress All the graphs are similar. So, what can you do to make sure that you have correctly written each inequality? Work with a partner. Write an inequality for each graph. Then, in words, describe all the values of x that make the inequality true. a. b. c d IN YOUR OWN WORDS How can you use a number line to represent solutions of an inequality? 5. STRUCTURE Is x 1. the same as 1. x? Explain. Use what you learned about writing and graphing inequalities to complete Exercises and 5 on page 18. Section.1 Writing and Graphing Inequalities 15

5 .1 Lesson Lesson Tutorials Key Vocabulary inequality, p. 16 solution of an inequality, p. 16 solution set, p. 16 graph of an inequality, p. 17 An inequality is a mathematical sentence that compares expressions. It contains the symbols <, >,, or. To write an inequality, look for the following phrases to determine where to place the inequality symbol. Inequality Symbols Symbol < > Key Phrases is less than is fewer than is greater than is more than is less than or equal to is at most is no more than is greater than or equal to is at least is no less than EXAMPLE 1 Writing an Inequality A number q plus 5 is greater than or equal to 7.9. Write this word sentence as an inequality. A number q plus 5 is greater than or equal to 7.9. q An inequality is q Exercises 6 9 Write the word sentence as an inequality. 1. A number x is at most 10.. Twice a number y is more than 5. A solution of an inequality is a value that makes the inequality true. An inequality can have more than one solution. The set of all solutions of an inequality is called the solution set. Value of x x + 1 Is the inequality true? Reading +? 1 0 / 1 no The symbol / means is not less than or equal to. +? yes +? 1 1 yes 16 Chapter Inequalities

6 EXAMPLE Checking Solutions Tell whether is a solution of each inequality. a. y 5 6 b. 5.5y < 1 y 5 6 Write the inequality. 5.5y < 1 5? 6 Substitute for y. 5.5( ) <? 1 7 / 6 Simplify. 11 < 1 7 is not greater than or 11 is less than 1. equal to 6. So, is not a solution of the inequality. So, is a solution of the inequality. Exercises Tell whether 5 is a solution of the inequality.. x + 1 > 7. 1 p 9 5. n.5 The graph of an inequality shows all the solutions of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction. EXAMPLE Graphing an Inequality Graph y > 8. Use an open circle because 8 is not a solution Study Tip The graph in Example shows that the inequality has infinitely many solutions. Test a number to the left of 8. y 1 is not a solution. Test a number to the right of 8. y 0 is a solution. Shade the number line on the side where you found the solution Exercises 17 0 Graph the inequality on a number line. 6. x < 1 7. z 8. s < t Section.1 Writing and Graphing Inequalities 17

7 .1 Exercises Help with Homework 1. PRECISION Should you use an open circle or a closed circle in the graph of the inequality b? Explain.. DIFFERENT WORDS, SAME QUESTION Which is different? Write both inequalities. k is less than or equal to. k is no more than. k is at most. k is at least.. REASONING Do x < 5 and 5 < x represent the same inequality? Explain. 9+(-6)= +(-)= +(-9)= 9+(-1)= Write an inequality for the graph. Then, in words, describe all the values of x that make the inequality true Write the word sentence as an inequality. 6. A number y is no more than A number w added to. is more than A number t multiplied by is at least A number b minus. is less than ERROR ANALYSIS Describe and correct Twice a number x the error in writing the word sentence is at most. as an inequality. x Tell whether the given value is a solution of the inequality. 11. n + 8 1; n = 1. 5h > 15; h = 5 1. p ; p = a 6 > ; a = s 6; s = k < 1 ; k = 1 Graph the inequality on a number line. 17. r g > x 1 0. z < FOOD TRUCK Each day at lunchtime, at least 5 people buy food from a food truck. Write an inequality that represents this situation. 18 Chapter Inequalities

8 Tell whether the given value is a solution of the inequality.. k < k + 8; k =. w w 1; w = y > y + 1; y = 1 5. b b + 8; b = 6. MODELING A subway ride for a student costs \$1.5. A monthly pass costs \$5. a. Write an inequality that represents the number of times you must ride the subway for the monthly pass to be a better deal. b. You ride the subway about 5 times per month. Should you buy the monthly pass? Explain. 7. LOGIC Consider the inequality b >. a. Describe the values of b that are solutions of the inequality. b. Describe the values of b that are not solutions of the inequality. Write an inequality for these values. c. What do all the values in parts (a) and (b) represent? Is this true for any inequality? 8. A postal service says that a rectangular package can have a maximum combined length and girth of 108 inches. The girth of a package is the distance around the perimeter of a face that does not include the length. w h a. Write an inequality that represents the allowable dimensions for the package. b. Find three different sets of allowable dimensions that are reasonable for the package. Find the volume of each package. girth Solve the equation. Check your solution. (Section.) 9. p 8 = w = x = 9. MULTIPLE CHOICE Which expression has a value less than 5? (Section 1.) A B C 1 + ( 8) D 7 + ( ) Section.1 Writing and Graphing Inequalities 19

9 . Solving Inequalities Using Addition or Subtraction solve an inequality? How can you use addition or subtraction to 1 ACTIVITY: Writing an Inequality Work with a partner. Members of the Boy Scouts must be less than 18 years old. In years, your friend will still be eligible to be a scout. a. Which of the following represents your friend s situation? What does x represent? Explain your reasoning. x + > 18 x + < 18 x + 18 x + 18 b. Graph the possible ages of your friend on a number line. Explain how you decided what to graph. ACTIVITY: Writing an Inequality Work with a partner. Supercooling is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid. Water can be supercooled to 86 F below its normal freezing point ( F) and still not freeze. Inequalities In this lesson, you will solve inequalities using addition or subtraction. solve real-life problems. a. Let x represent the temperature of water. Which inequality represents the temperature at which water can be a liquid or a gas? Explain your reasoning. x > 86 x < 86 x 86 x 86 b. On a number line, graph the possible temperatures at which water can be a liquid or a gas. Explain how you decided what to graph. 10 Chapter Inequalities

10 ACTIVITY: Solving Inequalities Math Practice Interpret Results What does the solution of the inequality represent? Work with a partner. Complete the following steps for Activity 1. Then repeat the steps for Activity. Use your inequality from part (a). Replace the inequality symbol with an equal sign. Solve the equation. Replace the equal sign with the original inequality symbol. Graph this new inequality. Compare the graph with your graph in part (b). What do you notice? ACTIVITY: Temperatures of Continents Work with a partner. The table shows the lowest recorded temperature on each continent. Write an inequality that represents each statement. Then solve and graph the inequality. a. The temperature at a weather station in Asia is more than 150 F greater than the record low in Asia. b. The temperature at a research station in Antarctica is at least 80 F greater than the record low in Antarctica. Continent Africa Antarctica Asia Australia Europe North America South America Lowest Temperature 11 F 19 F 90 F 9. F 67 F 81. F 7 F 5. IN YOUR OWN WORDS How can you use addition or subtraction to solve an inequality? 6. Describe a real-life situation that you can represent with an inequality. Write the inequality. Graph the solution on a number line. Use what you learned about solving inequalities to complete Exercises 5 on page 1. Section. Solving Inequalities Using Addition or Subtraction 11

11 . Lesson Lesson Tutorials Study Tip You can solve inequalities in the same way you solve equations. Use inverse operations to get the variable by itself. Addition Property of Inequality Words When you add the same number to each side of an inequality, the inequality remains true. Numbers < Algebra If a < b, then a + c < b + c. + + If a > b, then a + c > b + c. < 5 Subtraction Property of Inequality Words When you subtract the same number from each side of an inequality, the inequality remains true. Numbers < Algebra If a < b, then a c < b c. If a > b, then a c > b c. 5 < 1 These properties are also true for and. EXAMPLE 1 Solving an Inequality Using Addition Undo the subtraction. Solve x 5 <. Graph the solution. x 5 < Write the inequality Addition Property of Inequality x < Simplify. Check: x = 0: 0 5 <? 5 < x = 5: 5 5 <? 0 </ The solution is x <. x x 0 is a solution. x 5 is not a solution. Solve the inequality. Graph the solution. 1. y 6 > 7. b > z 1 1 Chapter Inequalities

12 EXAMPLE Undo the addition. Solving an Inequality Using Subtraction Solve 1 x + 1. Graph the solution. 1 x + 1 Write the inequality. 1 1 Subtraction Property of Inequality 1 x Simplify. Reading The inequality 1 x is the same as x 1. The solution is x 1. x Solve the inequality. Graph the solution. Exercises 17. w d x + > 1 1 EXAMPLE Real-Life Application A person can be no taller than 6.5 feet to become an astronaut pilot for NASA. Your friend is 5 feet 9 inches tall. Write and solve an inequality that represents how much your friend can grow and still meet the requirement. Words Current plus amount your is no the height height friend can grow more than limit. Variable Let h be the possible amounts your friend can grow. 5 ft 9 in. = = 69 in. 69 in. 1 ft = 5.75 ft 1 in. Inequality h h 6.5 Write the inequality Subtraction Property of Inequality h 0.5 Simplify. So, your friend can grow no more than 0.5 foot, or 6 inches. 7. Your cousin is 5 feet inches tall. Write and solve an inequality that represents how much your cousin can grow and still meet the requirement. Section. Solving Inequalities Using Addition or Subtraction 1

13 . Exercises Help with Homework 1. REASONING Is the inequality c + > 5 the same as c > 5? Explain.. WHICH ONE DOESN T BELONG? Which inequality does not belong with the other three? Explain your reasoning. w + 7 > w > 7 w + 7 < < w (-6)= +(-)= +(-9)= 9+(-1)= 1 Solve the inequality. Graph the solution.. x a > g k 7. 1 < y 6 8. n < 5 9. t p > b z < d > s m r + 0. < h 6 8. ERROR ANALYSIS Describe and correct the error in solving the inequality or graphing the solution of the inequality. 18. x 7 > x > x + 5 x AIRPLANE A small airplane can hold passengers. Fifteen passengers board the plane. a. Write and solve an inequality that represents the additional number of passengers that can board the plane. b. Can 0 more passengers board the plane? Explain. 1 Chapter Inequalities

14 Write and solve an inequality that represents x. 1. The perimeter is less. The base is greater. The perimeter is less than than 8 feet. than the height. or equal to 51 meters. 8 m 8 m 7 ft x 8 in. 10 m 10 m 7 ft x + in. x. REASONING The solution of d + s > is d > 7. What is the value of s? 5. BIRDFEEDER The hole for a birdfeeder post is feet deep. The top of the post needs to be at least 5 feet above the ground. Write and solve an inequality that represents the required length of the post. 6. SHOPPING You can spend up to \$5 on a shopping trip. a. You want to buy a shirt that costs \$1. Write and solve an inequality that represents the amount of money you will have left if you buy the shirt. b. You notice that the shirt is on sale for 0% off. How does this change the inequality? c. Do you have enough money to buy the shirt that is on sale and a pair of pants that costs \$? Explain. 7. POWER A circuit overloads at 00 watts of electricity. A portable heater that uses 1050 watts of electricity is plugged into the circuit. a. Write and solve an inequality that represents the additional number of watts you can plug in without overloading the circuit. b. In addition to the portable heater, what two other items in the table can you plug in at the same time without overloading the circuit? Is there more than one possibility? Explain. Item Watts Aquarium 00 Hair dryer 100 Television 150 Vacuum cleaner The possible values of x are given by x What is the greatest possible value of 7x? Solve the equation. Check your solution. (Section.) 9. x = 6 0. w = 9 1. b =. 60 = h. MULTIPLE CHOICE Which fraction is equivalent to.? (Section.1) A 1 5 B 51 5 C 8 5 D 6 5 Section. Solving Inequalities Using Addition or Subtraction 15

15 Study Help Graphic Organizer You can use a Y chart to compare two topics. List differences in the branches and similarities in the base of the Y. Here is an example of a Y chart that compares solving equations and solving inequalities. Solving Equations Solving Inequalities The sign between two expressions is an equal sign, =. One number is the solution. More than one number can be a solution. Use inverse operations to group numbers on one side. Use inverse operations to group variables on one side. Solve for the variable. Make Y charts to help you study and compare these topics. 1. writing equations and writing inequalities. graphing the solution of an equation and graphing the solution of an inequality. graphing inequalities that use > and graphing inequalities that use <. graphing inequalities that use > or < and graphing inequalities that use or 5. solving inequalities using addition and solving inequalities using subtraction Hey Descartes, do you have any suggestions for the Y chart I am making? After you complete this chapter, make Y charts for the following topics. 6. solving inequalities using multiplication and solving inequalities using division 7. solving two-step equations and solving two-step inequalities 8. Pick two other topics that you studied earlier in this course and make a Y chart to compare them. 16 Chapter Inequalities

16 .1. Quiz Progress Check Write the word sentence as an inequality. (Section.1) 1. A number y plus is greater than 5.. A number s minus. is at least 8. Tell whether the given value is a solution of the inequality. (Section.1). 8p < ; p =. z + > ; z = 8 Graph the inequality on a number line. (Section.1) 5. x < 1 6. v > 5 7. b 1 8. q. Solve the inequality. Graph the solution. (Section.) 9. n t 7 > w y.6 <. 1. STUDYING You plan to study at least 1.5 hours for a geography test. Write an inequality that represents this situation. (Section.1) 1. FITNESS TEST The three requirements to pass a fitness test are shown. (Section.1) a. Write and graph three inequalities that represent the requirements. b. You can jog 500 meters, perform 0 push-ups, and perform 0 pull-ups. Do you satisfy the requirements of the test? Explain. 15. NUMBER LINE Use tape on the floor to make the number line shown. All units are in feet. You are standing at 7. You want to move to a number greater than. Write and solve an inequality that represents the distance you must move. (Section.) Sections.1. Quiz 17

17 . Solving Inequalities Using Multiplication or Division solve an inequality? How can you use multiplication or division to 1 ACTIVITY: Using a Table to Solve an Inequality Work with a partner. Copy and complete the table. Decide which graph represents the solution of the inequality. Write the solution of the inequality. a. x > 1 x x x >? b. x 9 x x x? 9 Inequalities In this lesson, you will solve inequalities using multiplication or division. solve real-life problems ACTIVITY: Solving an Inequality Work with a partner. a. Solve x 9 by adding x to each side of the inequality first. Then solve the resulting inequality. b. Compare the solution in part (a) with the solution in Activity 1(b). 18 Chapter Inequalities

18 ACTIVITY: Using a Table to Solve an Inequality Work with a partner. Copy and complete the table. Decide which graph represents the solution of the inequality. Write the solution of the inequality. a. x < 1 x x x <? b. x x x x? Math Practice Analyze Conjectures When you apply your rules to parts (a) (d), do you get the same solutions? Explain. ACTIVITY: Writing Rules Work with a partner. Use a table to solve each inequality. x a. x 10 b. 6x > 0 c. < 1 d. x Write a set of rules that describes how to solve inequalities like those in Activities 1 and. Then use your rules to solve each of the four inequalities above. 5. IN YOUR OWN WORDS How can you use multiplication or division to solve an inequality? Use what you learned about solving inequalities using multiplication or division to complete Exercises 9 on page 1. Section. Solving Inequalities Using Multiplication or Division 19

19 . Lesson Lesson Tutorials Remember Multiplication and division are inverse operations. Multiplication and Division Properties of Inequality (Case 1) Words When you multiply or divide each side of an inequality by the same positive number, the inequality remains true. Numbers < 6 > 6 ( ) < 6 8 < 1 Algebra If a < b and c is positive, then a c < b c and If a > b and c is positive, then a c > b c and These properties are also true for and. > 6 > a c < b c. a c > b c. EXAMPLE Undo the division. 1 Solving an Inequality Using Multiplication Solve x. Graph the solution. 5 x 5 Write the inequality. 5 x 5 5 ( ) Multiplication Property of Inequality x 15 The solution is x 15. Simplify. x x 0 is a solution. x 0 is not a solution. Solve the inequality. Graph the solution. 1. n < m 10. > p 10 Chapter Inequalities

20 EXAMPLE Undo the multiplication. Solving an Inequality Using Division Solve 6x > 18. Graph the solution. 6x > 18 Write the inequality. 6x 6 > 18 6 x > Division Property of Inequality Simplify. The solution is x >. x x 6 is not a solution. x 0 is a solution. Exercises Solve the inequality. Graph the solution.. b 5. 1k <.5q Multiplication and Division Properties of Inequality (Case ) Words When you multiply or divide each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. Common Error A negative sign in an inequality does not necessarily mean you must reverse the inequality symbol. Only reverse the inequality symbol when you multiply or divide both sides by a negative number. Numbers < 6 > 6 ( ) > 6 < 6 8 > 1 < Algebra If a < b and c is negative, then a c > b c and If a > b and c is negative, then a c < b c and These properties are also true for and. a c > b c. a c < b c. Section. Solving Inequalities Using Multiplication or Division 11

21 EXAMPLE Solving an Inequality Using Multiplication Solve n 6. Graph the solution. n 6 Write the inequality. ( ) n 6 n The solution is n. n Use the Multiplication Property of Inequality. Reverse the inequality symbol. Simplify n 6 is not a solution. n 0 is a solution. Solve the inequality. Graph the solution. 7. x > y m h 8 EXAMPLE Solving an Inequality Using Division Solve z >.5. Graph the solution. Undo the multiplication. z >.5 z <.5 z < 1.5 The solution is z < 1.5. z 1.5 Write the inequality. Use the Division Property of Inequality. Reverse the inequality symbol. Simplify z 0 is a solution. z is not a solution. Solve the inequality. Graph the solution. Exercises z < 5 1. a > < n w 1 Chapter Inequalities

22 . Exercises Help with Homework 1. WRITING Explain how to solve x <.. PRECISION Explain how solving x < 16 is different from solving x < 16.. OPEN-ENDED Write an inequality that you can solve using the Division Property of Inequality where the direction of the inequality symbol must be reversed. 9+(-6)= +(-)= +(-9)= 9+(-1)= Use a table to solve the inequality.. x < 5. x 6. 6x > x x 1 > 5 9. x 1 1 Solve the inequality. Graph the solution. 10. n > c 1..m < > x 1. 1 w <.5k x y > b. 19. ERROR ANALYSIS Describe and correct the error in solving the inequality. x < 9 x > ( 9) x > 7 Write the word sentence as an inequality. Then solve the inequality. 0. The quotient of a number and is at most A number divided by 7 is less than.. Six times a number is at least.. The product of and a number is greater than 0.. SMART PHONE You earn \$9.0 per hour at your summer job. Write and solve an inequality that represents the number of hours you need to work in order to buy a smart phone that costs \$99. Section. Solving Inequalities Using Multiplication or Division 1

23 5. AVOCADOS You have \$9.60 to buy avocados for a guacamole recipe. Avocados cost \$.0 each. a. Write and solve an inequality that represents the number of avocados you can buy. b. Are there infinitely many solutions in this context? Explain. 6. SCIENCE PROJECT Students in a science class are divided into 6 equal groups with at least students in each group for a project. Write and solve an inequality that represents the number of students in the class. Solve the inequality. Graph the solution. 7. 5n w > h < 1 x 1. y < 1. d > m 6. k > b.5 6. ERROR ANALYSIS Describe and correct the error in solving the inequality. 7. TEMPERATURE It is currently 0 C outside. The temperature is dropping.5 C every hour. Write and solve an inequality that represents the number of hours that must pass for the temperature to drop below 0 C. m 9 m 9 m 1.5 ft 7 in. 8. STORAGE You are moving some of your belongings into a storage facility. a. Write and solve an inequality that represents the number of boxes that you can stack vertically in the storage unit. b. Can you stack 6 boxes vertically in the storage unit? Explain. Write and solve an inequality that represents x. 9. Area 10 cm 0. Area < 0 ft 10 cm x x 8 ft 1 Chapter Inequalities

24 1. AMUSEMENT PARK You and four friends are planning a visit to an amusement park. You want to keep the cost below \$100 per person. Write and solve an inequality that represents the total cost of visiting the amusement park.. LOGIC When you multiply or divide each side of an inequality by the same negative number, you must reverse the direction of the inequality symbol. Explain why.. PROJECT Choose two novels to research. a. Use the Internet or a magazine to complete the table. b. Use the table to find and compare the average number of copies sold per month for each novel. Which novel do you consider to be the most successful? Explain. c. Assume each novel continues to sell at the average rate. Write and solve an inequality that represents the number of months it will take for the total number of copies sold to exceed twice the current number sold. Author Name of Novel Release Date Current Number of Copies Sold 1.. Describe all numbers that satisfy both inequalities. Include a graph with your description.. m > and m < n and n x 6 and x s > 7 and 1 s < 1 Solve the equation. Check your solution. (Section.5) 8. w + = 1 9. v 5 6 = 50. (x 1) = m + 00 = MULTIPLE CHOICE What is the value of ( + 5? (Section.) 7) A B 1 1 C 7 10 D Section. Solving Inequalities Using Multiplication or Division 15

25 . Solving Two-Step Inequalities the dimensions of a figure? How can you use an inequality to describe 1 ACTIVITY: Areas and Perimeters of Figures Work with a partner. Use the given condition to choose the inequality that you can use to find the possible values of the variable. Justify your answer. Write four values of the variable that satisfy the inequality you chose. a. You want to find the values of x so that the area of the rectangle is more than square units. x + 1 > x + > x x + 1 x + 1 > b. You want to find the values of x so that the perimeter of the rectangle is greater than or equal to 8 units. x x x x c. You want to find the values of y so that the area of the parallelogram is fewer than 1 square units. y 7 Inequalities In this lesson, you will solve multi-step inequalities. solve real-life problems. 5y + 7 < 1 5y + 5 < 1 5y y d. You want to find the values of z so that the area of the trapezoid is at most 100 square units z z z 6 5z + 0 < z + 0 < Chapter Inequalities

26 ACTIVITY: Volumes of Rectangular Prisms Work with a partner. Use the given condition to choose the inequality that you can use to find the possible values of the variable. Justify your answer. Write four values of the variable that satisfy the inequality you chose. Math Practice State the Meaning of Symbols What inequality symbols do the phrases at least and no more than represent? Explain. a. You want to find the values of x so that the volume of the rectangular prism is at least 50 cubic units. 5 x 15x + 0 > 50 x x x + 50 b. You want to find the values of x so that the volume of the rectangular prism is no more than 6 cubic units..5 x 1 8x + < 6 6x + 18 < 6 x x IN YOUR OWN WORDS How can you use an inequality to describe the dimensions of a figure?. Use what you know about solving equations and inequalities to describe how you can solve a two-step inequality. Give an example to support your explanation. Use what you learned about solving two-step inequalities to complete Exercises and on page 150. Section. Solving Two-Step Inequalities 17

27 . Lesson Lesson Tutorials You can solve two-step inequalities in the same way you solve two-step equations. EXAMPLE 1 Solving Two-Step Inequalities a. Solve 5x 11. Graph the solution. 5x 11 Write the inequality. Step 1: Undo the subtraction. Step : Undo the multiplication. + + Addition Property of Inequality 5x 15 Simplify. 5x x Division Property of Inequality Simplify. The solution is x. x x 0 is not a solution. x is a solution. Step 1: Undo the addition. Step : Undo the division. b. Solve b + < 1. Graph the solution. b + < 1 Write the inequality. Subtraction Property of Inequality b < 9 Simplify. b > 9 Use the Multiplication Property of Inequality. Reverse the inequality symbol. b > 7 Simplify. The solution is b > 7. b Exercises 5 10 Solve the inequality. Graph the solution. 1. 6y 7 > 5. d 19. w + 8 > 9 18 Chapter Inequalities

28 EXAMPLE Graphing an Inequality Which graph represents the solution of 7(x + ) 8? A B C D (x + ) 8 Write the inequality. 7x 1 8 Distributive Property Step 1: Undo the subtraction Addition Property of Inequality 7x 9 Simplify. Step : Undo the multiplication. 7x x 7 Use the Division Property of Inequality. Reverse the inequality symbol. Simplify. The correct answer is B. Remember EXAMPLE Progress Report Month Pounds Lost In Example, the average is equal to the sum of the pounds lost divided by the number of months. Real-Life Application A contestant in a weight-loss competition wants to lose an average of at least 8 pounds per month during a 5-month period. How many pounds must the contestant lose in the fifth month to meet the goal? Write and solve an inequality. Let x be the number of pounds lost in the fifth month x x x Simplify. + x 0 Simplify. x 6 The phrase at least means greater than or equal to. Multiplication Property of Inequality Subtract from each side. So, the contestant must lose at least 6 pounds to meet the goal. Exercises 1 17 Solve the inequality. Graph the solution.. (k 5) < 6 5. (n 10) < (8 + y) 7. WHAT IF? In Example, the contestant wants to lose an average of at least 9 pounds per month. How many pounds must the contestant lose in the fifth month to meet the goal? Section. Solving Two-Step Inequalities 19

29 . Exercises Help with Homework 1. WRITING Compare and contrast solving two-step inequalities and solving two-step equations.. OPEN-ENDED Describe how to solve the inequality (a + 5) < 9. 9+(-6)= +(-)= +(-9)= 9+(-1)= Match the inequality with its graph.. t 1. 5x + 7 A A B B C C Solve the inequality. Graph the solution y 5 < 6. p > 8 h m 8. > b r ERROR ANALYSIS Describe and correct x the error in solving the inequality. + < 6 x + < 18 Solve the inequality. Graph the solution. x < ( g + ) > (w 6) (k ) (d + 1) < > 0.9(n + 8.6) (c.) 10 cm 18. UNICYCLE The first jump in a unicycle high-jump contest is shown. The bar is raised centimeters after each jump. Solve the inequality n to find the number of additional jumps needed to meet or exceed the goal of clearing a height of 6 centimeters. 150 Chapter Inequalities

30 Solve the inequality. Graph the solution x x d 7d +.8 < SCUBA DIVER A scuba diver is at an elevation of 8 feet. The diver starts moving at a rate of 1 feet per minute. Write and solve an inequality that represents how long it will take the diver to reach an elevation deeper than 00 feet.. KILLER WHALES A killer whale has eaten 75 pounds of fish today. It needs to eat at least 10 pounds of fish each day. a. A bucket holds 15 pounds of fish. Write and solve an inequality that represents how many more buckets of fish the whale needs to eat. b. Should the whale eat four or five more buckets of fish? Explain.. REASONING A student theater charges \$9.50 per ticket. a. The theater has already sold 70 tickets. Write and solve an inequality that represents how many more tickets the theater needs to sell to earn at least \$1000. b. The theater increases the ticket price by \$1. Without solving an inequality, describe how this affects the total number of tickets needed to earn at least \$ Problem Solving For what values of r will the area of the shaded region be greater than or equal to 1 square units? r Find the missing values in the ratio table. Then write the equivalent ratios. (Skills Review Handbook) 5. Flutes 7 8 Clarinets 1 6. Boys 6 Girls MULTIPLE CHOICE What is the volume of the cube? (Skills Review Handbook) A 8 ft B 16 ft C ft D ft ft Section. Solving Two-Step Inequalities 151

31 .. Quiz Solve the inequality. Graph the solution. (Section. and Section.) ) 1. p 18. x > 5 Progress Check. r 5. z 8 < n < 5 k 7. 1.m.8 < (1 y ) Write the word sentence as an inequality. Then solve the inequality. (Section.) 9. The quotient of a number and 5 is less than. 10. Six times a number is at least PEPPERS You have \$18 to buy peppers. Peppers cost \$1.50 each. Write and solve an inequality that represents the number of peppers you can buy. (Section.) 1. MOVIES You have a gift card worth \$90. You want to buy several movies that cost \$1 each. Write and solve an inequality that represents the number of movies you can buy and still have at least \$0 on the gift card. (Section.) 1. CHOCOLATES Your class sells boxes of chocolates to raise \$500 for a field trip. You earn \$6.5 for each box of chocolates sold. Write and solve an inequality that represents the number of boxes your class must sell to meet or exceed the fundraising goal. (Section.) c ft 1 ft 1. FENCE You want to put up a fence that encloses a triangular region with an area greater than or equal to 60 square feet. What is the least possible value of c? Explain. (Section.) 15 Chapter Inequalities

32 Chapter Review Review Key Vocabulary inequality, p. 16 solution of an inequality, p. 16 Vocabulary Help solution set, p. 16 graph of an inequality, p. 17 Review Examples and Exercises.1 Writing and Graphing Inequalities (pp. 1 19) a. Six plus a number x is at most 1. Write this word sentence as an inequality. Six plus a number x is at most x 1 An inequality is 6 + x 1. b. Graph m >. Step 1: Use an open circle because is not a solution. Step : Shade the number line on the side where you found the solution Step : Test a number to the left of. m is not a solution. Step : Test a number to the right of. m is a solution. Write the word sentence as an inequality. 1. A number w is greater than.. A number y minus 1 is no more than. Tell whether the given value is a solution of the inequality j > 8; j = 7. 6 n 5; n = Graph the inequality on a number line. 5. q > s < 1 7. BUMPER CARS You must be at least inches tall to ride the bumper cars at an amusement park. Write an inequality that represents this situation. Chapter Review 15

33 . Solving Inequalities Using Addition or Subtraction (pp ) Solve 5 < m. Graph the solution. Undo the subtraction. 5 < m Write the inequality. + + Addition Property of Inequality < m Simplify. The solution is m >. m m is not a solution. m is a solution. Solve the inequality. Graph the solution. 8. d + 1 < t z + 6. Undo the division.. Solving Inequalities Using Multiplication or Division (pp ) c Solve. Graph the solution. c Write the inequality. c ( ) Use the Multiplication Property of Inequality. Reverse the inequality symbol. c 6 The solution is c 6. Simplify. c c is a solution. c 9 is not a solution. Solve the inequality. Graph the solution q < r 6 1. > s 15 Chapter Inequalities

34 . Solving Two-Step Inequalities (pp ) a. Solve 6x Graph the solution. Step 1: Undo the subtraction. Step : Undo the multiplication. 6x 8 10 Write the inequality Addition Property of Inequality 6x 18 6x x Simplify. Division Property of Inequality Simplify. The solution is x. x x 0 is a solution. x 5 is not a solution. b. Solve q + 7 < 11. Graph the solution. q + 7 < 11 Write the inequality. Step 1: Undo the addition. 7 7 Subtraction Property of Inequality Step : Undo the division. q < Simplify. q > Use the Multiplication Property of Inequality. Reverse the inequality symbol. q > 16 Simplify. The solution is q > 16. q q 0 is not a solution. q 1 is a solution. Solve the inequality. Graph the solution. 1. x + > z t 5 < (q + ) < ( p + 9) ( j +.5).8 Chapter Review 155

35 Chapter Test Test Practice Write the word sentence as an inequality. 1. A number k plus 19.5 is less than or equal to 0.. A number q multiplied by 1 is greater than 16. Tell whether the given value is a solution of the inequality.. n ; n = 7. m < 1; m = c 7; c = 6..m > 6.8; m = Solve the inequality. Graph the solution. 7. w + 8. x > y p 1. 7 b z 6 1. k > 0.(d + 6) 15. GUMBALLS You have \$.50. Each gumball in a gumball machine costs \$0.5. Write and solve an inequality that represents the number of gumballs you can buy. 16. PARTY You can spend no more than \$100 on a party you are hosting. The cost per guest is \$8. a. Write and solve an inequality that represents the number of guests you can invite to the party. b. What is the greatest number of guests that you can invite to the party? Explain your reasoning. 17. BASEBALL CARDS You have \$0 to buy baseball cards. Each pack of cards costs \$5. Write and solve an inequality that represents the number of packs of baseball cards you can buy and still have at least \$10 left. 156 Chapter Inequalities

36 Cumulative Assessment 1. What is the value of the expression below when x = 5, y =, and z = 1? Test-Taking Strategy Answer Easy Questions First x y z A. C. 16 B. 16 D.. What is the value of the expression below? F H Scan the test and answer the easy questions first. Because, A is correct. G I. 0. Which graph represents the inequality below? x 8 9 A C B D Which value of p makes the equation below true? 5(p + 6) = 5 F. 1 H. 11 G. 5 I. 1 Cumulative Assessment 157

37 5. You set up the lemonade stand. Your profit is equal to your revenue from lemonade sales minus your cost to operate the stand. Your cost is \$8. How many cups of lemonade must you sell to earn a profit of \$0? Lemonade 50 per cup A. C. 60 B. D Which value is a solution of the inequality below? y < 7 F. 6 H. G. I What value of y makes the equation below true? 1 y = 6 8. What is the mean distance of the four points from? A. 1 B. 1 C. D Chapter Inequalities

38 9. Martin graphed the solution of the inequality x + 18 > 6 in the box below What should Martin do to correct the error that he made? F. Use an open circle at and shade to the left of. G. Use an open circle at and shade to the right of. H. Use a closed circle and shade to the right of. I. Use an open circle and shade to the left of. 10. What is the value of the expression below? You are selling T-shirts to raise money for a charity. You sell the T-shirts for \$10 each. Part A Part B Part C You have already sold T-shirts. How many more T-shirts must you sell to raise at least \$500? Explain. Your friend is raising money for the same charity and has not sold any T-shirts previously. He sells the T-shirts for \$8 each. What is the total number of T-shirts he must sell to raise at least \$500? Explain. Who has to sell more T-shirts in total? How many more? Explain. 1. Which expression is equivalent to the expression below? A ( 9) C B. ( 1 ) D. ( 1 ) Cumulative Assessment 159

### Writing and Graphing Inequalities

.1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle

### Writing and Graphing Inequalities

4.1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle

### How can you use multiplication or division to solve an inequality? ACTIVITY: Using a Table to Solve an Inequality

. Solving Inequalities Using Multiplication or Division How can you use multiplication or division to solve an inequality? 1 ACTIVITY: Using a Table to Solve an Inequality Work with a partner. Copy and

### How can you use addition or subtraction to solve an equation?

7.2 Solving Equations Using Addition or Subtraction How can you use addition or subtraction to solve an equation? When two sides of a scale weigh the same, the scale will balance. When you add or subtract

### Chapter 2. Worked-Out Solutions. Chapter 2 Mathematical Practices (p. 52) Chapter 2 Maintaining Mathematical Proficiency (p. 51)

Chapter Chapter Maintaining Mathematical Proficiency (p. ). 6 8 Chapter Mathematical Practices (p. ). x + < x. x > x +.. = 6. =. The solution is x .. x + x +. + = + =. = = 6. + =

### Evaluate and Simplify Algebraic Expressions

TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So

### 2 Solving Linear Inequalities

2 Solving Linear Inequalities 2.1 Writing and Graphing Inequalities 2.2 Solving Inequalities Using Addition or Subtraction 2.3 Solving Inequalities Using Multiplication or Division 2.4 Solving Multi-Step

### Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and

### Chapter 1. Worked-Out Solutions. Chapter 1 Maintaining Mathematical Proficiency (p. 1)

Chapter Maintaining Mathematical Proficiency (p. ). + ( ) = 7. 0 + ( ) =. 6 + = 8. 9 ( ) = 9 + =. 6 = + ( 6) = 7 6. ( 7) = + 7 = 7. 7 + = 8. 8 + ( ) = 9. = + ( ) = 0. (8) =. 7 ( 9) = 6. ( 7) = 8. ( 6)

### Writing Equations in One Variable

7.1 Writing Equations in One Variable solve the word problem? How does rewriting a word problem help you 1 ACTIVITY: Rewriting a Word Problem Work with a partner. Read the problem several times. Think

### Unit 4: Inequalities. Inequality Symbols. Algebraic Inequality. Compound Inequality. Interval Notation

Section 4.1: Linear Inequalities Section 4.2: Solving Linear Inequalities Section 4.3: Solving Inequalities Applications Section 4.4: Compound Inequalities Section 4.5: Absolute Value Equations and Inequalities

### ACTIVITY: Simplifying Algebraic Expressions

. Algebraic Expressions How can you simplify an algebraic expression? ACTIVITY: Simplifying Algebraic Expressions Work with a partner. a. Evaluate each algebraic expression when x = 0 and when x =. Use

### Lesson 7: Literal Equations, Inequalities, and Absolute Value

, and Absolute Value In this lesson, we first look at literal equations, which are equations that have more than one variable. Many of the formulas we use in everyday life are literal equations. We then

### Chapter Four: Linear Expressions, Equations, and Inequalities

Chapter Four: Linear Expressions, Equations, and Inequalities Index: A: Intro to Inequalities (U2L8) B: Solving Linear Inequalities (U2L9) C: Compound Inequalities (And) (U2L10/11) D: Compound Inequalities

### 1.1 Integers and Absolute Value 1.2 Adding Integers

1 Integers 1.1 Integers and Absolute Value 1.2 Adding Integers 1. Subtracting Integers 1.4 Multiplying Integers 1.5 Dividing Integers Look, subtraction is not that difficult. Imagine that you have five

### Solving Systems of Linear Equations by Substitution

5.2 Solving Systems of Linear Equations by Substitution a system of linear equations? How can you use substitution to solve 1 ACTIVITY: Using Substitution to Solve a System Work with a partner. Solve each

### BIG IDEAS MATH. A Bridge to Success. Ron Larson Laurie Boswell. Erie, Pennsylvania BigIdeasLearning.com

BIG IDEAS MATH A Bridge to Success Ron Larson Laurie Boswell Erie, Pennsylvania BigIdeasLearning.com 2 Solving Linear Inequalities 2.1 Writing and Graphing Inequalities 2.2 Solving Inequalities Using Addition

### How can you write and evaluate an expression that represents a real-life problem? ACTIVITY: Reading and Re-Reading

.1 Algebraic Expressions How can you write and evaluate an expression that represents a real-life problem? 1 ACTIVITY: Reading and Re-Reading Work with a partner. a. You babysit for hours. You receive

### ACTIVITY: Quarterback Passing Efficiency

3. Solving Inequalities Using Addition or Subtraction solve an inequality? How can you use addition or subtraction to 1 ACTIVITY: Quarterback Passing Efficiency Work with a partner. The National Collegiate

### Chapter Two B: Linear Expressions, Equations, and Inequalities

Chapter Two B: Linear Expressions, Equations, and Inequalities Index: A: Intro to Inequalities (U2L8) Page 1 B: Solving Linear Inequalities (U2L9) Page 7 C: Compound Inequalities (And) (U2L10/11) Page

### Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities

Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,

### Solving Linear Inequalities

Solving Linear Inequalities. Writing and Graphing Inequalities. Solving Inequalities Using Addition or Subtraction. Solving Inequalities Using Multiplication or Division. Solving Multi-Step Inequalities.5

### Solving Equations. A: Solving One-Variable Equations. One Step x + 6 = 9-3y = 15. Two Step 2a 3 6. Algebra 2 Chapter 1 Notes 1.4 Solving Equations

Algebra 2 Chapter 1 Notes 1.4 Solving Equations 1.4 Solving Equations Topics: Solving Equations Translating Words into Algebra Solving Word Problems A: Solving One-Variable Equations The equations below

### Why? Speed Skating Tracks offi cial track short track

Applying Systems of Linear Equations Then You solved systems of equations by using substitution and elimination. (Lessons 6-2, 6-3, and 6-4) Now 1Determine the best method for solving systems of 2Apply

### Think of three math problems using the four operations where order matters and three where order doesn t matter. ACTIVITY: Commutative Properties

3.3 Properties of Addition and Multiplication operation matter? Does the order in which you perform an ACTIVITY: Does Order Matter? Work with a partner. Place each statement in the correct oval. a. Fasten

### Unit 1: Introduction to Variables

Section 1.1: Writing Algebraic Expressions Section 1.2: The Story of x Section 1.3: Evaluating Algebraic Expressions Section 1.4: Applications Section 1.5: Geometric Formulas KEY TERMS AND CONCEPTS Look

### Goal: Write variable expressions and equations. a. A number increased by 3. c. 1 more than three times a number

S E S N Writing Expressions and Equations Goal: Write variable expressions and equations. Vocabulary Verbal model: EXAMPE 1 Translating Verbal Phrases Verbal phrase Expression a. A number increased by

### CCGPS Coordinate Algebra. EOCT Review Units 1 and 2

CCGPS Coordinate Algebra EOCT Review Units 1 and 2 Unit 1: Relationships Among Quantities Key Ideas Unit Conversions A quantity is a an exact amount or measurement. A quantity can be exact or approximate

### How can you use algebra tiles to solve addition or subtraction equations?

. Solving Equations Using Addition or Subtraction How can you use algebra tiles to solve addition or subtraction equations? ACTIVITY: Solving Equations Work with a partner. Use algebra tiles to model and

### SOL Review Items. 7.1 The student will. 7.1a 1. Which fraction and decimal are equivalent to A. and B. and C. and D. and 0.

7.1 The student will a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals,

### Questions # 1-53 Provide review for the Mid-Term Questions # Provide review for the Final

Central Carolina Technical College MAT 031 - Developmental Math Exam Review Questions # 1-53 Provide review for the Mid-Term Questions # 1-105 Provide review for the Final SHORT ANSWER. Write the word

### Study Guide For use with pages 63 68

2.1 For use with pages 63 68 GOAL Use properties of addition and multiplication. VOCABULARY Lesson 2.1 Commutative Property of Addition: In a sum, you can add the numbers in any order. Associative Property

### Name: Class: Date: ID: A

Name: Class: Date: 8th Grade Advanced Topic III, Linear Equations and Systems of Linear Equations, MA.8.A.1.1, MA.8.1.1.2, MA.8.A.1.3, *MA.8.A.1.4, MA.8.A.1.5, MA.8.A.1.6 Multiple Choice Identify the choice

### Course 1 Benchmark Test End of Year

Course 1 Benchmark Test End of Year 1. Which rule best describes the relationship shown in the function table below? Input A. subtract 2 Output 1 3 2 6 3 9 4 12 5 15 4. What is the least common multiple

### Silver Spring International Middle School Algebra Summer Packet

Name: Silver Spring International Middle School Algebra Summer Packet It is NOT mandatory to complete but, STRONGLY encouraged. MONTGOMERY COUNTY PUBLIC SCHOOLS SILVER SPRING INTERNATIONAL MIDDLE SCHOOL

### Why? 0.10d 2x z _

Variables and Expressions Then You performed operations on integers. (Lesson 0-3) Now 1Write verbal expressions for algebraic expressions. 2Write algebraic expressions for verbal expressions. Why? Cassie

### Expressions and Equations

Descartes, if you solve for in the equation, what do you get? Expressions and Equations. Algebraic Expressions. Adding and Subtracting Linear Expressions. Solving Equations Using Addition or Subtraction.

### Why is the product of two negative rational numbers positive?

. Multiplying and Dividing Rational Numbers Why is the product of two negative rational numbers positive? In Section., you used a table to see that the product of two negative integers is a positive integer.

### Solving Equations by Adding and Subtracting

SECTION 2.1 Solving Equations by Adding and Subtracting 2.1 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the addition property to solve equations 3. Determine whether

### Essential Question How can you use substitution to solve a system of linear equations?

5.2 Solving Systems of Linear Equations by Substitution Essential Question How can you use substitution to solve a system of linear equations? Using Substitution to Solve Systems Work with a partner. Solve

### Write an inequality for the graph. Then, in words, describe all the values of x that make the inequality true

Name Date. Practice A Write an inequality for the graph. Then, in words, describe all the values of that make the inequality true... 6 8 0 6 Write the word sentence as an inequality.. A number is at most..

### Write an equation for each relationship. Then make a table of input-output pairs and tell whether the function is proportional.

Functions Reteaching 41 Math Course, Lesson 41 A function is a rule that identifies a relationship between a set of input numbers and a set of output numbers. A function rule can be described in words,

### Maintaining Mathematical Proficiency

Name Date Chapter 1 Maintaining Mathematical Proficiency Add or subtract. 1. 1 + ( 3) 2. 0 + ( 12) 3. 5 ( 2) 4. 4 7 5. Find two pairs of integers whose sum is 6. 6. In a city, the record monthly high temperature

### Algebra 1 Spencer Unit 4 Notes: Inequalities and Graphing Linear Equations. Unit Calendar

Algebra 1 Spencer Unit 4 Notes: Inequalities and Graphing Linear Equations Unit Calendar Date Topic Homework Nov 5 (A ) 6.1 Solving Linear Inequalities +/- 6.2 Solving Linear Inequalities x/ 6.3 Solving

### 1Solve systems of. 2Apply Systems of. Then. Why? Now. New Vocabulary system of inequalities

Systems of Inequalities Then You graphed and solved linear (Lesson 5-6) Now 1Solve systems of linear inequalities by graphing. 2Apply Systems of Why? Jacui is beginning an exercise program that involves

### ACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.

7.9 Factoring Special Products special products? How can you recognize and factor 1 ACTIVITY: Factoring Special Products Work with a partner. Six different algebra tiles are shown below. 1 1 x x x 2 x

### 2. In Exercise 1, suppose the two pricing plans changed as follows. Complete parts (a) (d) based on these two plans.

A C E Applications Connections Extensions Applications 1. A school is planning a Saturday Back-to-School Festival to raise funds for the school art and music programs. Some of the planned activities are

### Inequalities Chapter Test

Inequalities Chapter Test Part 1: For questions 1-9, circle the answer that best answers the question. 1. Which graph best represents the solution of 8 4x < 4 A. B. C. D. 2. Which of the following inequalities

### 3 Mathematics SOL. Practice. Virginia

3 Mathematics Virginia SOL Practice To the Student This book provides practice with the math skills that students in grade 3 should master. The book has three practice tests, each with 50 multiple-choice

### Essential Question How can you solve an absolute value inequality? Work with a partner. Consider the absolute value inequality x

Learning Standards HSA-CED.A.1 HSA-REI.B.3.6 Essential Question How can you solve an absolute value inequality? COMMON CORE Solving an Absolute Value Inequality Algebraically MAKING SENSE OF PROBLEMS To

### 1 to 4; 1:4; 1_ 4. 1 to 3; 1:3; 1_ 3. 3 to 8; 3:8; 3_. 8 to 4; 8:4; 8_ 4

Understanding Ratios Reteaching 12-1 A ratio is a pair of numbers that compares two quantities. Count to find the ratio of squares to circles. Reteaching 12-1 4 to 3 The ratio 4 to 3 can also be written

### THANKS AND HAVE A GREAT SUMMER!

6 th Grade to 7 th Grade Summer Math Packet For this Math packet please show as much work as you can. The concepts you are going to be working on are those of the Common Core Standards for 6 th Grade that

### Math 7, Unit 4: Expressions, Equations and Inequalities Notes

Math 7, Unit 4: Epressions, Equations and Inequalities Notes Prep for 7.EE.B.4 A numerical epression is simply a name for a number. For eample, 4 + 6 is a numerical epression for 10, and 400 4 is a numerical

### Algebra 1 Fall Semester Final Review Name

It is very important that you review for the Algebra Final. Here are a few pieces of information you want to know. Your Final is worth 20% of your overall grade The final covers concepts from the entire

### Relationships Between Quantities

Algebra 1 Relationships Between Quantities Relationships Between Quantities Everyone loves math until there are letters (known as variables) in problems!! Do students complain about reading when they come

### 6th Grade Practice Exam

6th Grade Practice Exam Short Answer 1. The table gives the ratio of teachers to students at Jefferson Middle School. Jefferson Middle School Students Teachers 24 1 48 2 72 3 96 4 At Hamilton Middle School,

### You discovered in Lesson 4.1 that when two powers with the same base are multiplied, the base remains the

Division Properties of Exponents Lesson 4.2 You discovered in Lesson 4.1 that when two powers with the same base are multiplied, the base remains the same and the exponents are added together. Examine

### Grade Common Core Math

th 5 Grade Common Core Math Operations and Algebraic Thinking Printable Review Problems Standards Included: -Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with

### Unit 12: Systems of Equations

Section 12.1: Systems of Linear Equations Section 12.2: The Substitution Method Section 12.3: The Addition (Elimination) Method Section 12.4: Applications KEY TERMS AND CONCEPTS Look for the following

### 2. Mrs. Johnson asked her 6th-grade students to form a number pattern using these rules.

6 th Grade Practice Test Objective 1.1 1. Dale used these steps to form a number pattern. 1. The first term is 3. 2. The second term is 5. 3. Each term after the second is the sum of the two terms just

### Solving Multi-Step Equations 1.2. ACTIVITY: Solving for the Angles of a Triangle

. Solving Multi-Step Equations How can you solve a multi-step equation? How can you check the reasonableness of your solution? ACTIVITY: Solving for the Angles of a Triangle Work with a partner. Write

### Linear Functions, Equations, and Inequalities

CHAPTER Linear Functions, Equations, and Inequalities Inventory is the list of items that businesses stock in stores and warehouses to supply customers. Businesses in the United States keep about.5 trillion

### _ 4x 4 = _-8. Model. Activity. 90 Explore 2-3 Algebra Lab: Solving Multi-Step Equations

Algebra Lab Solving Multi-Step Equations You can use algebra tiles to model solving multi-step equations. Virginia SOL A.4 The student will solve multistep linear and quadratic equations in two variables,

### Algebra 1 End-of-Course Assessment Practice Test with Solutions

Algebra 1 End-of-Course Assessment Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit

### Algebraic expression is formed from variables and constants using different operations. NCERT

UNIT 10 ALGEBRAIC EXPRESSIONS (A) Main Concepts and Results Algebraic expression is formed from variables and constants using different operations. Expressions are made up of terms. A term is the product

### Math 11 - Systems of Equations and Inequalities

Name: Period: /30 ID: A Math 11 - Systems of Equations and Inequalities Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (1 point) What system of equations

### Solve Linear Systems by Substitution

5.2 Solve Linear Systems by Substitution Systems of linear equations can be used to help determine the best selling price for products and services. Investigate A Class Problem Tools graphing calculator

### Lesson 8: Using If-Then Moves in Solving Equations

Classwork Opening Exercise Recall and summarize the if-then moves. Write + 5 = 8 in as many true equations as you can using the if-then moves. Identify which if-then move you used. Example 1 Julia, Keller,

### Test Booklet. Subject: MA, Grade: HS CAHSEE Math Practice Test. Student name:

Test Booklet Subject: MA, Grade: HS CAHSEE Math Practice Test Student name: Author: California District: California Released Tests Printed: Friday December 16, 2011 1 Which number has the greatest absolute

### Comparing and Ordering Integers

6.2 Comparing and Ordering Integers to order real-life events? How can you use a number line 1 ACTIVITY: Seconds to Takeoff Work with a partner. You are listening to a command center before the of a rocket.

### Why? Step 3 Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair.

Substitution Then You solved systems of equations by graphing. (Lesson 6-1) Now 1Solve systems of equations by using substitution. 2Solve real-world problems involving systems of equations by using substitution.

### Indiana Core 40 End-of-Course Assessment Algebra I Blueprint*

Types of items on the Algebra I End-of-Course Assessment: Multiple-choice 1 point per problem The answer to the question can be found in one of four answer choices provided. Numeric response 1 point per

### Name: Period: Date: Algebra 1 1st Semester Review Which best describes the solution(s) for this equation? 3 ( 8x 12) = 33 2x

Name: Period: ate: lgebra 1 1st Semester Review 2011 1 Which algebraic expression could NOT match the pictorial representation below? 5 Which best describes the solution(s) for this equation? 3 ( 8x 12)

### 1 Ramón read books that were on his reading list. Which of the following improper fractions is equivalent to A B C D

5 th Grade Math -STR-STR-M omparison Spacing has been deleted and graphics minimized to fit table. (5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations.

### QUESTIONS 1-46 REVIEW THE OBJECTIVES OF CHAPTER 2.

MAT 101 Course Review Questions Valid for Fall 2014, Spring 2015 and Summer 2015 MIDTERM EXAM FINAL EXAM Questions 1-86 are covered on the Midterm. There are 25 questions on the midterm, all multiple choice,

### UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES. Solving Equations and Inequalities in One Variable

UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES This unit investigates linear equations and inequalities. Students create linear equations and inequalities and use them to solve problems. They

### Algebra I Notes Linear Inequalities in One Variable and Unit 3 Absolute Value Equations and Inequalities

PREREQUISITE SKILLS: students must have a clear understanding of signed numbers and their operations students must understand meaning of operations and how they relate to one another students must be able

### Section 2 Equations and Inequalities

Section 2 Equations and Inequalities The following Mathematics Florida Standards will be covered in this section: MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite it. MAFS.912.A-REI.1.1

### Functions. Essential Question What is a function? Work with a partner. Functions can be described in many ways.

. Functions Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation

### Algebra Practice Set. *Evaluate number and algebraic expressions using rational numbers and Order of Operations

Algebra Practice Set Expressions *Evaluate number and algebraic expressions using rational numbers and Order of Operations *Translate algebraic expressions into words using appropriate language *Write

### 12.2 Adding Rational Numbers

Rational Numbers. Rational Numbers. Adding Rational Numbers. Subtracting Rational Numbers. Multiplying and Dividing Rational Numbers On the count of, I m going to give you half of my dog biscuits.,,,,,,,...

### Name Class Date. Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.

Practice - Solving Two-Step Equations Solve each equation. Check your answer.. a +. +. b +. 9 + t. a +. -t + Write an equation to model each situation. Then solve.. You want to buy a bouquet of yellow

### Module 1-A Linear Inequalities. C. x > 3 D. x < 3 A. 4.4 B. 4.5 C. 4.6 D. 4.7

Name: Date: 1. The inequality 3x + 2 > x + 8 is equivalent to. x > 3 2. x > 3 2 C. x > 3 D. x < 3 2. The inequality 2x > x + 7 is equivalent to. x > 7. x < 7 C. x = 7 D. x > 7 3 3. Which number is not

### Honors Math 2 Unit 5 Exponential Functions. *Quiz* Common Logs Solving for Exponents Review and Practice

Honors Math 2 Unit 5 Exponential Functions Notes and Activities Name: Date: Pd: Unit Objectives: Objectives: N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of

### Algebra 1 PAP Fall Exam Review

Name: Pd: 2016-2017 Algebra 1 PAP Fall Exam Review 1. A collection of nickels and quarters has a value of \$7.30. The value of the quarters is \$0.80 less than triple the value of the nickels. Which system

### Review: Expressions and Equations

Review: Expressions and Equations Expressions Order of Operations Combine Like Terms Distributive Property Equations & Inequalities Graphs and Tables Independent/Dependent Variables Constant: a number

### How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots

. Approximating Square Roots How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots Work with a partner. Archimedes was a Greek mathematician,

### Algebra 1 Semester Exam

Algebra 1 Semester Exam 015-016 30 questions Topics: Expressions, Equations, and Inequalities Functions inear Function Systems of Equations and Inequalities Reporting Category 1: (50% of the Semester Exam)

### GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using

GRADE 7 MATH LEARNING GUIDE Lesson 26: Solving Linear Equations and Inequalities in One Variable Using Guess and Check Time: 1 hour Prerequisite Concepts: Evaluation of algebraic expressions given values

### Fair Game Review. Chapter. Name Date. Simplify the expression. Explain each step. 2. ( ) Big Ideas Math Red Record and Practice Journal

Name Date Chapter 1 Fair Game Review Simplify the expression. Explain each step. 1. 2 + ( 5 + y) 2. ( ) c + 1 + 9 3. ( 2.3 + n) + 1.4 4. 7 + ( d + 5) 5. 10( 7t ) 6. 84k ( ) Copyright Big Ideas Learning,

### Chapter 3 ( ) ( ) Section 3.1. Chapter 3 Opener. Big Ideas Math Red Worked-Out Solutions. 3.1 Activity (pp ) Try It Yourself (p.

Chapter Chapter Opener Try It Yourself (p. 9). y (). y () 9. y + + +. y () 9. q + 6. n 9. 6 p. h 9. t + 0. c Section.. Activity (pp. 0 ). a. Epression Value when 0 A. + + B. ( ) + C. + ( + ) D. + + E.

### CCGPS UNIT 1 Semester 1 COORDINATE ALGEBRA Page 1 of 33. Relationships Between Quantities Name:

CCGPS UNIT 1 Semester 1 COORDINATE ALGEBRA Page 1 of 33 Relationships Between Quantities Name: Date: Reason quantitatively and use units to solve problems. MCC9-12.N.Q.1 Use units as a way to understand

### Reteaching Subtracting Real Numbers

Name Date Class Reteaching Subtracting Real Numbers 6 You have added real numbers. Now you will subtract real numbers. Two numbers with the same absolute value but different signs are called opposites.

### 2.6 The Distributive Property

Page 1 of 8 2.6 The Distributive Property What you should learn GOAL 1 Use the distributive property. GOAL 2 Simplify epressions by combining like terms. Why you should learn it To solve real-life problems

### CHAPTER FIVE. g(t) = t, h(n) = n, v(z) = z, w(c) = c, u(k) = ( 0.003)k,

CHAPTER FIVE 5.1 SOLUTIONS 121 Solutions for Section 5.1 EXERCISES 1. Since the distance is decreasing, the rate of change is negative. The initial value of D is 1000 and it decreases by 50 each day, so

### 5. Determine the discriminant for each and describe the nature of the roots.

4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following

### For Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c

9.5 Factor x2 1 bx 1 c Before You factored out the greatest common monomial factor. Now You will factor trinomials of the form x 2 1 bx 1 c. Why So you can find the dimensions of figures, as in Ex. 61.