SAMPLE. Inequalities and linear programming. 9.1 Linear inequalities in one variable
|
|
- Darrell Wade
- 6 years ago
- Views:
Transcription
1 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, ou learned how linear equations and their graphs are used to model practical situations, such as plant growth, service charges and flow problems. In this chapter ou will learn how linear inequalities and their graphs can be used to model a different set of practical situations, such as determining the mi of products in a supermarket to maimise profit, or designing a diet to provide maimum nutrition for minimum cost. This is known as linear programming. Linear programming requires ou to solve both linear equations and linear inequalities. You learned how to solve linear equations in Chapter 2, Linear relations and equations. You now need to learn how to solve linear inequalities. 9.1 Linear inequalities in one variable Linear inequalities in one variable and the number line An epression such as is called a linear inequalit in one variable. It is an inequalit, not an equation, because it involves an inequalit sign ( ) rather than an equals sign ( = ). The sign means less than or equal to. The solution to the linear equation 3 = 9is = 3, and the solution to the linear equation 3 = 21 is 7. We can represent these solutions on a number line b putting a closed circle ( )onthe number line at = 3 and = 7asshown. = 3 = When solving an inequalit and graphing its solution on a number line, we need to be careful about whether the end values of the solution are included in the range of possible values. 376 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
2 End values included To solve the linear inequalit Chapter 9 Inequalities and linear programming 377 we divide through b 3 and get or 3 7 There is no single solution to this inequalit. An value of from 3 to 7 is a solution. For eample, = 3, = 3.5, = 4.95 and = 7 are all possible solutions. In fact, it is impossible to list ever possible solution, as there are an infinite number of solutions. However, we can represent all the possible solutions on a number line. This is done b marking the points = 3 and = 7 with a closed circle ( ) onthe number line. These points are then joined b drawing a solid line to indicate that all the values between = 3 and = 7 are also solutions, as shown below End values not included To solve the linear inequalit 9 < 3 < 21 we divide through b 3 to obtain the solution 3 < < 7 The sign < means less than. This means that = 3 and = 7 are not solutions, but all values between = 3 and = 7 are possible solutions. To represent this solution on a number line, mark in the points = 3 and = 7 with an open circle ( ). These two open circles are then joined b a solid line to indicate that all the values between 3 and 7 are solutions, but not = 3 and = 7. 3 < < Note that 7 > > 3 represents the same values of as 3 < < 7. A galler of signs = as in a = b reads as a equals b > as in a > b reads as a is greater than b as in a b reads as a is greater than or equal to b < as in a < b reads as a is less than b as in a b reads as a is less than or equal to b Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
3 378 Essential Standard General Mathematics Eample 1 Solve the inequalit 10 < 5 40 Solving an inequalit and graphing the solution for and displa the solution on a number line. Solution 1 Write the inequalit. 10 < Solve the inequalit for b dividing through b 5. or 10 5 < Displa the solution on a number line. or 2 < 8 Draw a number line to include 2 and Mark the point = 2with an open circle Mark the point = 8 with a closed circle. Join the two points with a solid line. Write in the solution inequalit on the graph. Eample 2 Solve the inequalit 10 > 20 Solving an inequalit and graphing the solution for and displa the solution on a number line < Solution 1 Write the inequalit. 10 > Solve the inequalit for b dividing through b 10. or 10 > Displa the solution on a number line. or > 2 Draw a number line to include Mark in the point = 2 with an open circle To indicate all values of greater than 2, draw a solid line from this point to the right that potentiall goes on forever. > 2 Write in the solution inequalit on the graph. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
4 Chapter 9 Inequalities and linear programming 379 Linear inequalities in one variable and the coordinate plane We can also represent linear inequalities in one variable on the coordinate plane. If the equation = 3isplotted on a set of aes we will have a vertical straight line, located at = 3. While the value of changes along the line, for ever point on this line the value of is 3. Just as with graphing the solution of an inequalit on a number line, we need to be careful about whether the boundar lines (end values) of the solution are included in the range of possible values. Boundar line included If we tried to plot the inequalit 3, we would have to plot all the points in the plane that have an -value greater than or equal to 3. f course we cannot show each individual point. What we do is shade in the containing these points. The shaded starts at the vertical line = 3 and etends right forever. Some representative points that satisf the condition 3, and which are found in the shaded, have also been plotted. Boundar line not included The plot of the inequalit > 3issimilar to the plot of 3, but the line = 3isdrawn as a dashed line to indicate that it is not included in the. Some representative points that satisf the condition > 3have also been plotted. Note: For (5, 6), 5 > 3 (4, 1), 4 > 3 Therefore both points satisf the inequalit > 3. = 3 = 3 (3, 6) (3, 3) (3, 0) (3, 1) (3, 0) Required = 3 Required (5, 6) (4, 1) > 3 (5, 6) Eample 3 n the coordinate plane, plot the graphs of: a 4 b 1 < < 3 (4, 1) > 3 Plotting a linear inequalit in one variable on the coordinate plane Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
5 380 Essential Standard General Mathematics Solution a 4 1 Draw in a solid line = 4todefine the boundar of the shaded. 2 Shade the on and below the line = 4to represent all the points defined b 4. b 1 < < 3 1 Draw in a dashed line = 3todefine the upper boundar of the shaded. 2 Draw in a dashed line = 1todefine the lower boundar of the shaded. 3 Shade the between the lines = 3 and = 1to represent all the points defined b 1 < < 3. Eercise 9A (0, 4) (0, 3) (0, 1) < 4 = 4 = 3 1 < < 3 = 1 1 Which of the smbols <, = or > should be placed in the bo in each of the following? a 7 9 b 3 2 c d e 8 4 f 3 1 g 2 1 h Represent each of the following inequalities on a number line. a 1 4 b 0 < < 4 c < 4 d 4 e 1 < 4 f 3 < 5 3 Write down an inequalit represented b each of the following graphs. a b c d e Solve each of the following inequalities and represent its solution on a number line. a 3 15 b 20 < 100 c 2 > 4 d 9 36 e 12 6 < 24 f 10 < A person becomes a teenager when the turn 13. The stop being a teenager when the turn 20. Let be the variable age (in ears). Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
6 Chapter 9 Inequalities and linear programming 381 a Write down an inequalit in terms of that defines a teenager. b Graph this inequalit on a number line. 6 Carr-on luggage in most passenger aircraft can weigh no more than 5 kg. Let be the variable weight (in kg). a Write down an inequalit in terms of that defines the acceptable weight for carr-on luggage. b Graph this inequalit on a number line. 7 Graph the following inequalities on the coordinate plane. a 1 b > 2 c 5 d > 1 e < 2 f 2 2 g 1 < < 2 h 3 < 5 i 3 < Linear inequalities in two variables The inequalities > 2 2 < 2 2 are linear inequalities in two variables, and. To help us interpret these inequalities, we have drawn the graph of = 2. This line (red) separates the coordinate plane into two s (2, 8) = 2 (2, 4) 4 (4, 4) The line = 2 The line is defined b the equation = 2 and coloured red. Itincludes all the points that lie on the line. From the graph above we can see that the point (2, 4) lies on the line. The points (2, 8) and (4, 4) clearl do not lie on the line. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
7 382 Essential Standard General Mathematics We can also show this b carring out the following tests, using the equation of the line. Test: (2, 4): = 4 2 = 2; so the point (2, 4) lies on the line = 2. (2, 8): = 8 2 = 6; 6 is greater than 2, so (8, 9) does not lie on = 2. (4, 4): = 4 4 = 0; 0 is less than 2, so (4, 4) does not lie on the line = 2. The s >2and 2 This is defined b the inequalit > 2 and coloured light blue. Itincludes all the points that lie above the line; the point (2, 8) is an eample. B including the line in this, we have a wa ofrepresenting the inequalit 2 This includes all the points on and above 10 8 (2, 8) the line. From the graph on the right we can see that the points (2, 4) and (2, 8) are eamples of points that lie in this (2, 4) (4, 4) the point (4, 4) clearl does not lie in the We can also show this b carring out the 2 following tests, using the equation of the line. Test: (2, 4): = 4 2 = 2; so the point (2, 4) lies in the 2. (2, 8): = 8 2 = 6; 6 is greater than 2, so the point (2, 8) lies in the 2 (4, 4): = 4 4 = 0; 0 is less than 2, so (4, 4) does not lie in the 2. = 2 The s <2and 2 This is defined b the inequalit < 2 and coloured purple. Itincludes all the points that lie below the line; the point (4, 4) is an eample. B including the line in this, we have a wa ofrepresenting the inequalit 2 This includes all the points on and below the line. From the graph on the right we can see that the points (2, 4) and (4, 4) are eamples of points that lie in this (2, 8) (2, 4) (4, 4) the point (2, 8) clearl does not lie in the = 2 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
8 Chapter 9 Inequalities and linear programming 383 We can also show this b carring out the following tests using the equation of the line. Test: (2, 4): = 4 2 = 2; so the point (2, 4) lies in the 2. (2, 8): = 8 2 = 6; 6 is greater than 2, so (8, 9) does not lie in the 2. (4, 4): = 4 4 = 0; 0 is less than 2, so (4, 4) lies in the 2. We now have a graphical wa of representing inequalities. Linear inequalities can be represented b s in the coordinate plane If the inequalit sign is: or, the line defining the is included, indicated b using a solid line to indicate the boundar <or>,the line defining the is not included, indicated b using a dashed line to indicate the boundar. Graphing a linear inequalit in two variables Eample 4 Graphing a linear inequalit in two variables Sketch the graph of the Solution 1 Find the intercepts for the boundar line = 18. Find the -intercept. Substitute = 0 into the equation and solve for. Find the -intercept. Substitute = 0 into the equation and solve for. 2 n a labelled set of aes, draw a straight line through the two intercepts. Use a solid line to indicate that the line is included in the. Label the line = 18 When = 0, 2 = 18 = 9 -intercept is (0, 9). When = 0, 3 = 18 = 6 -intercept is (6, 0). 3 Use a test point to determine whether the required lies above or below the line. Note: The origin (0, 0) is usuall a good point to test. (0, 9) = 18 (6, 0) Test (0, 0): = 3(0) + 2(0) = 0 0 < 18, so (0, 0) lies in the Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
9 384 Essential Standard General Mathematics 4 As (0, 0) is below the line, the required lies on and below the line. Shade in the on and below the line. Label the. Eample 5 (0, 9) Graphing a linear inequalit in two variables Sketch the graph of the 4 5 > 20. Solution 1 Find the intercepts for the boundar line 4 5 = 20. Find the -intercept. Substitute = 0 into the equation and solve for. Find the -intercept. Substitute = 0 into the equation and solve for. 2 n a labelled set of aes, draw a straight line through the two intercepts. Use a dashed line to indicate that the boundar line is not included in the. Label the line. 3 Use a test point to determine whether the required lies above or below the line. 4 As (0, 0) is above the line, the required lies below the line. Shade in the on and below the line. Label the = 18 (6, 0) 4 5 = 20 When = 0, 5 = 20 = 4 -intercept is (0, 4). When = 0, 4 = 20 = 5 -intercept is (5, 0). (0, 4) 4 5 = 20 (5, 0) Test (0, 0): 4 5 = 4(0) 5(0) = 0 0 < 20, so (0, 0) does not lie in the 4 5 > 20. (0, 4) 4 5 = 20 (5, 0) 4 5 > 20 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
10 Chapter 9 Inequalities and linear programming 385 Summar: Plotted linear inequalities Graph the inequalit as if it contained an equals (=) sign. Draw a solid line if the inequalit is or. Draw a dashed line if the inequalit is < or >. Pick a point not on the line to use as a test point. The origin is a good test point, provided the boundar line does not pass through the origin. Substitute the test point into the inequalit. If the point makes the inequalit true, shade the containing the test point. If not, shade the not containing the test point. Eercise 9B 1 Test to see whether the point (0, 0) lies in the following s. a + 0 b + < 4 c 2 + > 2 d e 2 > 5 f 3 < 6 2 Test to see whether the point (1, 2) lies in the following s. a + 0 b + < 0 c 2 + > 2 d e > 5 f Graph the following inequalities. a 5 b 2 4 c < 3 d + 10 e f g 3 5 < 15 h 2 5 > 5 i > s In Chapter 2, Linear relations and equations, ou learned how to solve pairs of simultaneous linear equations graphicall. Foreample, to solve the pair of linear equations = 18 (0, 9) = = 18 graphicall, we simpl plot their graphs and find the point of intersection. (0, 3) (4, 3) = 3 The solution is the point on the coordinate plane that (6, 0) is common to both graphs. This is the point (4, 3), the point where the two lines intersect. From this, we concludes that = 4 and = 3. When we tr to solve the pair of simultaneous linear inequalities Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
11 386 Essential Standard General Mathematics graphicall, there is not a single solution, but man solutions. The solutions are all the points that lie in the in the coordinate plane that is common to both inequalities. The common to both inequalities is called the feasible.it is called the feasible because all the points in this are possible solutions of the pair of simultaneous linear inequalities. The feasible (solution ) for a set of inequalities is determined b finding the common to all of the inequalities involved. This process is illustrated below for the inequalities (0, 9) and (6, 0) The shaded pink is defined b the inequalit (0, 3) > 3 The shaded blue is defined b the inequalit 3. (0, 9) (0, 3) (4, 3) (6, 0) The shaded purple is the feasible. It is the common to the inequalities and 3. The method we have used to graphicall determine the feasible is called shading in. Sometimes this method of finding the common to a set of inequalities can quickl become mess and impractical when we have too man inequalities. Fortunatel, for the sort of applications ou will meet in this chapter, the required will lie in the first quadrant and involve onl a small number of inequalities so that the shading in method is appropriate. Eample 6 Graphing a feasible Graph the feasible for the following four simultaneous inequalities: 0, 0, + 8, Solution Because 0 and 0, the feasible is restricted to the first quadrant. 1 Graph the inequalit + 8inthe first quadrant. Plot the boundar line + = 8, (0, 8) marking and labelling the -intercept (0, 8) and the -intercept (8, 0). + = 8 Shade in the bounded b the - and -aes and the line. Here it has been shaded blue. (8, 0) Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
12 2 Graph the inequalit in the first quadrant. Plot the boundar line = 30, marking and labelling the -intercept (0, 6) and the -intercept (10, 0). Shade in the bounded b the - and -aes and the line. Here it has been shaded pink, but it becomes purple where it overlaps the blue. 3 The overlap (purple) is the feasible. Label the overlap the. To complete the feasible, find the coordinates of the point where the two boundar lines intersect, b solving the simultaneous equations + = = 30 The lines intersect at the point (5, 3). Mark this point on the graph. Eample 7 Graphing a feasible Chapter 9 Inequalities and linear programming 387 (0, 8) (0, 6) (0, 8) (0, 6) + = = 30 (8, 0) + = 8 Graph the feasible for the following four simultaneous inequalities: 0, 0, , Solution Because 0 and 0, the feasible is restricted to the first quadrant. 1 Graph the inequalit in the first quadrant. Plot the boundar line + 2 = 10, marking and labelling the -intercept (0, 5) and the -intercept (10, 0). (0, 5) Shade in the bounded b the - and -aes and the line. Here it + 2 = 10 has been shaded blue. (10, 0) (5, 3) = 30 (8, 0) (10, 0) (10, 0) Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
13 388 Essential Standard General Mathematics 2 Graph the inequalit in the first quadrant. Plot the boundar line = 36, marking and labelling the -intercept (0, 9) and the -intercept (6, 0). Shade in the bounded b the - and -aes and the line. Here it has been shaded pink, but it becomes purple where it overlaps the blue. 3 The overlap (purple) is the feasible. Label the overlap the. To complete the feasible, find the coordinates of the point where the two boundar lines intersect, b solving the simultaneous equations + 2 = = 36 The lines intersect at the point (4, 3). Mark this point on the graph. (0, 9) (0, 5) + 2 = 10 (0, 9) How to graph a feasible using a TI-Nspire CAS = 36 (6, 0) (0, 5) + 2 = 10 (4, 3) = 36 (6, 0) (10, 0) (10, 0) Graph the feasible for the following four simultaneous inequalities: 0, 0, , Because 0 and 0, the feasible is restricted to the first quadrant. We take this into account when setting the viewing window on the calculator. Steps 1 To graph the inequalities and using a graphics calculator, first we need to rearrange both inequalities so that is the subject. Hence, (10 ) becomes 2 (36 6) becomes 4 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
14 Chapter 9 Inequalities and linear programming pen a new document (b pressing / + N) and select 2: Graphs & Geometr. a Use the backspace ke ( )to delete the f 1() = and tpe in >= (10 ) 2. Press enter. This plots the inequalit b Repeat the above but this time tpe in >= (36 6) 4. Press enter. This plots the inequalit c Press / + to hide the entr line. d The inequalities 0 and 0 indicate that the feasible is restricted to the first quadrant. This is best achieved b resetting the viewing window. 3 To reset the viewing window, press b/4:window/1:window Settings. Using to move between the entr boes, enter the following values: XMin: 0 XMa: 12 XScale: Auto YMin: 0 YMa: 10 YScale: Auto 4 Pressing enter confines the plot to the first quadrant. The graphs will appear as shown. The feasible is the more heavil shaded. Note: It ma be necessar to grab and move the graph labels if the overlap with other labels. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
15 390 Essential Standard General Mathematics 5 To complete the feasible, we need to know the coordinates of the corner points. a Press b/6:points & Lines/3:Intersection Point/s. b Move the cursor to one of the graphs and press.now move to the other graph and press. The point of intersection (4, 3) will be displaed. Press to eit the Intersection Point tool. The other two points, (0, 9) and (10, 0), can be determined from the equations of the boundar lines. How to graph a feasible using the ClassPad Graph the feasible for the following four simultaneous inequalities: 0, 0, , Because 0 and 0, the feasible is restricted to the first quadrant. We take this into account when setting the viewing window on the calculator. Steps 1 From the application menu, locate and open the Graph and Table ( ) built-in application. To graph the inequalities and , first we need to rearrange both inequalities so that is the subject. Hence, (10 ) becomes 2 (36 6) becomes 4 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
16 Chapter 9 Inequalities and linear programming To set the calculator to draw the correct inequalit, tap the down arrow ( ) adjacent to in the toolbar and select. Adjacent to 1: tpe (10 )/2. Press E. Adjacent to 2: tpe (36 6)/4. Press E. Adjacent to 3: tpe 0. Press E. 3 To enter the 0 inequalit, tap the down arrow ( ) adjacent to in the toolbar and select. Adjacent to 4: tpe 0. Press E. Tap on the View Window icon (6)inthe toolbar to set the graph viewing window. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
17 392 Essential Standard General Mathematics 4 To complete the, the corner points need to be found. From the Analsis menu item, select G-solve, then Intersect. Use the up and directions from the blue oval directional button on the front of the calculator to select the equations for 1 and 2. When an equation has been selected, press E to confirm its choice. The equation of each line is displaed in a window at the bottom of the graphing screen. 5 After the second equation has been selected and confirmed, the intersection point will be displaed on the screen, indicated b a cursor in the shape of a small cross. In this case, (4, 3). The other two boundar points are (0, 9) and (10, 0). Eercise ********* 9C Graph the feasible for each of the following sets of linear inequalities. 1 0, 0, , 0, , 0, , 0, + 6, , 0, 3 + 6, Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
18 6 0, 0, , , 0, , , 0, 2 0, Linear programming bjective functions and constraints Chapter 9 Inequalities and linear programming 393 An objective function is a quantit that ou are tring to make as large as possible (for eample, profits) or as small as possible (for eample, the amount of material needed to make a dress). f course, there are alwas factors, such as the resources available or the requirements of the dress pattern, that limit how much profit ou can make or how little material ou can use to make a dress. These are called constraints. The linear programming problem The process of maimising or minimising a linear quantit, subject to a set of constraints, is at the heart of linear programming. The linear programming problem From the mathematical point of view, linear programming can be viewed as finding the point, or points, in a feasible that gives the maimum or minimum value of some linear epression. Finding the maimum value of an objective function The aim is to find the maimum value of the objective function P = 2 + 3, subject to the constraints: 0, as shown b the feasible opposite. (0, 8) B(0, 6) A(0, 0) + = 8 C(5, 3) = 30 (10, 0) D(8, 0) At first, this seems like an insurmountable problem, as there is an infinite number of points in the to choose from. Fortunatel, we can make use of the corner point principle to help us solve the problem. The corner point principle In linear programming problems, the maimum or minimum value of a linear objective function will occur at one of the corners of the feasible. Note: If two corners give the same maimum or minimum value, then all points along a line joining the two points will also give the same maimum or minimum value. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
19 394 Essential Standard General Mathematics This means that we onl need to evaluate the objective function at each of the corner points, labelled A, B, C and D, and find which gives the maimum value. It helps to set up a table as follows. bjective function Points P =2 +3 A (0, 0) P = = 0 B (0, 6) P = = 18 C (5, 3) P = = 19 D (8, 0) P = = 16 Thus, the maimum value of the objective function, P = 19, occurs when = 5 and = 3. Eample 8 Find the minimum value of the objective function C = 5 + 2, subject to the constraints: 0, as displaed in the feasible opposite. Solution 1 Set up a table for the objective function. 2 Evaluate the objective function at each of the corners A, B and C. 3 Identif the corner point giving the minimum value and write our answer. Eercise 9D Finding the minimum value of an objective function A (0, 9) (0, 5) + 2 = 10 B(4, 3) = 36 (0, 6) C (10, 0) bjective function Points C = A(0, 9) C = = 18 B(4, 3) C = = 26 C(10, 0) C = = 50 The minimum value is C = 18, which occurs when = 0and = 9. For each of the following objective functions and feasible s, find the maimum or minimum value (as required) and the point at which it occurs. 1 P = (maimum) 2 P = (maimum) B(0, 5) C(3, 3) A(0, 0) D(6, 0) B(0, 10) A(0, 0) C(2, 12) D(6, 8) E(12, 0) Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
20 Chapter 9 Inequalities and linear programming C = (minimum) 4 C = + (minimum) A(0, 10) B(3, 2.5) C(6, 0) 5 P = + 2 (maimum) A(0, 0) B(10, 20) C(30, 0) A(0, 12) B(2, 4) C(10, 0) 6 C = (minimum) A(0, 5) 9.5 Linear programming applications B(5, 0) Younow have all the technical skills necessar to set up and solve a basic linear programming problem. Eample 9 Setting up and solving a maimising problem A manufacturer makes two sorts of orange-flavoured chocolates: House Brand and range Delights. 1kgofHouse Brand contains 0.3 kg of chocolate and 0.7 kg of orange fill. 1kgofrange Delights contains 0.5 kg of chocolate and 0.5 kg of orange fill. 300 kg of chocolate and 350 kg of orange fill are available to the manufacturer each da. The profit is $7.50 per kilogram on House Brand and $10 per kilogram on range Delights. How much of each tpe of orange-flavoured chocolate should be made each da to maimise profit? Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
21 396 Essential Standard General Mathematics Solution 1 Define and. Let be the amount (in kg) of House Brand made each da. Let be the amount (in kg) of range Delights made each da. 2 Write down the constraints. and cannot be negative. 300 kg of chocolate is available. 350 kg of orange fill is available. 3 Graph the feasible defined b the constraints. Mark in each of the corner points and label with their coordinates. Use a calculator to determine the point of intersection. 4 Write down the objective function (in dollars). Call it P, for profit. 5 Determine the maimum profit b evaluating the objective function at each corner of the feasible. Constraints: 0, (chocolate) (orange fill) B(0, 600) A(0, 0) bjective function: P = (0, 700) C (125, 525) = 300 D(500, 0) (1000, 0) = 350 bjective function Point P = A(0, 0) P = = $0 B(0, 600) P = = $6000 C(125, 525) P = = $ D(500, 0) P = = $ Write our answer to the question. Themaimum profit is $ , which is obtained b making 125 kg of House Brand and 525 kg of range Delights. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
22 Chapter 9 Inequalities and linear programming 397 Eample 10 Setting up and solving a minimising problem SpeedGro and Powerfeed are two popular brands of home garden fertiliser. The both contain the nutrients X, Y and Z, needed for health plant growth. 1kgofSpeedGro contains 30 units of X, 50units of Y and 10 units of Z. 1kgofPowerfeed contains 20 units of X, 20units of Y and 20 units of Z. Agardener calculates that he needs a fertiliser containing at least 160 units of nutrient X, 200 units of nutrient Y and 80 units of nutrient Z. Speedgro costs $8 per kg and Powerfeed costs $6 per kg. How much of each tpe of fertiliser should he bu to meet his needs at the minimum cost? Solution 1 Define and. Let be the amount (in kg) of SpeedGro needed. Let be the amount (in kg) of Powerfeed needed. 2 Write down the constraints. Constraints: and cannot be negative. 0, 0 At least 160 units of X are needed. At least 200 units of Y are needed. At least 80 units of Z are needed (nutrient X) (nutrient Y) (nutrient Z) 3 Graph the feasible defined b the constraints. Mark in each of the corner A(0, 10) points and label with their coordinates. (0, 8) Use a calculator to determine the points of intersection. B(2, 5) (0, 4) C (4, 2) D(8, 0) (4, 0) (5.3, 0) = = = Write down the objective function (in dollars). Call it C, for cost. bjective function: C = Determine the minimum cost b bjective function evaluating the objective function at Point C = each corner of the feasible. A(0, 10) C = = $60 B(2, 5) C = = $46 C(4, 2) C = = $44 D(8, 0) C = = $64 6 Write our answer to the question. Theminimum cost is $44, which is achieved b buing 4 kg of SpeedGro and 2 kg of Powerfeed. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
23 398 Essential Standard General Mathematics Eercise 9E 1 Afactor makes two products: Wigits and Gigits. Two different machines are used. To make a Wigit takes 1 hour on Machine 1 and 2 hours on Machine 2. To make a Gigit takes 1 hour on Machine 1 and 4 hours on Machine 2. Up to 8 hours of Machine 1 time and up to 24 hours of Machine 2 time are available each da. The factor makes a profit of $200 for each Wigit and $360 for each Gigit it produces. a Let be the number of Wigits made each da. Let be the number of Gigits made each da. The constraints for this problem are: 0, (Machine 1 time) (Machine 2 time) Determine the missing information. b The feasible is shown on the right. Some information is missing. Determine the missing information. (0, 8) + = 8 B C A(0, 0) D(8, 0) = 24 ( ) c The objective function is give b P = 200 +, where P stands for profit (in dollars). Determine the missing information. d How man Wigits and Gigits should be made each da to maimise profit, and what is this profit? 2 An outdoor clothing manufacturer makes two sorts of jackets: Polarbear and Polarfo. To make a Polarbear jacket takes2mofmaterial. The time taken to make a Polarbear jacket is 2.4 hours. To make a Polarfo jacket takes2mofmaterial. The time taken to make a Polarfo jacket is 3.2 hours. The manufacturer has 520 m of material available and 672 hours of worker time to make the jackets. The manufacturer makes a profit of $36 for each Polarbear jacket and $42 for each Polarfo jacket it produces. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
24 a Let be the number of Polarbear jackets made. Let be the number of Polarfo jackets made. The constraints for this problem are:, + 2 (material availabilit) (worker time availabilit) Determine the missing information. b The feasible is shown on the right. Some information is missing. Determine the missing information. Chapter 9 Inequalities and linear programming 399 B(0, 210) = 520 C ( ) = 672 E ( ) A(0, 0) D ( ) c The objective function is given b P = +, where P stands for profit (in dollars) Determine the missing information. d What is the maimum profit that can be made, and how man Polarbear jackets and Polarfo jackets should be made each da to achieve this profit? 3 Following a natural disaster, the arm plans to use helicopters to transport medical teams and their equipment into a remote area. The have two tpes of helicopter: Redhawks and Blackjets. Redhawks carr 45 people and 3 tonnes of equipment. Blackjets carr 30 people and 4 tonnes of equipment. At least 450 people and 36 tonnes of equipment need to be transported. Redhawks cost $3600 per hour to run and Blackjets cost $3200 per hour to run. a Let be the number of Redhawks. Let be the number of Blackjets. The constraints for this problem are: 0, 0 (people) (equipment) Determine the missing information. Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
25 400 Essential Standard General Mathematics b The feasible is shown on the right. Some information is missing. Determine the missing information. ( ) C(12, 0) ( ) = 36 = 450 c The objective function is given b C = +, where C stands for cost (in dollars). Determine the missing information. d How man Redhawks and Blackjets should be used to minimise the cost per hour, and what is this cost? A(0, 15) B( ) 4 Asawmill produces both construction grade and furniture grade timber. To produce 1 cubic metre of construction grade timber takes 2 hours of sawing and 3 hours of planing. To produce 1 cubic metre of furniture grade timber takes 2 hours of sawing and 6 hours of planing. Up to 8 hours of sawing time and 18 hours of planing time are available each da. The sawmill makes a profit of $500 per cubic metre of construction grade timber and $600 per cubic metre of furniture grade timber it produces. a Write down the constraints and profit function for this problem. b Draw a diagram. c Find how much construction grade and furniture grade timber the sawmill should make each da to maimise its profit. What is this profit? 5 Two breakfast cereal mies, Healthstart and Wakeup, are available in bulk. Each kilogram of Healthstart contains 12 mg of vitamin B1 and 40 mg of vitamin B2. Each kilogram of Wakeup contains 20 mg of vitamin B1 and 25 mg of vitamin B2. Youwant a mi of the two that contains at least 15 mg of vitamin B1 and 30 mg of vitamin B2. Healthstart costs $5 a kilogram and Wakeup costs $4.50 per kilogram. a Write down the constraints and cost function for this problem. b Draw a diagram. c Find the miture of these two cereals that will meet our needs at minimum cost. What is this cost? Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
26 Chapter 9 Inequalities and linear programming 401 Ke ideas and chapter summar Linear inequalit Displaing linear inequalities in one variable on a number line Displaing linear inequalities in one variable on the coordinate plane Displaing linear inequalities in two variables on the coordinate plane Linear programming A linear inequalit involves one or two of the signs >,, <or, but not an equals sign ( = ). A linear inequalit in one variable can be represented on a number line b a solid coloured line ending at one or two circles. The line represents all the possible solutions of the inequalit. An open circle ( ) indicates that the end value is not included in the inequalit (for < or >). A closed circle ( ) indicates that the end value is included in the inequalit (for or ). Linear inequalities in one variable can be represented on a coordinate plane b a shaded bounded b one or two lines parallel to the -or-aes. The represents all the possible solutions of the inequalit. A dashed line indicates that the line is not included in the inequalit (for < or >). A solid line indicates that the line is included in the inequalit (for or ). A linear inequalit in two variables can be represented on a coordinate plane b a shaded bounded b a line at an angle to the - and -aes. The represents all the possible solutions of the inequalit. The boundar line is dashed if it is not included in the inequalit (for < or >), but solid if it is included (for or ). A reference point, often the origin (0, 0), can be used to help decide whether the required lies above or below the line. When solving simultaneous inequalities, the in the coordinate plane that is common to all the inequalities is called the feasible.itrepresents all the possible solutions to the simultaneous inequalities. The feasible can be found graphicall (for a small number of inequalities) b shading in the required s for all the inequalities and determining where the all overlap. Agraphics calculator can be used to graph a feasible. Linear programming involves maimising or minimising a linear quantit subject to the constraints represented b a set of linear inequalities. The constraints (e.g. requirements, resources) define the feasible in which the quantit is to be maimised or minimised. Review Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
27 402 Essential Standard General Mathematics Review bjective function Corner point principle Skills check The constraints 0 and 0together restrict the feasible to the positive (first) quadrant. The objective function is a linear epression representing the quantit to be maimised (e.g. profit) or minimised (e.g. cost) in a linear programming problem. The corner point principle states that, in linear programming problems, the maimum or minimum value of a linear objective function will occur at one of the corners of the feasible, or on a line on the boundar of the feasible joining two of the corners. Having completed this topic ou should be able to: represent a linear inequalit in one variable on a number line represent a linear inequalit in one or two variables on the coordinate plane know the meaning of the terms feasible, constraint and objective function as the relate to linear programming determine the maimum or minimum value of an objective function for a given feasible set up and solve basic linear programming problems. Multiple-choice questions 1 The inequalit displaed on the number line on the right is: A 1 7 B 1 < < 7 C 1 < 7 D 1 < 7 E 1 > > 7 2 The inequalit displaed on the number line on the right is: A < 5 B 5 C > 5 D 5 E 0 > > 5 3 The inequalit displaed on the coordinate plane on the right is: A < 8 B 8 C < 8 D 8 E 0 > > (0, 8) = 8 Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
28 Chapter 9 Inequalities and linear programming The inequalit displaed on the coordinate plane on the right is: A 3 < < 10 B 3 < 10 C 3 10 D 3 < < 10 E 3 < 10 5 The equation of the line displaed on the right is: A = 4 B 4 5 = 4 C = 20 D = 20 E 4 5 = 20 6 The equation of the line displaed on the right is: A 4 5 = 0 B = 0 C 5 4 = 20 D = 20 E 5 = 20 7 The displaed on the right (including the line) represents the inequalit: A < 20 B C > 20 D E > 20 = 3 (0, 10) = 10 (3, 0) (10, 0) (0, 4) (0, 0) = 20 8 The displaed on the right (not including the line) represents the inequalit: A 3 6 B 3 < 6 C 3 6 D 3 > 6 E 3 > 6 (4, 0) (0, 6) ( 2, 0) (5, 0) (5, 4) 3 = 6 Review Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
29 404 Essential Standard General Mathematics Review 9 The two lines shown on the right intersect at the point: A (1, 1.3) B (2, 1.5) C (1.2, 1.2) D (2.4, 1.2) E (3, 2.4) (0, 3) (0, 2) 10 The feasible displaed on the right (including the line) is defined b the inequalities: (0, 5) A 0, 0, < 5 B 0, 0, 5 C 0, 0, + < 5 D 0, 0, + 5 E 0, 0, The feasible displaed on the right (including the lines) is defined b the inequalities: A 0, 0, , B 0, 0, , C 0, 0, + 2 > 10, 4 + > 12 D 0, 0, + 2 < 10, 4 + < 12 E 0, 0, , (0, 12) = 12 (0, 5) (2, 4) (3, 0) 4 + = = 6 (4, 0) (6, 0) (5, 0) 12 For the feasible displaed in Question 11, the minimum value of the objective function, C = 2 +, is: A 5 B 6 C 8 D 12 E For the feasible displaed on the right, the maimum value of the objective function, P = 4 + 3, is: A 0 B 40 C 42 D 48 E 60 (0, 10) (0, 12) (6, 6) (0, 0) (12, 0) (10, 0) + 2 = 10 The following information relates to Questions 14 to 16 An outdoor clothing manufacturer makes two stles of all-weather coats: long and short. To make a short coat,2mofmaterial are required. The time taken to make a short coat is 2.5 hours. To make a long coat,3mofmaterial are required. The time taken to make a long coat is 3.5 hours. (15, 0) Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
30 Chapter 9 Inequalities and linear programming 405 The manufacturer has 450 m of material available and 700 hours of worker time to make the coats. The manufacturer makes a profit of $40 for each short coat and $48 for each long coat. Let be the number of short coats made. Let be the number of long coats made. 14 The constraints that relate to the amount of material available are: A 0, 0, B 0, 0, C 0, 0, D 0, 0, E 0, 0, The constraints that relate to the amount of time available are: A 0, 0, B 0, 0, C 0, 0, D 0, 0, E 0, 0, The objective function P is: A P = B P = C P = D P = E P = Short-answer questions 1 Plot the inequalit 2 < 4onanumber line. 2 Plot the inequalit 1 < 5onthe coordinate plane. 3 Plot the inequalit < 40 on the coordinate plane. 4 Plot the defined b the inequalities: 0, 0, Plot the defined b the inequalities: 0, 0, , Etended-response questions 1 Agarden products compan makes two sorts of fertiliser: Standard Grade and Premium Grade. There are two main ingredients: nitrate and phosphate. To make a tonne of Standard Grade fertiliser takes 0.8 tonnes of nitrate and 0.2 tonnes of phosphate. To make a tonne of Premium Grade fertiliser takes 0.7 tonnes of nitrate and 0.3 tonnes of phosphate. The compan has 56 tonnes of nitrate and 21 tonnes of phosphate. The compan makes a profit of $600 per tonne on Standard Grade fertiliser and $750 per tonne on Premium Grade fertiliser. Review Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
31 406 Essential Standard General Mathematics Review a Write the constraints and profit function for this problem. b Draw a diagram. c Find how much of each tpe of fertiliser the compan should make to maimise its profit. What will this profit be? 2 Two foods fed to animals contain both vitamin A and vitamin B. 1kgofFood A contains 3 units of vitamin A and 4 units of vitamin B. 1kgofFood B contains 5 units of vitamin A and 3 units of vitamin B. The dail vitamin requirement of each animal is at least 15 units of vitamin A and at least 12 units of vitamin B. Food A costs $0.30 per kg and Food B costs $0.24 per kg. a Write the constraints and cost function for this problem. b Draw a diagram. c Find how much of each tpe of food should be fed to the animals each da to minimise cost. What is this cost? Cambridge Universit Press Uncorrected Sample Pages Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
SAMPLE. A Gallery of Graphs. To recognise the rules of a number of common algebraic relationships: y = x 1,
Objectives C H A P T E R 5 A Galler of Graphs To recognise the rules of a number of common algebraic relationships: =, =, = / and + =. To be able to sketch the graphs and simple transformations of these
More informationIntermediate 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
More informationLinear 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,
More informationUnit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations)
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
More informationF6 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
More informationUNIT 6 MODELING GEOMETRY Lesson 1: Deriving Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: appling the Pthagorean Theorem representing horizontal and vertical distances in a coordinate plane simplifing square roots writing
More informationData transformation. Core: Data analysis. Chapter 5
Chapter 5 5 Core: Data analsis Data transformation ISBN 978--7-56757-3 Jones et al. 6 66 Core Chapter 5 Data transformation 5A Introduction You first encountered data transformation in Chapter where ou
More information74 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
More informationMEP 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. :
More informationUNCORRECTED 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.
More information5A Exponential functions
Chapter 5 5 Eponential and logarithmic functions bjectives To graph eponential and logarithmic functions and transformations of these functions. To introduce Euler s number e. To revise the inde and logarithm
More informationx. 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
More informationP.4 Lines in the Plane
28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables
More informationINVESTIGATE the Math
. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.
More information8.4. 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,
More informationMaintaining 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
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationReady 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
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More information10.3 Solving Nonlinear Systems of Equations
60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6
More informationMt. 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.
More informationSAMPLE. Revision. Revision of Chapters 1 7. The implied (largest possible) domain for the function with the rule y = is: 2 x. 1 a
C H P T R 8 of Chapters 7 8. Multiple-choice questions The domain of the function whose graph is shown is: 3, ] B, 3] C [, 3] D [, 3), 3) 3 3 Which of the following sets of ordered pairs does not represent
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INEQUALITY = a mathematical statement that contains one of these four inequalit signs: ,.
More informationSection 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
More informationSAMPLE. Exponential and logarithmic functions
Objectives C H A P T E R 5 Eponential and logarithmic functions To graph eponential and logarithmic functions. To graph transformations of the graphs of eponential and logarithmic functions. To introduce
More informationSAMPLE. Exponential Functions and Logarithms
Objectives C H A P T E R 5 Eponential Functions and Logarithms To define and understand eponential functions. To sketch graphs of the various tpes of eponential functions. To understand the rules for manipulating
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - 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
More information10.4 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 + =
More information7.1 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
More informationLESSON #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
More informationChapter 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.
More information6.1 Solving Quadratic Equations by Graphing Algebra 2
10.1 Solving Quadratic Equations b Graphing Algebra Goal 1: Write functions in quadratic form Goal : Graph quadratic functions Goal 3: Solve quadratic equations b graphing. Quadratic Function: Eample 1:
More informationFALL 2012 MATH 1324 REVIEW EXAM 2
FALL 0 MATH 3 REVIEW EXAM MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the order of the matri product AB and the product BA, whenever the
More informationChapter 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
More informationINEQUALITIES
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
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationModule 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
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationThe standard form of the equation of a circle is based on the distance formula. The distance formula, in turn, is based on the Pythagorean Theorem.
Unit, Lesson. Deriving the Equation of a Circle The graph of an equation in and is the set of all points (, ) in a coordinate plane that satisf the equation. Some equations have graphs with precise geometric
More informationCubic and quartic functions
3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving
More informationSAMPLE. Quadratics. Objectives
Objectives C H A P T E R 4 Quadratics To recognise and sketch the graphs of quadratic relations. To determine the maimum or minimum values of a quadratic relation. To solve quadratic equations b factorising,
More informationNAME DATE PERIOD. Study Guide and Intervention
NAME DATE PERID Stud Guide and Intervention Graph To graph a quadratic inequalit in two variables, use the following steps: 1. Graph the related quadratic equation, = a 2 + b + c. Use a dashed line for
More information3.2. Properties of Graphs of Quadratic Relations. LEARN ABOUT the Math. Reasoning from a table of values and a graph of a quadratic model
3. Properties of Graphs of Quadratic Relations YOU WILL NEED grid paper ruler graphing calculator GOAL Describe the ke features of the graphs of quadratic relations, and use the graphs to solve problems.
More informationMATHEMATICS: PAPER I. 4. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
GRADE 11 EXEMPLAR PAPERS NOVEMBER 007 MATHEMATICS: PAPER I Time: 3 hours 150 marks Instructions to candidates 1. This eamination consists of 9 pages.. Read the questions carefull. 3. Answer all the questions
More informationChapter 6: Systems of Equations and Inequalities
Chapter 6: Sstems of Equations and Inequalities 6-1: Solving Sstems b Graphing Objectives: Identif solutions of sstems of linear equation in two variables. Solve sstems of linear equation in two variables
More informationLESSON #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
More informationLESSON #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
More informationChapter 6 Class Notes 6-1 Solving Inequalities Using Addition and Subtraction p n 1
Chapter Class Notes Alg. CP - Solving Inequalities Using Addition and Subtraction p.. t. a. n. r r - Solving Inequalities Using Multiplication and Division p. 0-0 A) n B) n A) B) p When ou multipl or divide
More informationCh 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
More informationSAMPLE. Rates of Change
Objectives C H A P T E R 18 Rates of Change To recognise relationships between variables. To calculate average rates of change. To estimate gradients of curves. To estimate instantaneous rates of change.
More informationChapter 11. Systems of Equations Solving Systems of Linear Equations by Graphing
Chapter 11 Sstems of Equations 11.1 Solving Sstems of Linear Equations b Graphing Learning Objectives: A. Decide whether an ordered pair is a solution of a sstem of linear equations. B. Solve a sstem of
More informationSection 5.1: Functions
Objective: Identif functions and use correct notation to evaluate functions at numerical and variable values. A relationship is a matching of elements between two sets with the first set called the domain
More informationReteaching (continued)
Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression
More informationLESSON 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
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
More information1.7 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,
More informationMini-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.
More information136 Maths Quest 10 for Victoria
Quadratic graphs 5 Barr is a basketball plaer. He passes the ball to a team mate. When the ball is thrown, the path traced b the ball is a parabola. Barr s throw follows the quadratic equation =.5 +. +.5.
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationCONSUMER CHOICES Madison is thinking about leasing a car for. Example 1 Solve the system of equations by graphing.
2-1 BJECTIVES Solve sstems of equations graphicall. Solve sstems of equations algebraicall. Solving Sstems of Equations in Two Variables CNSUMER CHICES Madison is thinking about leasing a car for two ears.
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationSystems 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
More informationName: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig*
Name: Richard Montgomer High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website
More information5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS
CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif
More informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More informationy x can be solved using the quadratic equation Y1 ( x 5), then the other is
Math 0 Precalculus Sstem of Equation Review Questions Multiple Choice. The sstem of equations A. 7 0 7 0 0 0 and can be solved using the quadratic equation. In solving the quadratic equation 0 ( ) intersection
More informationSolving 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
More informationMATH 115: Review for Chapter 6
MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING
More informationFunctions. 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
More informationUnit 4 Relations and Functions. 4.1 An Overview of Relations and Functions. January 26, Smart Board Notes Unit 4.notebook
Unit 4 Relations and Functions 4.1 An Overview of Relations and Functions Jan 26 5:56 PM Jan 26 6:25 PM A Relation associates the elements of one set of objects with the elements of another set. Relations
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More informationMATH 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 $
More informationFair 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
More information3.7 Linear and Quadratic Models
3.7. Linear and Quadratic Models www.ck12.org 3.7 Linear and Quadratic Models Learning Objectives Identif functions using differences and ratios. Write equations for functions. Perform eponential and quadratic
More informationSystems of Linear Inequalities
. Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit
More informationFunctions. 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
More informationUnit 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
More informationThe American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet
The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More information5.7 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
More informationPower Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n
1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr,
More informationReady 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
More informationLinear Equation Theory - 2
Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite
More informationAlgebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.
Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph
More informationEssential Question How can you use a scatter plot and a line of fit to make conclusions about data?
. Scatter Plots and Lines of Fit Essential Question How can ou use a scatter plot and a line of fit to make conclusions about data? A scatter plot is a graph that shows the relationship between two data
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper Practice Paper A Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion
More informationFair 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)
More informationWhat Did You Learn? Key Terms. Key Concepts. 158 Chapter 1 Functions and Their Graphs
333371_010R.qxp 12/27/0 10:37 AM Page 158 158 Chapter 1 Functions and Their Graphs Ke Terms What Did You Learn? equation, p. 77 solution point, p. 77 intercepts, p. 78 slope, p. 88 point-slope form, p.
More informationAdvanced Algebra 2 Final Review Packet KG Page 1 of Find the slope of the line passing through (3, -1) and (6, 4).
Advanced Algebra Final Review Packet KG 0 Page of 8. Evaluate (7 ) 0 when and. 7 7. Solve the equation.. Solve the equation.. Solve the equation. 6. An awards dinner costs $ plus $ for each person making
More information3.2 Introduction to Functions
8 CHAPTER Graphs and Functions Write each statement as an equation in two variables. Then graph each equation. 97. The -value is more than three times the -value. 98. The -value is - decreased b twice
More informationAnswers to All Exercises. Appendix C ( ) ( ) Section C.1 (page C7) APPENDICES. Answers to All Exercises Ans1
Answers to All Eercises Ans Answers to All Eercises Appendi C Section C. (page C). Cartesian. Distance Formula. Midpoint Formula. ( h) + ( k) = r, center, radius. c. f. a. d. e. b. A: (, ); B: (, ); C:
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More informationChapter 6. Exploring Data: Relationships
Chapter 6 Exploring Data: Relationships For All Practical Purposes: Effective Teaching A characteristic of an effective instructor is fairness and consistenc in grading and evaluating student performance.
More informationMATH GRADE 8 UNIT 4 LINEAR RELATIONSHIPS EXERCISES
MATH GRADE 8 UNIT LINEAR RELATIONSHIPS Copright 01 Pearson Education, Inc., or its affiliate(s). All Rights Reserved. Printed in the United States of America. This publication is protected b copright,
More information7.1 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....
More informationLESSON #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,
More informationGraphs of Nonlinear Inequalities
3-3 BJECTIVES Graph polnomial, absolute value, and radical inequalities in two variables. Solve absolute value inequalities. Graphs of Nonlinear Inequalities PHARMACLGY Pharmacists label medication as
More informationUNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism
More informationGet Discount Coupons for our Coaching institute and FREE Stud Material at www.pickmycaching.com MDULE - I 6 LINEAR INEQUATINS AND THEIR APPLICATINS You have alread read about linear and quadratic equations
More informationMA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman)
MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) Chapter Goals: Solve an equation for one variable in terms of another. What is a function? Find inverse functions. What is a graph?
More information