Multi-Step Equations and. Inequalities

Transcription

1 A Solving Multi-Step Equations - Properties of Rational Numbers - Simplifying Algebraic Expressions - Solving Multi-Step Equations LAB Model Equations with Variables on Both Sides - Solving Equations with Variables On Both Sides Multi-Step Equations and Inequalities B Solving Inequalities - Inequalities - Solving Inequalities by Adding or Subtracting - Solving Inequalities by Multiplying or Dividing - Solving Two-Step Inequalities KEYWORD: MTCA Ch Equations and inequalities can describe how fast water is flowing in streams and rivers. Merced River, Yosemite National Park Chapter

2 Vocabulary Choose the best term from the list to complete each sentence. algebraic expression. A letter that represents a value that can change is equation called a(n)?. numerical expression. A(n)? has one or more variables. variable. A(n)? is a mathematical sentence that uses an equal sign to show that two expressions have the same value.. A mathematical phrase that contains operations and numbers is called a(n)?. It does not have an equal sign. Complete these exercises to review skills you will need for this chapter. Operations with Fractions Evaluate each expression Solve One-Step Equations Use mental math to solve each equation.. x. p. v. b. t. w Connect Words and Equations Write an equation to represent each situation.. The perimeter P of a rectangle is the sum of twice the length and twice the width w.. The volume V of a rectangular prism is the product of its three dimensions: length, width w, and height h.. The surface area S of a sphere is the product of π and the square of the radius r.. The cost c of a telegram of words is the cost f of the first words plus the cost a of each additional word. Multi-Step Equations and Inequalities

3 The information below unpacks the standards. The academic vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard AF. Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). (Lesson -) Academic Vocabulary variable a symbol, usually a letter, used to show an amount that can change Example: x verbal using words Chapter Concept You rewrite a verbal statement using mathematical symbols. Example: a number is greater than n > AF. Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. (Lessons - and -) AF. Students solve simple linear equations and inequalities over the rational numbers. (Lessons - and -) property a characteristic of numbers, operations, or equations Example: One property of addition is that you can add numbers in any order without changing the sum. justify give a reason for solve find the value or values of an unknown quantity that make one side of an equation equal to the other side (make the equation true) You use mathematical properties to simplify expressions. You give reasons for each step when you simplify expressions. You find the values of a variable that make an inequality true. AF. Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solutions in the context from which they arose, and verify the reasonableness of the results. (Lesson -) interpret to understand and explain the meaning of context in this case, a real-world situation You understand and can explain the meaning of solutions to inequalities. Chapter

4 Reading Strategy: Read a Lesson for Understanding Before you begin reading a lesson, find out which standard or standards are the main focus of the lesson. These standards are located at the top of the first page of the lesson. Reading with the standards in mind will help guide you through the lesson material. You can use the following tips to help you follow the math as you read. Lesson Features Reading Tips Identify the standard or standards of the lesson. Then skim through the lesson to get a sense of how the standards are covered. Work through each example. The examples help to demonstrate the standards. Check your understanding of the lesson by answering the Think and Discuss questions. Try This Use Lesson - in your textbook to answer each question.. What is the standard of the lesson?. What questions or problems did you have when you read the lesson?. Write your own example problem similar to Example.. What skill is being practiced in the second Think and Discuss question? Multi-Step Equations and Inequalities

5 - Properties of Rational Numbers California Standards AF. Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Vocabulary Commutative Property Associative Property Distributive Property Why learn this? You can use mental math and properties of rational numbers to calculate costs when shopping. (See Exercises and.) In Chapter, you performed operations with rational numbers. The following properties are useful when you simplify expressions that contain rational numbers. PROPERTIES OF ADDITION AND MULTIPLICATION Words Commutative Property You can add numbers in any order. You can multiply numbers in any order. Associative Property When you are only adding or only multiplying, changing the grouping will not affect the sum or product. Numbers ( ) ( ) Algebra a b b a ab ba (a b) c a (b c) (a b) c a (b c) EXAMPLE Identifying Properties of Addition and Multiplication Name the property that is illustrated in each equation. ( x) ( ) x ( x) ( ) x The factors are grouped differently. Associative Property of Multiplication () () () () The order of the numbers changed. Commutative Property of Addition You can use the properties of rational numbers to rearrange or regroup numbers in a way that helps you do math mentally. Chapter Multi-Step Equations and Inequalities

6 EXAMPLE Using the Commutative and Associative Properties Simplify each expression. Justify each step. Compatible numbers help you do math mentally. Try to make multiples of or. They are simpler to use when multiplying. Commutative Property of Addition ( ) Associative Property of Addition Add. Commutative Property of Multiplication Associative Property of Multiplication Multiply. The Distributive Property is also helpful when you do math mentally. DISTRIBUTIVE PROPERTY Numbers Algebra ( ) ( ) a(b c) a b a c a(b c) a b a c When you need to find the product of two numbers, write one of the numbers as a sum or difference. Then use the Distributive Property to help you find the product mentally. EXAMPLE Break the larger factor into a sum or difference that contains a multiple of. Using the Distributive Property Write each product using the Distributive Property. Then simplify. () () ( ) Rewrite as a sum. () Distributive Property Multiply. Then add. () ( ) Rewrite as a difference. Distributive Property Multiply. Then subtract. Think and Discuss. Explain which property you would use to simplify. (. ).. Describe two ways to use the Distributive Property to find. - Properties of Rational Numbers

7 - Exercises California Standards Practice AF. KEYWORD: MTCA - KEYWORD: MTCA Parent See Example See Example GUIDED PRACTICE Name the property that is illustrated in each equation.. y y. () () Simplify each expression. Justify each step ( ). See Example Write each product using the Distributive Property. Then simplify.. (). (). (). (). (). () See Example See Example INDEPENDENT PRACTICE Name the property that is illustrated in each equation.. x x. (.) (. ) Simplify each expression. Justify each step... (. ).... (.). See Example Write each product using the Distributive Property. Then simplify.. (). (). (). (). (). () Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Name the property that is illustrated in each equation.. ( x) x.. ( y) z (y z). =. m n n m. ( t) t Simplify each expression. Justify each step..... Consumer Math Mikiko is buying five DVDs at SaveMart. How can she use the Distributive Property to find the total cost of the DVDs before tax?. Consumer Math Jerome is buying a DVD, a pair of jeans, and a t-shirt at SaveMart. Show how he can use properties of rational numbers to find the total cost of the items before tax. SaveMart Price List Item Price DVD $ Jeans $ T-Shirt $ Chapter Multi-Step Equations and Inequalities

8 Write an example of each property using rational numbers.. Distributive Property. Associative Property of Multiplication. Commutative Property of Addition Complete each equation. Then name the property that is illustrated in each.. ( )... (x y) x ( ). ( z) z. Weather Leann wants to know the total amount of rainfall in Berkeley, California, from through. Explain how she can use mental math and properties of rational numbers to calculate this amount.. Reasoning Make a conjecture: Is division of rational numbers commutative? Explain your thinking.. What s the Error? A student writes, You can use the Associative Property of Addition to change the order of two numbers before you add them. What is the student s error?. Write About It A case of cat food has cans. Explain how to use mental math and the Distributive Property to find the number of cans in cases.. Challenge Simplify the expression. Justify each step. Total rainfall (in.) Annual Rainfall, Berkeley, CA MCS_C_L A Year NS., NS., AF.. Multiple Choice The equation ( x) (x ) is an example of which property? A B Associative Property of Addition Commutative Property of Addition Commutative Property of Multiplication Distributive Property. Multiple Choice Which is an example of the Associative Property of Multiplication? A () y y () C ( ) C D B ( ) D ( ) Write each decimal as a fraction in simplest form. (Lesson -) Divide. Write each answer in simplest form. (Lesson -) Properties of Rational Numbers

9 - Simplifying Algebraic Expressions California Standards AF. Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Vocabulary term like terms coefficient constant equivalent expressions Who uses this? Consumers can simplify algebraic expressions to find the total cost of tickets. (See Exercise.) In the expression below, x,, y, and x are terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. Constant Coefficients Like terms, such as x and x, can be grouped together because they have the same variable raised to the same power. Often, like terms have different coefficients. A coefficient is a number that is multiplied by a variable in an algebraic expression. A constant is a number that does not change. Constants, such as,., and, are also like terms. When you combine like terms, you change the way an expression looks but not the value of the expression. Equivalent expressions have the same value for all values of the variables. EXAMPLE When you rearrange terms, move the operation in front of each term with that term. Combining Like Terms in One-Variable Expressions Combine like terms. x x x x Identify like terms. x Combine coefficients:. m m m m m m m m m m Identify like terms. Commutative Property Combine coefficients. Simplify. Chapter Multi-Step Equations and Inequalities

10 EXAMPLE Combining Like Terms in Two-Variable Expressions Combine like terms. a a b a a b Identify like terms. a b Combine coefficients:. k n n k k n n k Identify like terms; the coefficient of k is because k k. k k n n Commutative Property k n Combine coefficients: ; f g. f g No like terms f g To simplify an expression, perform all possible operations, including combining like terms. You may need to use the Associative, Commutative, or Distributive Properties. EXAMPLE Using the Distributive Property to Simplify Simplify (y ) y. (y ) y (y) () y y y y y y y Distributive Property Multiply. Identify like terms. Commutative Property y Combine coefficients:. y y y Think and Discuss. Describe the first step in simplifying the expression (y ) y.. Tell how many sets of like terms are in the expression in Example B. What are they? - Simplifying Algebraic Expressions

11 - Exercises California Standards Practice AF. KEYWORD: MTCA - KEYWORD: MTCA Parent See Example GUIDED PRACTICE Combine like terms.. x x. z z. f f f. g g. p p. x x x See Example. x y x y. x y y x. x y x y. p p z z. g h. h m h m See Example Simplify.. (r ) r. ( x) x. (t ) t. ( p) p. (y ). ( m) m See Example INDEPENDENT PRACTICE Combine like terms.. y y. z z. a a a. z z. x x. b b b See Example. z z b. a a z z. x y x y. x x q. d d e. a c a c See Example Simplify.. (y ) y. (y ) y. (x ) x. (y ). (x ) x. (x ) x Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING. Geometry A rectangle has length x and width x. Write and simplify an expression for the perimeter of the rectangle.. Hobbies Charlie has x state quarters. Ty has more quarters than Charlie has. Vinnie has times as many quarters as Ty has. Write and simplify an expression to show how many state quarters they have in all.. Reasoning Determine whether the expression r m is equivalent to (r m) (m r). Use properties to justify your answer. Simplify each expression. Justify each step.. ( k). d d d. x (y x). r r r r. y y z. (k ) k Chapter Multi-Step Equations and Inequalities

12 Write and simplify an expression for each situation.. Business A promoter charges $ for each adult ticket, plus an additional $ per ticket for tax and handling. What is the total cost of x tickets?. Sports Write an expression for the total number of medals won in the Summer Olympics by the countries shown below. United States Gold Silver Bronze Great Britain Gold Silver Bronze Brazil Gold Silver Bronze Write an algebraic expression for each verbal description. Then simplify the expression.. four times the sum of m and p, decreased by six times m. y squared minus twice the sum of x and y squared. the product of three and r, increased by the sum of nine, r, and one. What s the Error? A student said that x y can be simplified to xy by combining like terms. What error did the student make?. Write About It Write an expression that can be simplified by combining like terms. Then write an expression that cannot be simplified, and explain why it is already in simplest form.. Challenge Simplify the expression x x x. Lithuania Gold Silver Bronze AF., AF.. Multiple Choice Which expression is equivalent to p t p? A p t B p t C (p t) D t p. Gridded Response Simplify (x ) x. What is the coefficient of x? Solve. (Lesson -). x. a. w. Complete each equation. Then name the property that is illustrated in each. (Lesson -). (x ) x... ( ) ( ) - Simplifying Algebraic Expressions

13 - Solving Multi-Step Equations California Standards Extension of AF. Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Why learn this? You can solve problems about average speed by solving multi-step equations. (See Example.) A multi-step equation requires more than two steps to solve. To solve a multi-step equation, you may have to simplify the equation first by combining like terms. EXAMPLE Solving Equations That Contain Like Terms Solve x x. x x x x x x x x Commutative Property of Addition Combine like terms. Since is subtracted from x, add to both sides. Since x is multiplied by, divide both sides by. If an equation contains fractions, it may help to multiply both sides of the equation by the least common denominator (LCD) to clear the fractions before you isolate the variable. EXAMPLE Solving Equations That Contain Fractions Solve. y y y y Multiply both sides by. Distributive Property y y y y Simplify. Since is added to y, subtract from both sides. Since y is multiplied by, divide both sides by. Chapter Multi-Step Equations and Inequalities

14 The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly. See Skills Bank p. SB. Solve. p p p p p p p p p p p p p p Multiply both sides by, the LCD of the fractions. Distributive Property Simplify. Combine like terms. Since is subtracted from p, add to both sides. Since p is multiplied by, divide both sides by. EXAMPLE Travel Application On the first day of her vacation, Carly drove m miles in hours. On the second day, she drove twice as far in hours. If her average speed for the two days was. mi/h, how far did she drive on the first day? Round your answer to the nearest tenth of a mile. Carly s average speed is her total distance for the two days divided by the total time. total distance total time average speed m m Substitute m m for total distance. and for total time. m. Simplify. m (.) Multiply both sides by. m. m. Divide both sides by. m. Carly drove approximately. miles on the first day. Think and Discuss. List the steps required to solve x x.. Tell how you would clear the fractions in x x. - Solving Multi-Step Equations

15 - See Example See Example Exercises GUIDED PRACTICE California Standards Practice Extension of AF.; AF. Solve.. d d. y y. e e. c c. h h h. x x x y y... p. z KEYWORD: MTCA - KEYWORD: MTCA Parent See Example See Example See Example See Example Extra Practice See page EP.. Travel Barry s family drove mi to see his grandparents. On the first day, they drove mi. On the second day, how long did it take to reach Barry s grandparents house if they averaged mi/h? INDEPENDENT PRACTICE Solve.. n n n. k k. c c. w w. a a. y y. p. h h. g g. m m. b b. x x. Recreation Lydia rode miles in a three-day bike trip. On the first day, Lydia rode miles. On the second day, she rode miles. How many miles per hour did she average on the third day if she rode for hours? PRACTICE AND PROBLEM SOLVING Solve and check.. n. n n x. b b.. x x. r. y y. n n. Finance Alessia is paid. times her normal hourly rate for each hour she works over hours in a week. Last week she worked hours and earned $.. What is her normal hourly rate? Chapter Multi-Step Equations and Inequalities

16 Sports. Geometry The obtuse angle of an isosceles triangle measures. Write and solve an equation to find the measure of the base angles. (Hint: An isosceles triangle has two congruent angles. An obtuse angle measures more than but less than.). Reasoning The sum of two consecutive numbers is. What are the two numbers? Explain your solution. You can estimate the weight in pounds of a fish that is L inches long and G inches around at the thickest part by using the formula LG W.. Sports The average weight of the top fish caught at a fishing tournament was. pounds. The weights of the second-, third-, fourth-, and fifth-place fish are shown in the table. What was the weight of the heaviest fish?. Science The formula K F. is used to convert a temperature from degrees Fahrenheit to kelvins. Water boils at kelvins. Use the formula to find the boiling point of water in degrees Fahrenheit. Winning Entries Caught by Wayne S. Carla P. Deb N. Virgil W. Brian B. Weight (lb)..... What s the Error? A student s work in solving an equation is shown. What error has the student made, and what is the correct answer? x x x x x x. Write About It Compare the steps you would use to solve the equations x and (x ).. Challenge Solve the following equation. x x AF., AF., Ext. of AF.. Multiple Choice Solve k k. A k B k. C k D k.. Gridded Response Antonio s first four test grades were,,, and. What must he score on the next test to have an test average? Solve. (Lesson -). n. y x. Combine like terms. (Lesson -). m m m. t k. a b - Solving Multi-Step Equations

17 - Model Equations with Variables on Both Sides Use with Lesson - KEY Algebra tiles + x x + REMEMBER Adding or subtracting zero does not change the value of an expression. + + To solve an equation with the same variable on both sides of the equal sign, you must first add or subtract to eliminate the variable term from one side of the equation. KEYWORD: MTCA Lab California Standards Extension of AF. Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Activity Model and solve the equation x x x x Add x to both sides Remove zero. Add to both sides Think and Discuss Remove zero. Divide each side into equal groups. of each side is the solution.. How would you check the solution to x x using algebra tiles?. Why must you isolate the variable terms by having them on only one side of the equation? x Try This Model and solve each equation.. x x. x x. x x. x x x Chapter Multi-Step Equations and Inequalities

18 - Solving Equations with Variables on Both Sides California Standards Extension of AF. Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Also covered: AF. Who uses this? Consumers can use equations with variables on both sides to compare costs. (See Example.) The fees for two dogsitting services are shown at right. To find the number of hours for which the costs will be the same for both services, you can write and solve an equation with variables on both sides of the equal sign. To solve an equation like this, first use inverse operations to collect variable terms on one side of the equation. EXAMPLE You can always check your solution by substituting the value back into the original equation. Solving Equations with Variables on Both Sides Solve. a a a a a a a Check a a ()? ()?? To collect the variable terms on one side, subtract a from both sides. Substitute for a in the original equation. v v v v v v v v v v To collect the variable terms on one side, subtract v from both sides. Since is added to v, subtract from both sides. Since v is multiplied by, divide both sides by. - Solving Equations with Variables on Both Sides

19 If the variables in an equation are eliminated and the resulting statement is false, the equation has no solution. Solve. g g g g g g To collect the variable terms on one side, subtract g from both sides. There is no solution. There is no number that can be substituted for the variable g to make the equation true. To solve more complicated equations, you may need to first simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable. EXAMPLE Solving Multi-Step Equations with Variables on Both Sides Solve. c c c c c c c c c c c c c c Combine like terms. To collect the variable terms on one side, add c to both sides. Since is add to c, add to both sides. Since c is multiplied by, divide both sides by. w w w w w w w w w Multiply both sides by w w (w), the LCD. Distributive Property w w w w w Combine like terms. w w Add w to both sides. w Subtract from w both sides. w Divide both sides by. w Chapter Multi-Step Equations and Inequalities

20 A system of equations is a set of two or more equations that contain two or more variables. To solve a system of two equations, you can reduce the system to one equation that has only one variable. EXAMPLE Business Application Happy Paws charges a flat fee of $. plus $. per hour to keep a dog. Woof Watchers charges a flat fee of $. plus $. per hour. Find the number of hours for which you would pay the same amount for both services. What is the cost? Write an equation for each service. Let c represent the total cost and h represent the number of hours. total cost is flat fee plus cost per hour Happy Paws: Woof Watchers: c.. h c.. h Now write an equation showing that the costs are equal...h..h.h.h To collect the variable terms on one...h side, subtract.h from both sides... Subtract. from both sides...h..h Divide both sides by..... h The two services cost the same when used for. hours. To find the cost, substitute. for h in either equation. Happy Paws: Woof Watchers: c..h c..h c..(.) c..(.) c.. c.. c. c. The cost for. hours at either service is $.. Think and Discuss. Explain how you would solve the equation x x x x. What do you think the solution means?. Give a series of steps that you can use to solve any equation with variables on both sides of the equal sign. - Solving Equations with Variables on Both Sides

21 - Exercises See Example See Example GUIDED PRACTICE California Standards Practice AF., Extension of AF.; MG. Solve.. x x. a a. x x. y y KEYWORD: MTCA - KEYWORD: MTCA Parent. x x. t t. x x x. n n n. d d d. (x ) x See Example. Business A long-distance phone company charges $. per minute and a $ monthly fee. Another long-distance phone company charges $. per minute with no monthly fee. Find the number of minutes for which the charges for both companies would be the same. What is the cost? See Example See Example See Example INDEPENDENT PRACTICE Solve.. n n. x x. n n. (x ) x. x x. y y. p p p. (x ) x. x. (n ) n n. a. a a. Business Al s Rentals charges $ per hour to rent a Windsurfer and a wet suit. Wendy s charges $ per hour plus $ extra for a wet suit. Find the number of hours for which the total charges for both would be the same. What is the cost? Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Solve and check.. y y. n n. n n (n ). (x ) x. (x ) x..p...p. Find two consecutive whole numbers such that of the first number is more than the second number. (Hint: Let n represent the first number. Then n represents the next consecutive whole number.) Chapter Multi-Step Equations and Inequalities

22 Science Sodium and chlorine bond together to form sodium chloride, or salt. The atomic structure of sodium chloride causes it to form cubes. Write an equation to represent each relationship. Then solve the equation.. Six plus the product of and a number is the same as the product of and the number.. A number decreased by is the same as minus times the number.. Eight less than times a number is the same as the number increased by. The figures in each pair have the same perimeter. Find each perimeter.. x. x x x x x x x x x. Science An atom of chlorine (Cl) has more protons than an atom of sodium (Na). The atomic number of chlorine is less than twice the atomic number of sodium. The atomic number of an element is equal to the number of protons per atom. a. How many protons are in an atom of chlorine? b. What is the atomic number of sodium?. Choose a Strategy Solve the following equation for t. How can you determine the solution once you have combined like terms? (t ) t (t ). Write About It Two cars are traveling in the same direction. The first car is going mi/h, and the second car is going mi/h. The first car left hours before the second car. Explain how you could solve an equation to find how long it will take the second car to catch up to the first car.. Challenge Solve the equation x x. NS., AF.. Multiple Choice Find three consecutive integers (x, x, and x ) so that the sum of the first two integers is more than the third integer. A,, B,, C,, D,,. Multiple Choice Solve w w. A w B w C w D w Solve. (Lesson -) n. x.. g y. Compare. Write,, or. (Lesson -) Solving Equations with Variables on Both Sides

23 Quiz for Lessons - Through - - Properties of Rational Numbers Name the property that is illustrated in each equation... m n n m. x x Simplify each expression. Justify each step Write each product using the Distributive Property. Then simplify.. (). (). () - Simplifying Algebraic Expressions Simplify.. x x. p p. t t t. x y x y. (r ) r. n m n m - Solving Multi-Step Equations Solve.. c c. x t t.. m m. b b. r r. Marlene drove miles to visit a friend. She drove hours and stopped for gas. She then drove hours and stopped for lunch. How many more hours did she drive if her average speed for the trip was miles per hour? - Solving Equations with Variables on Both Sides Solve.. x x. q q. n n. m m. a a a. y y. The rectangle and the triangle have the same perimeter. x Find the perimeter of each figure. x x x x Chapter Multi-Step Equations and Inequalities

24 Make a Plan Write an equation California Standards MR. Analyze problems by identifying relationships, distinguishing relevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Also covered: AF., Extension of AF. Several steps may be needed to solve a problem. It often helps to write an equation that represents the steps. Example: Juan s first exam scores are,, and. What does he need to score on his next exam to average for the exams? Let x be the score on his next exam. The average of the exam scores is the sum of the scores, divided by. This amount must equal. Write the equation in words: Exam Exam Exam Exam Number of exams x x x () x x Juan needs a on his next exam. Read each problem and write an equation that could be used to solve it. The average of two numbers is. The first number is three times the second number. What are the two numbers? Nancy spends of her monthly salary on rent,. on her car payment, on food, and % on other bills. She has $ left for other expenses. What is Nancy s monthly salary? A vendor at a concert sells new and used CDs. The new CDs cost. times as much as the old CDs. If used CDs and new CDs cost $, what is the price of each item? Amanda and Rick have the same amount to spend on school supplies. Amanda buys notebooks and has $. left. Rick buys notebooks and has $. left. How much does each notebook cost?

25 - Inequalities California Standards AF. Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). Vocabulary inequality algebraic inequality solution set Why learn this? You can show the maximum capacity of an elevator using an inequality. An inequality compares two expressions using,,, or. Symbol Meaning Word Phrases Is less than Fewer than, below Is greater than More than, above Is less than or equal to At most, no more than Is greater than or equal to At least, no less than An inequality that contains a variable is an algebraic inequality. EXAMPLE Translating Word Phrases into Inequalities Write an inequality for each situation. The capacity of an elevator is at most people. Let c the capacity of the elevator. c At most means less than or equal to. There are more than books in the library. Let b the number of books in the library. b More than means greater than. EXAMPLE Writing Inequalities Write an inequality for each statement. A number x plus is greater than or equal to. A number x plus is greater than or equal to x x A number n decreased by is less than. A number n decreased by is less than n n Chapter Multi-Step Equations and Inequalities

26 A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set. You can graph the solution set on a number line. The symbols and indicate an open circle. This open circle shows that is not a solution. The symbols and indicate a closed circle. This closed circle shows that is a solution. EXAMPLE m is the same as m. Graphing Inequalities Graph each inequality. x m Draw an open circle at. The solutions are all values of x greater than, so shade to the right of. Draw a closed circle at. The solutions are and all values of m less than, so shade to the left of. A compound inequality is the result of combining two inequalities. The words and and or are used to describe how the two parts are related. EXAMPLE The compound inequality in Example B can also be written with the variable between the two endpoints. n. Writing Compound Inequalities Write a compound inequality for each statement. A number t is either less than or greater than or equal to. t or t A number n is both greater than or equal to and less than.. n and n. Think and Discuss. Explain how to write x is no less than as an inequality.. Compare the graphs of the inequalities x and x. - Inequalities

27 - Exercises AF. California Standards Practice KEYWORD: MTCA - KEYWORD: MTCA Parent See Example GUIDED PRACTICE Write an inequality for each situation.. There are no more than people in the theater.. The temperature of the water is above F. See Example Write an inequality for each statement.. A number m increased by is at least.. Twice a number x is less than. See Example See Example Graph each inequality.. x. w.. y. m Write a compound inequality for each statement.. A number s is either less than or greater than or equal to.. A number t is both greater than and less than. See Example INDEPENDENT PRACTICE Write an inequality for each situation.. Fewer than students rode their bikes to the game.. No more than people may ride the roller coaster at one time. See Example Write an inequality for each statement.. A number x decreased by is less than.. Three times a number n is greater than.. A number y divided by is at most. See Example Graph each inequality.. m. s.. x. y. b. x. n. c See Example Write a compound inequality for each statement.. A number x is both less than. and greater than or equal to.. A number c is either greater than or equal to or less than or equal to. Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING. Suly earned points on her first test and p points on her second test. She needs a total of at least points on the two tests to pass the class. Write an inequality for this situation. Chapter Multi-Step Equations and Inequalities

28 Write an inequality for each statement.. A number w multiplied by is no less than.. The sum of and a number g is greater than... A number m decreased by is at most. Write an inequality shown by each graph Business A cafe sells fruit smoothies for $. each. The manager of the cafe wants the total daily revenue from the smoothies to be at least $. Assume the cafe sells n smoothies per day. Write an inequality that represents the manager s goal.. Astronomy The diameter of Jupiter, the largest planet in the Solar System, is, miles. Let d be the diameter of any planet in the Solar System. Write an inequality for d.. What s the Error? A student was asked to graph the inequality n. Explain the student s error in the graph at right.. Write About It In mathematics, the conventional way to write an inequality is with the variable on the left, such as x. Explain how to rewrite the inequality x in the conventional way.. Challenge Write an inequality for the statement less than twice a number x is greater than three times the number. NS., NS., AF.. Multiple Choice Which inequality represents the statement, A number z decreased by is no more than? A z B z C z D z. Multiple Choice Which inequality is shown by the graph? A x B x C x D x Write each set of integers in order from least to greatest. (Lesson -).,,.,,,.,,,.,,, Add or subtract. Write each answer in simplest form. (Lesson -) Inequalities

29 - Solving Inequalities by Adding or Subtracting California Standards AF. Students solve simple linear equations and inequalities over the rational numbers. Why learn this? You can solve an inequality to find the amount of a nutrient that you should consume. (See Example.) When you add or subtract the same number on both sides of an inequality, the resulting inequality will still be true. You can use this idea to solve inequalities. You find solution sets of inequalities the same way you find solutions of equations, by isolating the variable. EXAMPLE Solving Inequalities by Adding or Subtracting Solve and graph. x x x Since is added to x, subtract from both sides. When checking your solution, choose numbers that are easy to work with. Remember to substitute the numbers into the original inequality. Check According to the graph, should be a solution and should not be a solution. x x? Substitute? Substitute? for x.? for x. So is a solution. So is not a solution. t t t Since is subtracted from t, add to both sides. Chapter Multi-Step Equations and Inequalities

30 Solve and graph. z z z Since is added to z, subtract from both sides. n n n Since is subtracted from n, add to both sides. EXAMPLE Nutrition Application Manganese is a mineral that is found in nuts and grains. It is recommended that men consume at least. mg of manganese each day. Eric has consumed. mg today. Write and solve an inequality to find how many additional milligrams he should consume. Let m the number of additional milligrams of manganese.. milligrams plus additional milligrams is at least. milligrams. m.. m... m. Since. is added to m, subtract. from both sides. Eric should consume at least. additional milligrams of manganese. Check is greater. m.. m..? than....?..? Substitute. for m..?. is less than.. Substitute for m. Think and Discuss. Explain how you know whether to use addition or subtraction to solve an inequality.. Describe how to check whether is a solution of t. - Solving Inequalities by Adding or Subtracting

31 - Exercises California Standards Practice NS., AF., AF. KEYWORD: MTCA - See Example See Example GUIDED PRACTICE KEYWORD: MTCA Parent Solve and graph.. x. b. f. z.. k.. x. A measuring cup can hold no more than fluid ounces of liquid. Rosa pours fluid ounces of water into the cup. Write and solve an inequality to determine how many additional fluid ounces of water she can add.. Paul s car can go at most miles on one tank of gas. Paul fills the tank and then drives miles. Write and solve an inequality to find out how many more miles Paul can drive before he will have to refill the tank. See Example See Example INDEPENDENT PRACTICE Solve and graph.. x. t. x.. y.. c. a (). Consumer Math A clothes store gives customers a free gift if they spend at least $ in the store. Stacey plans to buy a pair of jeans that cost $.. Write and solve an inequality to show how much more she must spend in order to get the free gift.. Consumer Math Latrell s cell-phone plan allows him to talk for no more than minutes per month. He has already used minutes this month. Write and solve an inequality to determine how many more minutes he can talk on the phone this month. Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Solve and graph.. z... x. y.. m (). k... g You can use set-builder notation to write the solution of an inequality. For example, {x : x } means the set of all real numbers x such that x is less than. Solve each inequality and write the solution using set-builder notation.. x. z.. b. Reasoning When a number is added to, the result is greater than. What are the possible values of the number? Graph them on a number line. Chapter Multi-Step Equations and Inequalities

32 California Language Arts Californian John Steinbeck, author of The Grapes of Wrath and Of Mice and Men, won the Nobel Prize for Literature in.. Business Toshi Business Solutions will make a profit for the current year if their total sales are greater than their operating costs. Their accountants estimate that the company will have operating costs of $, for the entire year. So far this year, the company has sales of $,. a. Write and solve an inequality to find out how much more money Toshi must earn in sales for the remainder of the year to show a profit. b. Check your answer. Then explain why your answer is reasonable.. Language Arts Danielle is reading one of the novels in the graph. She has already read pages. Write and solve two different inequalities to find out how many pages she has left to read. (Hint: Write one inequality based on the minimum number of pages and one inequality based on the maximum number of pages.). Reasoning Substitute the values,,,,, and for x in x. Use the results to make a conjecture about the solution of the inequality.. Write a Problem Write a word problem that can be answered by solving the inequality x.. Write About It Explain how to check the solution of an inequality.. Challenge The inequality y is missing a number. The solution of the inequality is shown on the number line. What is the missing number? Title Great Novels Number of pages NS., AF., AF.. Multiple Choice Solve m for m. A m B m C m D m. Multiple Choice In the inequality x, x is the length of a movie in minutes. Which phrase most accurately describes the length of the movie? A B At least minutes More than minutes At most minutes Less than minutes Add or subtract. (Lesson -).... Write an inequality for each statement. (Lesson -). A number t increased by is less than.. Twice a number w is no more than. C D - Solving Inequalities by Adding or Subtracting

33 - Solving Inequalities by Multiplying or Dividing California Standards AF. Students solve simple linear equations and inequalities over the rational numbers. Also covered: AF. Why learn this? You can solve an inequality to determine how many representatives voted on a bill. (See Exercise.) When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true. b a a b a b b a a b Multiply both sides by. b a Multiply both sides by. a b Use the number line to b a Use the number line to determine the direction determine the direction of the inequality symbol. of the inequality symbol. EXAMPLE When graphing an inequality on a number line, an open circle means that the point is not part of the solution and a closed circle means that the point is part of the solution. Solving Inequalities by Multiplying or Dividing Solve and graph. h h Multiply both sides by. h, or h Check According to the graph, should be a solution and should not be a solution. h h? Substitute??. for h.?. So is a solution. Substitute for h. So is not a solution. x x x Divide both sides by ; changes to. Chapter Multi-Step Equations and Inequalities

34 EXAMPLE PROBLEM SOLVING APPLICATION If all the sheets of paper used by personal computer printers each year were laid end to end, they would circle Earth more than times. Earth s circumference is about, mi (,,, in.), and one letter-size sheet of paper is in. long. About how many sheets of paper are used each year? Understand the Problem The answer is the number of sheets of paper used by personal computer printers in one year. List the important information: The amount of paper would circle the earth more than times. Once around Earth is approximately. billion in. One sheet of paper is in. long. Show the relationship of the information: the number of sheets of paper the length of one sheet the distance around Earth Make a Plan Use the relationship to write an inequality. Let x represent the number of sheets of paper. x in.. billion in. Solve x. x Simplify. x Divide both sides by. x. More than billion sheets of paper are used by personal computer printers in one year. Look Back To circle Earth once takes,,,,, sheets of paper; to circle it times would take,,,,, sheets. Think and Discuss. Give all the symbols that make true. Explain.. Explain how you would solve the inequality x. - Solving Inequalities by Multiplying or Dividing

35 - Exercises AF., California Standards Practice AF. KEYWORD: MTCA - See Example See Example See Example See Example Extra Practice See page EP. GUIDED PRACTICE KEYWORD: MTCA Parent Solve and graph. r.. w j.. r a r.. m.. x. The owner of a sandwich shop is selling the special of the week for $.. At this price, he makes a profit of $. on each sandwich sold. To make a total profit of at least $ from the special, what is the least number of sandwiches he must sell? INDEPENDENT PRACTICE Solve and graph. x p. r.. w. t a.. h.. q. Social Studies A bill in the U.S. House of Representatives passed because at least of the members present voted in favor of it. If the bill received votes, at least how many members of the House of Representatives were present for the vote? PRACTICE AND PROBLEM SOLVING Solve and graph. x p. r.. w. t.. h. a. q Write and solve an algebraic inequality.. Nine times a number is less than.. The quotient of a number and is at least.. The product of and a number is no more than.. The quotient of some number and is greater than. Write and solve an algebraic inequality. Then explain the solution.. A school receives a shipment of books. There are cartons, and each carton weighs pounds. The school s elevator can hold pounds. What is the greatest number of cartons that can be carried on the elevator at one time if no people ride with them?. Each evening, Marisol spends at least twice as much time reading as she spends doing homework. If Marisol works on her homework for minutes, how much time can she spend reading? Chapter Multi-Step Equations and Inequalities

36 Choose the graph that represents each inequality.. y A. B. C.. h A. B. C.. What s the Error? A student solved x and got an answer of x. What error did the student make?. Write About It The expressions no more than, at most, and less than or equal to all indicate the same relationship between values. Write a problem that uses this relationship. Write the problem using each of the three expressions.. Challenge Angel weighs times as much as his dog. When they stand on a scale together, the scale gives a reading of less than pounds. If both their weights are whole numbers, what is the most each can weigh? NS., AF., AF.. Multiple Choice Which inequality is shown by the graph? A w B w C w D w. Gridded Response In order to have the $ he needs for a bike, Kevin plans to put money away each week for the next weeks. What is the minimum amount in dollars that Kevin will need to average each week in order to reach his goal? Multiply. Write each answer in simplest form. (Lesson -)..... Frank needs to earn at least $. He earns $ for each hour h that he babysits. Write an inequality that represents Frank s goal. (Lesson -) - Solving Inequalities by Multiplying or Dividing

37 - Solving Two-Step Inequalities California Standards AF. Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Why learn this? Drama club members can use two-step inequalities to determine how many tickets they must sell to a musical to break even. (See Example.) When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities. EXAMPLE Math Builders For more on solving two-step inequalities, see the Step-by-Step Solution Builder on page MB. Solving Two-Step Inequalities Solve and graph. y y Since is subtracted from y, add to y both sides. y Since y is multiplied by, divide both sides by. y Check According to the graph, should be a solution and should not be a solution. y y ()? Substitute ()? Substitute? for y.? for y. So is a solution. So is not a solution. If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. x x Since is added to x, subtract from x both sides. x x Since x is multiplied by, divide both sides by. Change to. Chapter Multi-Step Equations and Inequalities

38 EXAMPLE When an inequality contains fractions, you may want to multiply both sides by the LCD to clear the fractions. Solving Inequalities That Contain Fractions Solve x and graph the solution. x Multiply by the LCD,. x Distributive Property x Since is added to x, subtract x from both sides. x x Since x is multiplied by, divide both sides by. Change to. x Simplify. EXAMPLE School Application The Drama Club is planning a spring musical. Club members estimate that the entire production will cost $.. If they have $. left from fund-raising, how many tickets must they sell to at least break even? In order to at least break even, ticket sales plus the money in the budget must be greater than or equal to the cost of the production..t.... Subtract. from both sides..t... t. Divide both sides by... t The drama club must sell at least tickets in order to break even. Think and Discuss. Compare solving a multi-step equation with solving a multi-step inequality.. Describe two situations in which you would have to reverse the inequality symbol when solving a multi-step inequality. - Solving Two-Step Inequalities

39 - Exercises California Standards Practice AF. KEYWORD: MTCA - See Example See Example See Example GUIDED PRACTICE KEYWORD: MTCA Parent Solve and graph.. k. z... y. x. y... k x.. b. h c.. d. m. The chess club is selling caps to raise $ for a trip. They have $ already. If the club members sell caps for $ each, at least how many caps do they need to sell to make enough money for their trip? See Example See Example See Example INDEPENDENT PRACTICE Solve and graph.. k. x. p. q...n.. x p.. a. n. k. n. r. Josef is on the planning committee for the eighth-grade party. The food, decoration, and entertainment costs a total of $. The committee has $ already. If the committee sells the tickets for $ each, at least how many tickets must be sold to cover the remaining cost of the party? Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Solve and graph.. p. n. w. x. a. y. q q. m. b... k. f. v. Reasoning What is the least whole number that is a solution of r..?. Entertainment A speech is being given in a gymnasium that can hold no more than people. A permanent bleacher will seat people. The event organizers are setting up rows with an equal number of chairs. At most, how many chairs can be in each row? Chapter Multi-Step Equations and Inequalities

40 . Katie and April are making a string of beads for pi day (March ). The string already has beads. If there are only more days until pi day, and they want to string beads by then, at least how many beads do they have to string each day?.. Sports The Astros have won and lost baseball games. They have games remaining. At least how many of the remaining games must the Astros win to have a winning season? (Hint: A winning season means they win more than % of their games.). Economics Satellite TV customers can either purchase a dish and receiver for $ or pay a $ fee and rent the equipment for $ a month. a. How much would it cost to rent the equipment for months? b. How many months would it take for the rental charges to be more than the purchase price?. Write a Problem Write and solve an inequality using the following shipping rates for orders from a mail-order catalog. Mail-Order Shipping Rates Merchandise $. $. $. $. $. Amount $.... and over Shipping Cost $. $. $. $. $.. Write About It Describe two different ways to solve the inequality x x. x x. Challenge Solve the inequality.... AF., AF.. Multiple Choice Solve g. A g B g C g D g. Short Response Solve and graph x. Name the property that is illustrated in each equation. (Lesson -). y y. a (b c) (a b) c. x y y x Simplify. (Lesson -). (x ) x. (r ) r. ( n) - Solving Two-Step Inequalities

41 Quiz for Lessons - Through - - Inequalities Write an inequality for each statement.. A number n decreased by is no more than.. The product of and a number x is above. Graph each inequality.. r. a. c. h Write a compound inequality for each statement.. A number m is both greater than and less than or equal to.. A number d is either less than or greater than. - Solving Inequalities by Adding or Subtracting Solve and graph.. n. y. x. t. d.. n. - Solving Inequalities by Multiplying or Dividing Solve and graph.. x. k. y. m. n. h. Rachael is serving lemonade from a pitcher that holds ounces. What are the possible numbers of -ounce juice glasses she can fill from one pitcher? - Solving Two-Step Inequalities Solve and graph.. k..z... x. t. x. m. Jillian must average at least on two quiz scores before she can move to the next skill level. Jillian got a on her first quiz. What scores could Jillian get on her second quiz in order to move to the next skill level? Chapter Multi-Step Equations and Inequalities

42 Skate Away Ms. Lucinda wants to treat her class of students to a skating party to celebrate the end of the school year. Item Rink rental Skate rental (per person) Cost $ plus $ per hour $. plus $. per hour Refreshments $. (per person). Ms. Lucinda considers renting the rink at Skate Away. How much would it cost to rent the rink for x hours?. Another rink, Skate Palace, charges $ plus $ per hour to rent the rink. Write and solve an equation to find the number of hours for which the cost of renting the rink at Skate Palace is the same as the cost of renting the rink at Skate Away.. Ms. Lucinda decides to take the class to Skate Away. How much will it cost to rent skates for students for x hours? How much will it cost to buy refreshments for students?. Ms. Lucinda has budgeted $ for the party. Write and solve an inequality to find the maximum number of hours the class can have its party at Skate Away. Be sure to include the cost of the rink, the skates, and the refreshments.. The final bill for the party was $. How long did the party last? Concept Connection

43 Trans-Plants Solve each equation below. Then use the values of the variables to decode the answer to the question. a n b b.o...o.c..c p p p d d d d q q e e r r f s s (g ) t h h u u i j v v w w w w w w k k x x x x l l y y m.z What happens to plants that live in a math classroom?,,,,,,,,,,,,,,, Points This traditional Chinese game is played using a deck of cards numbered, with four of each number. The cards are shuffled, and four cards are placed face up in the center. The winner is the first player who comes up with an expression that equals, using each of the numbers on the four cards once. Complete rules and a set of game cards are available online. KEYWORD: MTCA Games Chapter Multi-Step Equations and Inequalities

44 Materials magazine glue scissors index cards PROJECT Picture Envelopes A Make these picture-perfect envelopes in which to store your notes on the lessons of this chapter. Directions Flip through a magazine and carefully tear out eight pages with full-page pictures that you like. B Lay one of the pages in front of you with the picture face down. Fold the page into thirds as shown, and then unfold the page. Figure A Fold the sides in, about inch, and then unfold. Cut away the four rectangles at the corners of the page. Figure B Fold in the two middle flaps. Then fold up the bottom and glue it onto the flaps. Figure C Cut the corners of the top section at an angle to make a flap. Figure D Repeat the steps to make seven more envelopes. Label them so that there is one for each lesson of the chapter. C D Taking Note of the Math Use index cards to take notes on the lessons of the chapter. Store the cards in the appropriate envelopes.

45 Vocabulary algebraic inequality Associative Property coefficient Commutative Property constant Distributive Property equivalent expressions inequality like terms solution set term Complete the sentences below with vocabulary words from the list above.. A(n)? is a statement that two quantities are not equal..? states that two or more numbers can be added in any order or multiplied in any order..? in an expression are set apart by plus or minus signs. - Properties of Rational Numbers (pp. ) AF. EXAMPLE Name the property that is illustrated in the equation. (x y) x y Distributive Property EXERCISES Name the property that is illustrated in each equation... x ( y) (x ) y. ( n) n. - Simplifying Algebraic Expressions (pp. ) AF. EXAMPLE Simplify. (z ) z z () z z z z Distributive Property z and z are like terms. Combine coefficients. EXERCISES Simplify.. (m ) m. w (w ). x y x. t t t Chapter Multi-Step Equations and Inequalities

46 - Solving Multi-Step Equations (pp. ) Ext. of AF. EXAMPLE Solve. x x x x Multiply both sides by. x x Distributive Property x x Simplify. x Combine like terms. Subtract from x both sides. x x Divide both sides by. EXERCISES Solve.. y y. h h. t. r z z.. a a - Solving Equations with Variables on Both Sides (pp. ) Ext. of EXAMPLE EXERCISES AF. Solve. x x x x x x x x x x x Combine like terms. Add x to both sides. Add to both sides. Divide both sides by. Solve.. s (s ). c c c. x x. y y. n n. z z - Inequalities (pp. ) AF. EXAMPLE Write an inequality for the situation. The capacity of the elevator was at most pounds. Let c capacity of elevator c lb at most means less than or equal to Graph x. EXERCISES Write an inquality for each situation.. It is no more than a one mile walk from home to the school.. The cost of the trip will be at least $.. Fewer than students are expected to attend the workshop. Graph each inequality.. m. x. c Study Guide: Review

47 - EXAMPLE Solve and graph. n Solving Inequalities by Adding or Subtracting (pp. ) x x n Add to both sides. Subtract from both sides. EXERCISES Solve and graph.. r. n. x.. y AF.. Ellory budgets at most $ each week for lunch. She has spent $. so far this week. Write and solve an inequality to determine how much more Ellory can spend and stay within her lunch budget. - EXAMPLE Solving Inequalities by Multiplying or Dividing (pp. ) Solve and graph. z z () () z Multiply both sides by. Change to. EXERCISES Solve and graph.. m. n. t. p AF.. b. a - Solving Two-Step Inequalities (pp. ) AF. EXAMPLE Solve and graph x. x Add to both sides. x x Divide both sides by. Change to. x EXERCISES Solve and graph.. z. h. a x.. k. y Chapter Multi-Step Equations and Inequalities

48 Simplify each expression. Justify each step.... ( ).. (..) Simplify.. x x. m m. n n n. y z. (s ) s. b (b ) Solve.. x x. y y. c c. x. b b. g. On her last three quizzes, Elise scored,, and. What grade must she get on her next quiz to have an average of for all four quizzes? Solve.. x x. q q. n n. m m. a a. z z. The square and the equilateral triangle have the same perimeter. Find the perimeter of each figure. x x Solve and graph.. h. y. m. b. n. p. u. z. Glenda has a $ gift certificate to a café that sells her favorite tuna sandwich for $. after tax. What are the possible numbers of tuna sandwiches that Glenda can buy with her gift certificate? Solve and graph.. m. p. z x.. c. d Chapter Test

49 Gridded Response: Write Gridded Responses When responding to a test item that requires you to place your answer in a grid, examine the grid to be sure you know how to fill it in correctly. Grid formats may vary from test to test. The grid in this book is used often, but it is not used on every test that has gridded-response items. Gridded Response: Divide..... has decimal place, so multiply by., Divide. Simplify. Write your answer in the answer boxes at the top of the grid. Put only one digit in each box. Do not leave a blank box in the middle of an answer. Shade the bubble for each digit in the column beneath it. / Gridded Response: Solve x. x Add to both sides of the equation. x Find a common denominator. x,, or. Add. Mixed numbers and repeating decimals cannot be gridded, so you must grid the answer as. Write your answer in the answer boxes at the top of the grid. Put only one digit or symbol in each box. On some grids, the fraction bar and the decimal point have a designated box. Shade the bubble for each digit or symbol in the correct column. Chapter Multi-Step Equations and Inequalities

50 You cannot grid a negative number in a gridded-response item because the grid does not include the negative sign. If you get a negative answer to a test item, recalculate the problem because you probably made a math error. Read each statement and then answer the questions that follow. Item A A student correctly evaluated an expression and got as a result. Then the student filled in the grid as shown.. What error did the student make when filling in the grid?. Explain how to fill in the answer correctly. Item B A student added. and. and got an answer of.. This answer is displayed in the grid. /. Item C A student found. as the answer to (.). Then the student filled in the grid as shown.. What error does the grid show?. Another student got an answer of.. Explain why the student knew this answer was wrong. Item D A student found that x was the solution to the equation x. Then the student filled in the grid as shown.. What answer does the grid show?. Explain why you cannot fill in a mixed number... Write the answer in two forms that could be entered in the grid correctly. /. What errors did the student make when filling in the grid?. Explain how to fill in the answer correctly. Strategies for Success