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. : 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. 97
. MEP Pupil 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) (e) 98
MEP Pupil Tet (f) (g) (h) (i) (j). On a motorwa there is a minimum speed of m.p.h. and a maimum speed of 7 m.p.h. (a) Cop the diagram below and represent this information on it. The letter, V, is used to represent the speed. 7 8 V 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) < <. 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) < < < < 99
. MEP Pupil Tet (c) (d) < < < < 8. List all the possible integer values of n such that n <. (LON). of Linear Inequalities Inequalities such as 7 can be simplified before solving them. The process is similar to that used to solve equations, ecept that there should be no multiplication or division b negative numbers. 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 Begin with the inequalit ( ) >. ( ) >. First divide both sides of the inequalit b to give >. Then adding to both sides of the inequalit gives > 7.
MEP Pupil Tet 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 <. 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.
. MEP Pupil Tet () < + ( ) () ( ) + < ( ) ( ) < 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) + 8 8 > 7 (c) + 7 < 7 (d) 7 7 (e). Solve the following inequalities. ( ) < (f) 8 (a) < 9 7 + 9 (c) 8 (d) (e) + 7 < (f) 7 8 > ( ) (g) 8 (h) < 7 (i) 8 (j) 7 + 9 (k) > (l). 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) < < 7
MEP Pupil Tet. Chris runs a barber's shop. It costs him per da to cover his epenses and he charges for ever hair cut. (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.. The distance that a car can travel on a full tank of petrol varies between and miles. (a) If m represents the distance (in miles) travelled on a full tank of petrol, write down an inequalit involving m. Distances in kilometres, k, are related to distances in miles b m k = 8. Write down an alternative inequalit involving k instead of m. (c) How man kilometres can the car travel on a full tank of petrol? 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 p 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. 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. Ajaz said, "I thought of an integer, multiplied it b then subtracted. The answer was between 7 and." List the integers that Ajaz could have used.
. MEP Pupil Tet. (a) is a whole number such that <. (MEG) (i) Make a list of all the possible values of. (ii) What is the largest possible value of? Ever week Rucci has a test in Mathematics. It is marked out of. Rucci has alwas scored at least half the marks available. She has never quite managed to score full marks. Using to represent Rucci's marks, write this information in the form of two inequalities. (NEAB). 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 = ±. 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) <.
MEP Pupil Tet (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 >. Begin with the inequalit Adding 7 to both sides gives Dividing both sides b gives Then the solution is 7. 8... Investigation Find the number of points (, ) where and are positive integers which lie on the line + = 9.
. MEP Pupil Tet 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 >. 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
MEP Pupil Tet. 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 (a) 9 A. Find an inequalit which satisfies and solve it. What are 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 length of the rectangle? What is the minimum width of the rectangle?. Solve the following inequalities for. (a) + < 7 < (NEAB) Just for Fun 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? 7
MEP Pupil Tet. 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 below. + 7 7 The coordinates of an point in the shaded area satisf Note The coordinates of an point on the line satisf + =. +. 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 below. 7 7 8
MEP Pupil Tet Worked Eample Shade the region which satisfies the inequalit 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.) If the values, = and =, are substituted into the inequalit, we obtain 7 (, 7) ( ) 7 or. This statement is clearl false and will also be false for an point on that side of the line. 7 (, ) Therefore the other side of the line should be shaded, as shown. 7 9
. MEP Pupil 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
MEP Pupil Tet 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) Just for Fun Without using a calculator or a table, determine which is larger, ( + 9) or 7.
. MEP Pupil 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 +. 8 7 + 7 8 The triangle indicated b bold lines has been shaded three times. The points inside this region, including those points on each of the boundaries, satisf all three inequalities.
MEP Pupil 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. 7 7 7 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
. MEP Pupil 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. 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 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. >. 9 8 7 + > 7 8 9 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.
MEP Pupil Tet 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) (c) (d)
. MEP Pupil Tet (e) (f). At a certain shop, CDs cost and tapes cost 8. Andrew goes into the shop with to spend. (a) If = the number of CDs and = the number of tapes 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.. In organising the sizes of classes, a head teacher decides that the number of children 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.
MEP Pupil Tet 7. Ice cream sundaes are sold for either or. Veronica 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. Mrs. Brown's purse contains onl fift pence coins and pound coins. On the graph below, the region ABCD (including the boundar) contains all the possible combinations (, ) of p coins and coins in Mrs. Brown's purse. is the number of p coins, is the number of coins. coins 8 A D B C 8 8 p coins (a) (i) Write down the minimum possible number of p coins in the purse. (ii) Find the maimum possible amount of mone in the purse. (iii) Find the minimum possible amount of mone in the purse. (iv) Use the graph to find the maimum possible number of coins in the purse. Mrs. Brown now looks in her purse and notices that she has twice as man fift pence coins as pound coins. How man pound coins could her purse contain? (MEG) 7
. MEP Pupil 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 obtain the maimum profit. (ii) State this profit. (NEAB). (a) Find all integer values of n which satisf the inequalit n <. Cop the diagram below and label with the letter 'R' the single region which satisfies all the inequalities,, +, 8. 8
MEP Pupil Tet = = + = + = 8 = (SEG). 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. (LON) Just for fun Bag A contains black balls, Bag B contains white balls and Bag C contains black ball and white ball. John decides to switch all the three labels so that all three bags now have the wrong labels. If ou are allowed to draw one ball from an one of the three bags onl once and look at its colour, which bag would ou choose so that ou can determine the colours of the balls in each bag? Give a reason for our choice. 9