# Unit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations)

Size: px
Start display at page:

Download "Unit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations)"

Transcription

1 UNIT Solving Inequalities: Student Tet Contents STRAND G: Algebra Unit Solving Inequalities Student Tet Contents Section. Inequalities on a Number Line. of Linear Inequalities (Inequations). Inequalities Involving Quadratic Terms. Graphical Approach to Inequalities. Dealing With More Than One Inequalit CIMT and e-learning Jamaica

2 UNIT Solving Inequalities: Student Tet Solving Inequalities. Inequalities on a Number Line An inequalit involves one of the four smbols >,, < or The following statements illustrate the meaning of each of them. > : is greater than : is greater than or equal to < : is less than : is less than or equal to Inequalities can be represented on a number line, as shown in the following worked eamples. Worked Eample Represent the following inequalities on a number line. (a) < (c) < (a) The inequalit,, states that must be greater than or equal to. This is represented as shown. Note that solid mark,, is used at to show that this value is included. The inequalit < states that must be less than. This is represented as shown. Note that a hollow mark, o, is used at to show that this value is not included. (c) The inequalit < states that is greater than and less than or equal to. This is represented as shown. Note that o is used at because this value is not included and is used at because this value is included. CIMT and e-learning Jamaica

3 . UNIT Solving Inequalities: Student Tet Worked Eample Write an inequalit to describe the region represented on each number line below. (a) (a) The diagram indicates that the value of must be less than or equal to, which would be written as The diagram indicates that must be greater than or equal to and less than. This is written as < Eercises. Represent each of the inequalities below on a number line. (a) > < (c) > (d) < (e) (f) (g) (h) (i) (j) < (k) < < (l). Write down the inequalit which describes the region shown in each diagram. (a) (c) (d) CIMT and e-learning Jamaica

4 . UNIT Solving Inequalities: Student Tet (e) (f) (g) (h) (i) (j). The speed limits on a road require drivers to travel at a minimum speed of km/hr and a maimum speed of 7 km/hr. (a) Cop the diagram below and represent this information on it. The letter, V, is used to represent the speed. 7 8 Write down an inequalit to describe our diagram.. Frozen chickens will be sold b a major chain of supermarkets onl if their weight is at least. kg and not more than. kg. (a) Represent this information on a number line. Write an inequalit to describe the region which ou have marked.. List all the whole numbers which satisf the inequalities below. (a) 8 < < 7 (c) < (d) < < CIMT and e-learning Jamaica

5 . UNIT Solving Inequalities: Student Tet. List all the integers (positive or negative whole numbers) which satisf the inequalities below. (a) (c) < < (d) < 7. Write down one fraction which satisfies the inequalities below. (a) < < < < (c) < < (d) < < 8. List all the possible integer values of n such that n <. of Linear Inequalities (Inequations) Inequalities such as 7 can be simplified before solving them. The process is similar to that used to solve equations. Worked Eample Solve the inequalit and illustrate the result on a number line. 7 Begin with the inequalit Adding 7 to both sides gives Dividing both sides b gives 7 This is represented on the number line below. Worked Eample Solve the inequalit ( ) > CIMT and e-learning Jamaica

6 . UNIT Solving Inequalities: Student Tet Begin with the inequalit ( ) > First divide both sides of the inequalit b to give > Then adding to both sides of the inequalit gives > 7. Worked Eample Solve the inequalit Begin with the inequalit 9 9 In this case, note that the inequalit contains a ' ' term. The first step here is to add to both sides, giving 9 + Now 9 can be added to both sides to give Then dividing both sides b gives or Worked Eample Solve the inequalit < + Begin with the inequalit < + The same operation must be performed on each part of the inequalit. The first step is to subtract, which gives < Then dividing b gives < CIMT and e-learning Jamaica

7 . UNIT Solving Inequalities: Student Tet The result can then be represented on a number line as shown below. An alternative approach is to consider the inequalit as two separate inequalities: () < + and () + These can be solved as shown below. () < + ( ) () ( ) + < ( ) ( ) < Both inequalities can then be displaed as shown below. 7 8 Where the two lines overlap gives the solution as Eercises. Solve each inequalit below and illustrate the solution on a number line. (a) > 7 (c) + 7 < 7 (d) 7 7 (e). Solve the following inequalities. ( ) < (f) 8 (a) < (c) 8 (d) (e) + 7 < (f) 7 8 > ( ) (g) 8 (h) < 7 (i) 8 (j) (k) > (l) CIMT and e-learning Jamaica

8 . UNIT Solving Inequalities: Student Tet. Solve each of the following inequalities and illustrate them on a number line. (a) < + < (c) + 7 < 7 (d) < 7 < 9 (e) 9 (f) + < 7. Solve each of the following inequalities. (a) < + 7 < ( ) < ( ) (c) < + 7 (d) 8 (e) + (f) < <. Claton owns a barber's shop. It costs him \$ per da to cover his epenses and he charges \$ for ever haircut. (a) Eplain wh his profit for an da is \$ ( ), where is the number of haircuts in that da. He hopes to make at least \$ profit per da, but does not intend to make more than \$ profit. Write down an inequalit to describe this situation. (c) Solve the inequalit. 7. The distance that a car can travel on a full tank of gasoline varies between and miles. (a) If m represents the distance (in miles) travelled on a full tank of gasoline, write down an inequalit involving m. Distances in kilometres, k, are related to distances in miles b k m = 8 Write down an alternative inequalit involving k instead of m. (c) Write down an inequalit for the number of kilometres the car can travel on a full tank of gasoline. 7. A man finds that his electricit bill varies between \$ and \$9. (a) If C represents the size of his bill, write down an inequalit involving C. The bill is made up of a standing charge of \$ and a cost of cents per kilowatt hour of electricit. If n is the number of kilowatt hours used, write down a formula for C in terms of n. (c) Using our formula, write down an inequalit involving n and solve this inequalit. CIMT and e-learning Jamaica 7

9 . UNIT Solving Inequalities: Student Tet 8. In an office, the temperature, F (in degrees Fahrenheit), must satisf the inequalit F 7 The temperature, F, is related to the temperature, C (in degrees Centigrade), b 9 F = + C Write down an inequalit which involves C and solve this inequalit. 9. (a) List all the integers which satisf < n Scott said, "I thought of an integer, multiplied it b then subtracted. The answer was between 7 and." List the integers that Scott could have used.. (a) is an integer such that < (i) Make a list of all the possible values of. (ii) What is the largest possible value of? Ever week Rosie has a test in Mathematics. It is marked out of. Rosie has alwas scored at least half the marks available. She has never quite managed to score full marks. Using to represent Rosie's marks, write this information in the form of two inequalities.. Inequalities Involving Quadratic Terms Inequalities involving rather than can still be solved. For eample, the inequalit < 9 will be satisfied b an number between and. So the solution is written as < < If the inequalit had been > 9, then it would be satisfied if was greater than or if was less than. So the solution will be > or < The end points of the intervals are defined as 9 = ±. CIMT and e-learning Jamaica 8

10 . UNIT Solving Inequalities: Student Tet Note For this tpe of inequalit it is ver eas to find the end points but care must be taken when deciding whether it is the region between the points or the region outside the points which is required. Testing a point in a region will confirm whether our answer is correct. For eample, for > 9, test =, which gives > 9. This is not true, so the region between the points is the wrong region; the region outside the points is needed. Worked Eample Show on a number line the solutions to: (a) < (a) The solution to is which is shown below. or The solution of < is which is shown below. < < Worked Eample Find the solutions of the inequalities (a) + > 7 (a) B subtracting from both sides, the inequalit becomes Then the solution is + > > 9 < or > CIMT and e-learning Jamaica 9

11 . UNIT Solving Inequalities: Student Tet Begin with the inequalit Adding 7 to both sides gives Dividing both sides b gives Then the solution is 7 8 Worked Eample Solve the inequalit > The left-hand side of the inequalit can be factorised to give ( ) ( + ) > The inequalit will be equal to when = and =. This gives the end points of the region as = and =, as shown below. Points in each region can now be tested. = gives > or > This is not true. = gives > or > This is true. = gives > or >. This is true. So the inequalit is satisfied for values of greater than, or for values of less than. This gives the solution < or > CIMT and e-learning Jamaica

12 . UNIT Solving Inequalities: Student Tet Eercises. Illustrate the solutions to the following inequalities on a number line. (a) (c) (d) < 9 (e) > (f) > (g). (h) <. (i).. Find the solutions of the following inequalities: (a) + 8 (c) < (d) < (e) 9 (f) ( ) < (i) (g) + 7 (h) 8 + (j) > (k) (l) 8. Find the solutions of the following inequalities. ( )( + ) ( )( ) (a) (c) ( ) > (d) (e) 7 + < (f) + > (g) (h) +. The area, A, of the square shown satisfies the inequalit 9 A Find an inequalit which satisfies and solve it, giving an inequalit for the possible dimensions of the square.. (a) Write down an epression, in terms of, for the area, A, of the rectangle below. (c) (d) If the area, A, of the rectangle satisfies the inequalit A write down an inequalit for and solve it. What is the maimum (horizontal) length of the rectangle? What is the minimum (vertical) width of the rectangle? CIMT and e-learning Jamaica

13 . UNIT Solving Inequalities: Student Tet. Solve the following inequalities for. (a) + < 7 < 7. (a) Show that the roots of the equation + =, are ± Determine the set of values of for which + >. (CXC) Investigation Find the number of points (, ) where and are positive integers which lie on the line + = 9.. Graphical Approach to Inequalities When an inequalit involves two variables, the inequalit can be represented b a region on a graph. For eample, the inequalit + is illustrated on the graph on the right. 7 The coordinates of an point in the shaded area satisf +. + Note The coordinates of an point on the line satisf + =. 7 + = 7 If the inequalit had been + >, then a dashed line would have been used to show that points on the line do not satisf the inequalit, as in the second graph. + > 7 + = CIMT and e-learning Jamaica

14 . UNIT Solving Inequalities: Student Tet Worked Eample Shade the region which satisfies the inequalit = 7 7 (, ) The region has the line so first of all the line = 7 as a boundar, = 7 is drawn. (, ) The coordinates of points on this line are (, 7), (, ) and (, ) (, ) These points are plotted and a solid line is drawn through them. A solid line is drawn as the inequalit contains a ' ' sign which means that points on the boundar are included. Net, select a point such as (, ). (It does not matter on which side of the line the point lies.) 7 (, 7) If the values, = and =, are substituted into the inequalit, we obtain ( ) 7 or This statement is clearl false and will also be false for an point on that side of the line. = 7 7 (, ) Therefore the other side of the line should be shaded, as shown. 7 CIMT and e-learning Jamaica

15 . UNIT Solving Inequalities: Student Tet Worked Eample Shade the region which satisfies the inequalit + < The line + = will form the boundar of the region, but will not itself be included in the region. To show this, the line should be drawn as a dashed line. Before drawing the line, it helps to rearrange the equation as = Now points on the line can be calculated, for eample (, ),, This line is shown below. ( ) and, ( ). (, ) (, ) (, ) (, ) + = 7 Net, a point on one side of the line is selected, for eample (, ), where = and =. Substituting these values for and into the inequalit gives + < or 8 < This is clearl true and so points on this side of the line will satisf the inequalit. This side of the line can now be shaded, as below. + < (, ) + = 7 CIMT and e-learning Jamaica

16 . UNIT Solving Inequalities: Student Tet Challenge! Without using a calculator or a table, determine which is larger, ( + 9) or 7. Eercises. Use sets of aes with and values from to to show the regions which the following inequalities satisf. (a) > + (c) < (d) > + (e) (f) + (g) (h) > (i) + (j) + (k) + (l) + <. For each region below, (i) find the equation of the line which forms the boundar, and (ii) find the inequalit represented b the region. (a) (c) (d) CIMT and e-learning Jamaica

17 . UNIT Solving Inequalities: Student Tet (e) (f). (a) On the same set of aes, shade the regions which satisf the inequalities + and + Which inequalit is satisfied b the region shaded twice? Shade the region which satisfies the inequalit.. (a) Draw the graph of = and shade the region which satisfies the inequalit. On the same set of aes, draw the graphs of = + and = Shade the region which satisfies the inequalit, < < +.. Dealing With More Than One Inequalit If more than one inequalit has to be satisfied, then the required region will have more than one boundar. The diagram below shows the inequalities, and The triangle indicated b bold lines has all three shadings. The points inside this region, including those points on each of the boundaries, satisf all three inequalities. CIMT and e-learning Jamaica

18 . UNIT Solving Inequalities: Student Tet Worked Eample Find the region which satisfies the inequalities,, + Write down the coordinates of the vertices of this region. First shade the region which is satisfied b the inequalit Then add the region which satisfies using a different tpe of shading, as shown. 7 Finall, add the region which is satisfied b + using a third tpe of shading. The region which has been shaded in all three different was (the triangle outlined in bold) satisfies all three inequalities. 7 + The coordinates of its vertices can be seen from the diagram as (, ), (, ) and (, 8). 7 CIMT and e-learning Jamaica 7

19 . UNIT Solving Inequalities: Student Tet Note When a large number of inequalities are involved, and therefore a greater amount of shading, the required region becomes more difficult to see on the graph. Therefore it is better to shade out rather shade in, leaving the required region unshaded. This method is used in the following eample, where 'shadow' shading indicates the side of the line which does not satisf the relevant inequalit. The region where all the inequalities are true is called the feasible region. All points inside the feasible region satisf all the inequalities. Worked Eample A small factor emplos people at two rates of pa. The maimum number of people who can be emploed is. More workers are emploed on the lower rate than on the higher rate. Describe this situation using inequalities, and draw a graph to show the feasible region in which the are satisfied. Let and = number emploed at the lower rate of pa, = number emploed at the higher rate of pa. The maimum number of people who can be emploed is, so +. As more people are emploed at the lower rate than the higher rate, then >. As neither nor can be negative, then and. These inequalities are represented on the graph below > The triangle formed b the unshaded sides of each line is the region where all four inequalities are satisfied. The dots indicate all the possible emploment options. Note that onl integer values inside the region are possible solutions. CIMT and e-learning Jamaica 8

20 . UNIT Solving Inequalities: Student Tet Note Often there is a linear objective (e.g. profit or something similar) for which we need to find its optimal value (e.g. maimum or minimum), subject to a number of inequalities. This is called linear programming and it is an important topic. We can solve linear programming problems easil b finding the value of the objective function at each verte of the feasible region. The maimum and minimum values must occur at a verte of the feasible region. We will illustrate this method in Worked Eample, below. Worked Eample The shaded area in the diagram below shows the solution of a set of inequalities in and. The variable represents the number of bos in a cricket club and represents the number of girls in the cricket club. = feasible region = + Use the graph above to answer the questions which follow. (a) State, using arguments based on the graph, whether the cricket club can have as members: (i) bos and girls (ii) bos and girls. Write down the set of THREE inequalities that define the shaded region. (c) A compan sells uniforms for the club and makes a profit of \$. on a bo's uniform and \$. on a girl's uniform. (i) Write an epression in and that represents the total profit made b the compan on the sale of uniforms. (ii) Calculate the minimum profit the compan can make. (CXC) CIMT and e-learning Jamaica 9

21 . UNIT Solving Inequalities: Student Tet (a) (i) No, as point (, ) is not in the feasible region. (ii) Yes, as point (, ) is in the feasible region. + ; ; (c) (i) P = + (ii) The vertices are at (, ), ( 7, 8 7 ), (, ) and the corresponding values of P are \$, \$, \$ 7. 7 So the minimum profit is at (, ) of value \$. Eercises. On a suitable set of aes, show b shading the regions which satisf both the inequalities given below. (a) < 7 (c) < 8 (d) + (e) + (f) < + > > (g) (h) (i) + +. For each set of three inequalities, draw graphs to show the regions which the all satisf. List the coordinates of the points which form the vertices of each region. (a) (c) > + + (d) + < (e) + (f) > > >. Each diagram shows a region which satisfies inequalities. Find the three inequalities in each case. (a) CIMT and e-learning Jamaica

22 . UNIT Solving Inequalities: Student Tet (c) (d) (e) (f). At a certain shop, DVDs cost \$ and CDs cost \$8. Andrew goes into the shop with \$ to spend. (a) If = the number of DVDs and = the number of CDs which Andrew bus, eplain wh + 8 Eplain wh and. (c) Draw a graph to show the region which satisfies all three inequalities.. A securit firm emplos people to work on foot patrol or to patrol areas in cars. Ever night a maimum of people are emploed, with at least two people on foot patrol and one person patrolling in a car. (a) If = the number of people on foot patrol and = the number of people patrolling in cars, complete the inequalities below. (i) +? (ii)? (iii)? Draw a graph to show the region which satisfies these inequalities. CIMT and e-learning Jamaica

23 . UNIT Solving Inequalities: Student Tet. In organising the sizes of classes, a head teacher decides that the number of students in each class must never be more than, that there must never be more than bos in a class and that there must never be more than girls in a class. (a) If = the number of bos in a class and = the number of girls in a class, complete the inequalities below. (i) +? (ii)? (iii)? (c) The values of and can never be negative. Write down two further inequalities. Draw a diagram to show the region which satisfies all the inequalities above. 7. Ice cream sundaes are sold for either \$ or \$. Victoria is going to bu sundaes for some of the members of her famil, but onl has \$ to spend. Use and (a) = the number of \$ sundaes bought = the number of \$ sundaes bought. Write down inequalities which describe the situation above. Draw a diagram to show the region which satisfies all four inequalities. 8. The diagram below shows a triangular region bounded b the lines = +, = + and the line HK. 8 G H = + K = (a) Write the equation of the line HK. Write the set of three inequalities which define the shaded region GHK. (CXC) CIMT and e-learning Jamaica

24 . UNIT Solving Inequalities: Student Tet 9. The school hall seats a maimum audience of people for performances. Tickets for the Christmas concert cost \$ or \$ each. The school needs to raise at least \$ from this concert. It is decided that the number of \$ tickets must not be greater than twice the number of \$ tickets. There are tickets at \$ each and tickets at \$ each. (a) Eplain wh (i) + (ii) + (iii) The graphs of + =, + = and = are drawn on the grid below. = + = + = Cop the grid and show b shading the region of the grid which satisfies all three inequalities in (a). (c) (i) Hence find the number of \$ and \$ tickets which should be sold to (ii) obtain the maimum profit. State this profit.. (a) Find all integer values of n which satisf the inequalit n < Cop the following diagram and label with the letter 'R' the single region which satisfies all the inequalities,, +, 8 CIMT and e-learning Jamaica

25 . UNIT Solving Inequalities: Student Tet = = + = + = 8 =. At each performance of a school pla, the number of people in the audience must satisf the following conditions. (i) The number of children must be less than. (ii) The maimum size of the audience must be. (iii) There must be at least twice as man children as adults in the audience. On an one evening there are children and adults in the audience. (a) Write down the three inequalities which and must satisf, other than and. B drawing straight lines and shading on a suitable grid, indicate the region within which and must lie to satisf all the inequalities. Tickets for each performance cost \$ for a child and \$ for an adult. (c) Use our diagram to find the maimum possible income from ticket sales for one performance. To make a profit, the income from ticket sales must be at least \$. (d) Use our diagram to find the least number of children's tickets which must be sold for a performance to make a profit.. Rose makes hanging baskets which she sells at her local market. She makes two tpes, large and small. Rose makes large baskets and small baskets. Each large basket costs \$7 to make an each small basket costs \$ to make. Rose has \$ she can spend on making the baskets. (a) Write down an inequalit, in terms of and, to model this constraint. Two further constraints are and CIMT and e-learning Jamaica

26 . UNIT Solving Inequalities: Student Tet Use these two constraint to write down statements that describe the numbers of large and small baskets that Rose can make. (c) On a suitable grid, show these three constraints and,. Hence label the feasible region, R. Rose makes a profit of \$ on each large basket and \$ on each small basket. Rose wishes to maimise her profit, \$P. (d) Write down the objective function. (e) Use our graph to determine the optimal numbers of large and small baskets Rose should make, and state the optimal profit.. In order to supplement his dail diet Damien wishes to take some Xtravit and some Yeastalife tablets. Their contents of iron, calcium and vitamins (in milligrams per tablet) are shown in the table. Tablet Iron Calcium Vitamin Xtravit Yeastalife (a) (c) B taking tablets of Xtravit and tablets of Yeastalife Damien epects to receive at least 8 milligrams of iron, milligrams of calcium and milligrams of vitamins. Write these conditions down as three inequalities in and. In a coordinate plane illustrate the region of those points (, ) which simultaneousl satisf,, and the three inequalities in (a). If the Xtravit tablets cost cents each and the Yeastalife tablets cost cents each, how man tablets of each should Damien take in order to satisf the above requirements at the minimum cost? Challenge! Two travellers, one carring buns and the other buns, met a ver rich Arab in a desert. The Arab was ver hungr and, as he had no food, the two men shared their buns and each of the men had an equal share of the 8 buns. In return for their kindness, the Arab gave them 8 gold coins and told them to share the mone fairl. The second traveller, who had contributed buns, said that he should receive gold coins and the other gold coins should go to the first traveller. However the latter said that he should get more than gold coins as he had given the Arab more of his buns. The could not agree and so a fight started. Can ou help them to solve their problem? CIMT and e-learning Jamaica

### MEP Pupil Text 16. The following statements illustrate the meaning of each of them.

MEP Pupil Tet Inequalities. Inequalities on a Number Line An inequalit involves one of the four smbols >,, < or. The following statements illustrate the meaning of each of them. > : is greater than. :

### F6 Solving Inequalities

UNIT F6 Solving Inequalities: Tet F6 Solving Inequalities F6. Inequalities on a Number Line An inequalit involves one of the four smbols >,, < or The following statements illustrate the meaning of each

### x. 4. 2x 10 4x. 10 x

CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain

### Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear

### THOMAS WHITHAM SIXTH FORM

THOMAS WHITHAM SIXTH FORM Algebra Foundation & Higher Tier Units & thomaswhitham.pbworks.com Algebra () Collection of like terms. Simplif each of the following epressions a) a a a b) m m m c) d) d d 6d

### Maintaining Mathematical Proficiency

Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

### Mt. Douglas Secondary

Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms.

### Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7

Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. =- + 3 = - (1, ). = 1 - (, 1) =-3 + 5 3. = 3 + + = 1 (, 3). =-5 = - 7. = 3-5 3 - = 0 (1, 5) 5. -3 + = 5 =-7 (, 7).

### 8.4. If we let x denote the number of gallons pumped, then the price y in dollars can \$ \$1.70 \$ \$1.70 \$ \$1.70 \$ \$1.

8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

### Module 3, Section 4 Analytic Geometry II

Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

### 74 Maths Quest 10 for Victoria

Linear graphs Maria is working in the kitchen making some high energ biscuits using peanuts and chocolate chips. She wants to make less than g of biscuits but wants the biscuits to contain at least 8 g

### Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

### LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

### Chapter 5: Systems of Equations

Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

### N5 R1.1 Linear Equations - Revision

N5 R Linear Equations - Revision This revision pack covers the skills at Unit Assessment and eam level for Linear Equations so ou can evaluate our learning of this outcome. It is important that ou prepare

### UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

### Name Date PD. Systems of Equations and Inequalities

Name Date PD Sstems of Equations and Inequalities Sstems of Equations Vocabular: A sstem of linear equations is A solution of a sstem of linear equations is Points of Intersection (POI) are the same thing

### Fair Game Review. Chapter = How many calculators are sold when the profit is \$425? Solve the equation. Check your solution.

Name Date Chapter 4 Fair Game Review Solve the equation. Check our solution.. 8 3 = 3 2. 4a + a = 2 3. 9 = 4( 3k 4) 7k 4. ( m) 2 5 6 2 = 8 5. 5 t + 8t = 3 6. 3 5h 2 h + 4 = 0 2 7. The profit P (in dollars)

### LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

### Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

### Name: Period: QVMS GTA FALL FINAL EXAM REVIEW PRE-AP ALGEBRA 1

Name: Period: QVMS GTA FALL FINAL EXAM REVIEW PRE-AP ALGEBRA ) When simplifing an epression, ou perform operations inside grouping smbols first. a. alwas b. sometimes c. never ) The opposite of a negative

### THIS IS A CLASS SET - DO NOT WRITE ON THIS PAPER

THIS IS A CLASS SET - DO NOT WRITE ON THIS PAPER ALGEBRA EOC PRACTICE Which situation can be represented b =? A The number of eggs,, in dozen eggs for sale after dozen eggs are sold B The cost,, of buing

### LESSON 4.3 GRAPHING INEQUALITIES

LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear

### (c) ( 5) 2. (d) 3. (c) 3(5 7) 2 6(3) (d) (9 13) ( 3) Question 4. Multiply using the distributive property and collect like terms if possible.

Name: Chapter 1 Question 1. Evaluate the following epressions. (a) 5 (c) ( 5) (b) 5 (d) ( 1 ) 3 3 Question. Evaluate the following epressions. (a) 0 5() 3 4 (c) 3(5 7) 6(3) (b) 9 + (8 5) (d) (9 13) + 15

### Section 3.1 Solving Linear Systems by Graphing

Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem

### Bridge-Thickness Experiment. Student 2

Applications 1. Below are some results from the bridge-thickness eperiment. Bridge-Thickness Eperiment Thickness (laers) Breaking Weight (pennies) 15 5 5 a. Plot the (thickness, breaking weight) data.

### CHAPTER 6: LINEAR SYSTEMS AND THEIR GRAPHS

Name: Date: Period: CHAPTER : LINEAR SYSTEMS AND THEIR GRAPHS Notes #: Section.: Solving Linear Sstems b Substitution The solution to a sstem of equations represents the where the. It would be great if

### Intermediate Math Circles Wednesday November Inequalities and Linear Optimization

WWW.CEMC.UWATERLOO.CA The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Intermediate Math Circles Wednesda November 21 2012 Inequalities and Linear Optimization Review: Our goal is to solve sstems

### b(n) = 4n, where n represents the number of students in the class. What is the independent

Which situation can be represented b =? A The number of eggs,, in dozen eggs for sale after dozen eggs are sold B The cost,, of buing movie tickets that sell for \$ each C The cost,, after a \$ discount,

### CHAPTER 6: LINEAR SYSTEMS AND THEIR GRAPHS

Name: Date: Period: CHAPTER : LINEAR SYSTEMS AND THEIR GRAPHS Notes #: Section.: Solving Linear Sstems b Substitution The solution to a sstem of equations represents the where the. It would be great if

### Fair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.

Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

### Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

### MATH 1710 College Algebra Final Exam Review

MATH 7 College Algebra Final Eam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) There were 80 people at a pla. The admission price was \$

### Inequalities and Multiplication

Lesson 3-6 Inequalities and Multiplication BIG IDEA Multipling each side of an inequalit b a positive number keeps the direction of the inequalit; multipling each side b a negative number reverses the

### A calculator may be used on the exam.

The Algebra Semester A eamination has the following tpes of questions: Selected Response Student Produced Response (Grid-in) Brief Constructed Response (BCR) Etended Constructed Response (ECR) Short Answer

### Algebra 2 Unit 2 Practice

Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of

### ACTIVITY: Using a Table to Plot Points

.5 Graphing Linear Equations in Standard Form equation a + b = c? How can ou describe the graph of the ACTIVITY: Using a Table to Plot Points Work with a partner. You sold a total of \$6 worth of tickets

### 5 Linear Graphs and Equations

Linear Graphs and Equations. Coordinates Firstl, we recap the concept of (, ) coordinates, illustrated in the following eamples. Eample On a set of coordinate aes, plot the points A (, ), B (0, ), C (,

### Summary and Vocabulary

Chapter Chapter Summar and Vocabular Equations involving percents ma be written in the form p q = r, where p is the decimal form of the percent, q is the initial quantit, and r is the resulting quantit.

88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:

### c. x x < 60 d. x x =9. What are the first four terms of the sequence? a. 12, 21, 30, 39 b.

Algebra I Unit Reasoning with Linear Equations and Inequalities Post Test... A famil s cell phone plan costs \$ per month for, minutes and cents per minute over the limit. This month, the famil paid \$..

### INEQUALITIES

Chapter 4 INEQUALITIES 4.2.1 4.2.4 Once the students understand the notion of a solution, the can etend their understanding to inequalities and sstems of inequalities. Inequalities tpicall have infinitel

### Lesson 5.1 Solving Systems of Equations

Lesson 5.1 Solving Sstems of Equations 1. Verif whether or not the given ordered pair is a solution to the sstem. If it is not a solution, eplain wh not. a. (, 3) b. (, 0) c. (5, 3) 0.5 1 0.5 2 0.75 0.75

### Applications. 60 Say It With Symbols. g = 25 -

Applications 1. A pump is used to empt a swimming pool. The equation w =-275t + 1,925 represents the gallons of water w that remain in the pool t hours after pumping starts. a. How man gallons of water

### 5.7 Start Thinking. 5.7 Warm Up. 5.7 Cumulative Review Warm Up

.7 Start Thinking Graph the linear inequalities < + and > 9 on the same coordinate plane. What does the area shaded for both inequalities represent? What does the area shaded for just one of the inequalities

### Name Class Date. Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Travels in Air. Distance (miles) Time (seconds)

Practice - Rate of Change and Slope Find the slope of each line.... O O O Find the slope of the line that passes through each pair of points.. (, ), (7, 5) 5. (8, ), (, ). (, 5), (, 7) 7. (-, 7), (, -)

### 3.2 Understanding Relations and Functions-NOTES

Name Class Date. Understanding Relations and Functions-NOTES Essential Question: How do ou represent relations and functions? Eplore A1.1.A decide whether relations represented verball, tabularl, graphicall,

### MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

### A calculator may be used on the exam.

The Algebra Semester A eamination will have the following tpes of questions: Selected Response Student Produced Response (Grid-in) Brief Constructed Response (BCR) Etended Constructed Response (ECR) Short

### A.7A Mini-Assessment

LGER I (.) Linear Functions. The formulates equations and inequalities based on linear functions, uses a variet of methods to solve them, and analzes the solutions in terms of the situation. The student

### Systems of Linear Equations

Sstems of Linear Equations Monetar Sstems Overload Lesson 3-1 Learning Targets: Use graphing, substitution, and elimination to solve sstems of linear equations in two variables. Formulate sstems of linear

### REVIEW PACKET FOR END OF COURSE EXAM

Math H REVIEW PACKET FOR END OF COURSE EXAM DO NOT WRITE ON PACKET! Do on binder paper, show support work. On this packet leave all fractional answers in improper fractional form (ecept where appropriate

### An assessment of these skills will occur the first week of school.

Fort Zumwalt West High School Pre-AP Algebra SUMMER REVIEW PACKET Name: * This packet is to be handed in to our Pre AP Algebra teacher on the first da of the school ear. *All work must be shown in the

### Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?

3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear

### Unit 2: Linear Equations and Inequalities

Mr. Thurlwell's Assignment Sheet Algebra 1 Unit 2: Linear Equations and Inequalities Name: Assignment #1 (3.3) pg 177 4-22e Assignment #2 (4.3) pg 235 2-10e, 24,30,47,50 Assignment #3 (4.1) pg 219 2-14e,15,59

### A. Real numbers greater than 2 B. Real numbers less than or equal to 2. C. Real numbers between 1 and 3 D. Real numbers greater than or equal to 2

39 CHAPTER 9 DAY 0 DAY 0 Opportunities To Learn You are what ou are when nobod Is looking. - Ann Landers 6. Match the graph with its description. A. Real numbers greater than B. Real numbers less than

### Active Maths 2 Old Syllabus Strand 5

Junior certificate HIGHER LEVEL Active Maths Old Sllabus Strand 5 πr m = - - πr Oliver Murph Contents. Functions.... Functions and Graphs...5.3 Graphs...7.4 Linear Graphs...7.5 Quadratic Graphs...9.6 Real-Life

### 7.1 Solving Linear Systems by Graphing

7.1 Solving Linear Sstems b Graphing Objectives: Learn how to solve a sstem of linear equations b graphing Learn how to model a real-life situation using a sstem of linear equations With an equation, an

### 7.1 Guided Practice (p. 401) 1. to find an ordered pair that satisfies each of the equations in the system. solution of the system.

CHAPTER 7 Think and Discuss (p. 9). 6,00,000 units. 0,00,000 6,00,000 4,400,000 renters Skill Review (p. 96) 9r 4r 6r. 8.. 0.d.d d 4. w 4 w 4 w 4 w 4 w. 6. 7 g g 9 g 7 g 6 g 0 7 8 40 40 40 7. 6 8. 8 9....

### 2.3 Start Thinking. 2.3 Warm Up. 2.3 Cumulative Review Warm Up

. Start Thinking Choose an two numbers and compare them with an inequalit smbol ( < or > ). Multipl each number b 1. Is the new inequalit still true? Continue this eercise b dividing the original inequalit

### Review of Elementary Algebra Content

Review of Elementar Algebra Content 0 1 Table of Contents Fractions...1 Integers...5 Order of Operations...9 Eponents...11 Polnomials...18 Factoring... Solving Linear Equations...1 Solving Linear Inequalities...

### Maintaining Mathematical Proficiency

Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on

### Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.

Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems

### a. In the statement "Height is a function of weight," which is the independent variable and which is the dependent variable?

1. The weights and heights of si mathematics students are given in the table. Answer parts a through e. Weight (lb.) Height (cm) 157 19 11 155 1 11 175 17 157 15 17 17 a. In the statement "Height is a

### (3) ( ) UNIT #11 A FINAL LOOK AT FUNCTIONS AND MODELING REVIEW QUESTIONS. Part I Questions. = 3 + 2, then 1.

Name: Date: UNIT # A FINAL LOOK AT FUNCTIONS AND MODELING REVIEW QUESTIONS Part I Questions. If a quadratic function, f ( ), has a turning point at (, ), and g( ) f ( ) where does g( ) have a turning point?

### Algebra 2 Unit 1 Practice

Algebra Unit Practice LESSON - Use this information for Items. Aaron has \$ to rent a bike in the cit. It costs \$ per hour to rent a bike. The additional fee for a helmet is \$ for the entire ride.. Write

### Functions. Introduction

Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

### Intermediate Algebra / MAT 135 Spring 2017 Master ( Master Templates)

Test 1 Review #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: 1. Solve for. = 8 18. Solve for. = + a b 3. Solve for. a b = L 30. Two trains leave stations miles

### Chapter 9 BUILD YOUR VOCABULARY

C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular

### Equations and Inequalities

Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types

### SAMPLE. Inequalities and linear programming. 9.1 Linear inequalities in one variable

C H A P T E R 9 Inequalities and linear programming What is a linear inequalit? How do we solve linear inequalities? What is linear programming and how is it used? In Chapter 3, Linear graphs and models,

### Linear Programming. Maximize the function. P = Ax + By + C. subject to the constraints. a 1 x + b 1 y < c 1 a 2 x + b 2 y < c 2

Linear Programming Man real world problems require the optimization of some function subject to a collection of constraints. Note: Think of optimizing as maimizing or minimizing for MATH1010. For eample,

### Unit 3: Relations and Functions

Unit 3: Relations and Functions 5-1: Binar Relations Binar Relation: - a set ordered pairs (coordinates) that include two variables (elements). (, ) = horizontal = vertical Domain: - all the -values (first

### 6-1 Study Guide and Intervention

NAME DATE PERID 6- Stud Guide and Intervention Graphing Sstems of Equations Possible Number of Solutions Two or more linear equations involving the same variables form a sstem of equations. A solution

### Solving Systems Using Tables and Graphs

3-1 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slope-intercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee

### Name Period Date DRAFT

Name Period Date Equations and Inequalities Student Packet 4: Inequalities EQ4.1 EQ4.2 EQ4.3 Linear Inequalities in One Variable Add, subtract, multiply, and divide integers. Write expressions, equations,

### Part I. Hours

Part I Answer all questions in this part. Each correct answer will receive credits. No partial credit will be allowed. For each question, write on the separate answer sheet the numeral preceding the word

### Functions. Introduction CHAPTER OUTLINE

Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

### 10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.

Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =

### PreAP Algebra 2 Unit 1 and Unit 2 Review Name A#

PreAP Algebra Unit and Unit Review Name A# Domain and Range:. The graph of the square root function f is shown below. - - - - - - - - - - Interval Notation Domain: Range: Inequalit Notation Domain: Range:.

### Systems of Linear Equations

Sstems of Linear Equations Monetar Sstems Overload Lesson 3-1 Learning Targets: Use graphing, substitution, and elimination to solve sstems of linear equations in two variables. Formulate sstems of linear

### c. Find the slope and y-intercept of the graph of the linear equation. Then sketch its graph.

Name Solve. End-of-Course. 7 =. 5 c =. One cell phone plan charges \$0 per month plus \$0.5 per minute used. A second cell phone plan charges \$5 per month plus \$0.0 per minute used. Write and solve an equation

### Systems of Linear Equations: Solving by Graphing

8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

### LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

### ALGEBRA I END-OF-COURSE EXAM: PRACTICE TEST

Page 1 ALGEBRA I END-OF-COURSE EXAM: PRACTICE TEST 1. Order the following numbers from least to greatest:, 6, 8.7 10 0, 19 b. 19,, 8.7 100, 6 6, 8.7 10 0,, 19 c. d. 8.7 10 0,, 19, 6, 6, 19, 8.7 100. If

### Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square

Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.

### 1.7 Inverse Functions

71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

### Math 154A Elementary Algebra Fall 2014 Final Exam Study Guide

Math A Elementar Algebra Fall 0 Final Eam Stud Guide The eam is on Tuesda, December 6 th from 6:00pm 8:0pm. You are allowed a scientific calculator and a " b 6" inde card for notes. On our inde card be

3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

### Name % Correct % MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

1 Pre Test Unit Name % Correct % MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide if the given number is a solution to the given equation. 1)

### 15.4 Equation of a Circle

Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle

### California State University, Northridge

California State Universit, Northridge MATH 09 HYBRID WORKBOOKS Spring 00 Chapter Equations, Inequalities and Applications. The Addition Propert of Equalit Learning Objectives:. Use the Addition Propert

### Domain, Range, and End Behavior

Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship

### 3.1 Solving Linear Systems by Graphing 1. Graph and solve systems of linear equations in two variables. Solution of a system of linear equations

3.1 Solving Linear Systems by Graphing Objectives 1. Graph and solve systems of linear equations in two variables. Key Terms System of linear equations Solution of a system of linear equations Check whether

### LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### 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