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


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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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 lefthand 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning 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 elearning Jamaica
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