Linear Inequality in Two Variables


 Nickolas Lee
 1 years ago
 Views:
Transcription
1 90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter. In this section we etend the ideas of linear equations in two variables to stud linear inequalities in two variables. Definition Graph of a Linear Inequalit Using a Test Point to Graph an Inequalit Applications Definition Linear inequalities in two variables have the same form as linear equations in two variables. An inequalit smbol is used in place of the equal sign. Linear Inequalit in Two Variables If A, B, and C are real numbers with A and B not both zero, then A B C is called a linear inequalit in two variables. In place of, we can also use,, or. The inequalities 4 8,, and 9 0 are linear inequalities. Not all of these are in the form of the definition, but the could all be rewritten in that form. An ordered pair is a solution to an inequalit in two variables if the ordered pair satisfies the inequalit. E X A M P L E Satisfing a linear inequalit Determine whether each point satisfies the inequalit 6. a) (4, ) b) (, 0) c) (, ) stud tip a) To determine whether (4, ) is a solution to the inequalit, we replace b 4 and b in the inequalit 6: Write about what ou read in the tet. Sum things up in our own words. Write out important facts on note cards. When ou have a few spare minutes in between classes review our note cards. Tr to get the information on the cards into our memor. (4) () Incorrect So (4, ) does not satisf the inequalit 6. b) Replace b and b 0: () (0) Correct So the point (, 0) satisfies the inequalit 6. c) Replace b and b : () ( ) Correct So the point (, ) satisfies the inequalit 6.
2 7.4 Graphing Linear Inequalities in Two Variables (7) 9 Graph of a Linear Inequalit The graph of a linear inequalit in two variables consists of all points in the rectangular coordinate sstem that satisf the inequalit. For eample, the graph of the inequalit consists of all points where the coordinate is larger than the coordinate plus. Consider the point (, 5) on the line. The coordinate of (, 5) is equal to the coordinate plus. If we choose a point with a larger coordinate, such as (, 6), it satisfies the inequalit and it is above the line. In fact, an point above the line satisfies. Likewise, all points below the line satisf the inequalit. See Fig helpful hint Wh do we keep drawing graphs? When we solve 7, we don t bother to draw a graph showing, because the solution set is so simple. However, the solution set to a linear inequalit is a ver large set of ordered pairs. Graphing gives us a wa to visualize the solution set. > + Above the line (, 6) = + (, 5) < + Below the line FIGURE 7.6 To graph the inequalit, we shade all points above the line. To indicate that the line is not included in the graph of, we use a dashed line. The procedure for graphing linear inequalities is summarized as follows. Strateg for Graphing a Linear Inequalit in Two Variables. Solve the inequalit for, then graph m b. m b is the region above the line. m b is the line itself. m b is the region below the line.. If the inequalit involves onl, then graph the vertical line k. k is the region to the right of the line. k is the line itself. k is the region to the left of the line.
3 9 (74) Chapter 7 Sstems of Linear Equations and Inequalities E X A M P L E Graphing a linear inequalit Graph each inequalit. a) b) c) 6 a) The set of points satisfing this inequalit is the region below the line. To show this region, we first graph the boundar line. The slope of the line is, and the intercept is (0, ). We draw the line dashed because it is not part of the graph of. In Fig. 7.7 the graph is the shaded region. + > < + FIGURE 7.7 FIGURE 7.8 FIGURE 7.9 b) Because the inequalit smbol is, ever point on or above the line satisfies this inequalit. We use the fact that the slope of this line is and the intercept is (0, ) to draw the graph of the line. To show that the line is included in the graph, we make it a solid line and shade the region above. See Fig c) First solve for : 6 6 Divide b and reverse the inequalit. To graph this inequalit, we first graph the line with slope and intercept (0, ).We use a dashed line for the boundar because it is not included, and we shade the region above the line. Remember, less than means below the line and greater than means above the line onl when the inequalit is solved for. See Fig. 7.9 for the graph. E X A M P L E Horizontal and vertical boundar lines Graph each inequalit. a) 4 b)
4 7.4 Graphing Linear Inequalities in Two Variables (75) 9 a) The line 4 is the horizontal line with intercept (0, 4). We draw a solid horizontal line and shade below it as in Fig b) The line is a vertical line through (, 0). An point to the right of this line has an coordinate larger than. The graph is shown in Fig > FIGURE 7.0 FIGURE 7. Using a Test Point to Graph an Inequalit The graph of a linear equation such as 6 separates the coordinate plane into two regions. One region satisfies the inequalit 6, and the other region satisfies the inequalit 6. We can tell which region satisfies which inequalit b testing a point in one region. With this method it is not necessar to solve the inequalit for. E X A M P L E 4 helpful hint Some people alwas like to choose (0, 0) as the test point for lines that do not go through (0, 0). The arithmetic for testing (0, 0) is generall easier than for an other point. Using a test point Graph the inequalit 6. First graph the equation 6 using the intercept (, 0) and the intercept (0, ) as shown in Fig. 7.. Select a point on one side of the line, sa (0, ), to test in the inequalit. Because (0) () 6 is false, the region on the other side of the line satisfies the inequalit. The graph of 6 is shown in Fig. 7.. Test point (0, ) > 6 FIGURE 7. FIGURE 7.
5 94 (76) Chapter 7 Sstems of Linear Equations and Inequalities Applications The values of variables used in applications are often restricted to nonnegative numbers. So solutions to inequalities in these applications are graphed in the first quadrant onl. E X A M P L E 5 Manufacturing tables The Ozark Furniture Compan can obtain at most 8000 board feet of oak lumber for making two tpes of tables. It takes 50 board feet to make a round table and 80 board feet to make a rectangular table. Write an inequalit that limits the possible number of tables of each tpe that can be made. Draw a graph showing all possibilities for the number of tables that can be made. If is the number of round tables and is the number of rectangular tables, then and satisf the inequalit Now find the intercepts for the line : Draw the line through (0, 00) and (60, 0). Because (0, 0) satisfies the inequalit, the number of tables must be below the line. Since the number of tables cannot be negative, the number of tables made must be below the line and in the first quadrant as shown in Fig Assuming that Ozark will not make a fraction of a table, onl points in Fig. 7.4 with wholenumber coordinates are practical FIGURE 7.4 WARMUPS True or false? Eplain our answer.. The point (, 4) satisfies the inequalit. True. The point (, ) satisfies the inequalit. True. The graph of the inequalit 9 is the region above the line 9. True 4. The graph of the inequalit is the region below the line. False 5. The graph of is a single point on the ais. False 6. The graph of 5 is the region below the horizontal line 5. False
6 7.4 Graphing Linear Inequalities in Two Variables (77) 95 WARMUPS (continued) 7. The graph of is the region to the left of the vertical line. True 8. In graphing the inequalit we use a dashed boundar line. False 9. The point (0, 0) is on the graph of the inequalit. True 0. The point (0, 0) lies above the line. False 7. 4 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is a linear inequalit in two variables? A linear inequalit has the same form as a linear equation ecept that an inequalit smbol is used.. How can ou tell if an ordered pair satisfies a linear inequalit in two variables? An ordered pair satisfies a linear inequalit if the inequalit is correct when the variables are replaced b the coordinates of the ordered pair.. How do ou determine whether to draw the boundar line of the graph of a linear inequalit dashed or solid? If the inequalit smbol includes equalit, then the boundar line is solid; otherwise it is dashed. 4. How do ou decide which side of the boundar line to shade? We shade the side that satisfies the inequalit. 5. What is the test point method? In the test point method we test a point to see which side of the boundar line satisfies the inequalit. 6. What is the advantage of the test point method? With the test point method ou can use the inequalit in an form. Determine which of the points following each inequalit satisf that inequalit. See Eample (, ), (, 9), (8, ) (, 9) 8. (, 6), (0, ), (, 0) (, 6) 9. 5 (, 0), (, ), (, 5) (, 0), (, ) 0. 6 (, 0), (, 9), ( 4, ) (, 0), (, 9). 4 (, ), (7, ), (0, 5) (, ), (0, 5). (, ), (, 4), (0, ) (, 4) Graph each inequalit. See Eamples and
7 96 (78) Chapter 7 Sstems of Linear Equations and Inequalities Graph each inequalit. Use the test point method of Eample
8 7.4 Graphing Linear Inequalities in Two Variables (79) FIGURE FOR EXERCISE 50 rocker requires board feet of maple. write an inequalit that limits the possible number of maple rockers of each tpe that can be made, and graph the inequalit in the first quadrant Solve each problem. See Eample Storing the tables. Ozark Furniture Compan must store its oak tables before shipping. A round table is packaged in a carton with a volume of 5 cubic feet (ft ), and a rectangular table is packaged in a carton with a volume of 5 ft. The warehouse has at most 850 ft of space available for these tables. Write an inequalit that limits the possible number of tables of each tpe that can be stored, and graph the inequalit in the first quadrant Enzme concentration. A food chemist tests enzmes for their abilit to break down pectin in fruit juices (Dennis Callas, Snapshots of Applications in Mathematics). Ecess pectin makes juice cloud. In one test, the chemist measures the concentration of the enzme, c, in milligrams per milliliter and the fraction of light absorbed b the liquid, a. If a 0.07c 0.0, then the enzme is working as it should. Graph the inequalit for 0 c Maple rockers. Ozark Furniture Compan can obtain at most 000 board feet of maple lumber for making its classic and modern maple rocking chairs. A classic maple rocker requires 5 board feet of maple, and a modern
9 98 (70) Chapter 7 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student then drew a dashed line through these two points and shaded the region below the line. What is wrong with this method? Do all of the points graphed b this student satisf the inequalit? 5. Writing. Compare and contrast the two methods presented in this section for graphing linear inequalities. What are the advantages and disadvantages of each method? How do ou choose which method to use? In this section The to a Sstem of Inequalities Graphing a Sstem of Inequalities E X A M P L E stud tip Read the tet and recite to ourself what ou have read. Ask questions and answer them out loud. Listen to our answers to see if the are complete and correct. Would other students understand our answers? 7.5 GRAPHING SYSTEMS OF LINEAR INEQUALITIES In Section 7.4 ou learned how to solve a linear inequalit. In this section ou will solve sstems of linear inequalities. The to a Sstem of Inequalities A point is a solution to a sstem of equations if it satisfies both equations. Similarl, a point is a solution to a sstem of inequalities if it satisfies both inequalities. Satisfing a sstem of inequalities Determine whether each point is a solution to the sstem of inequalities: 6 a) (, ) b) (4, ) c) (5, ) a) The point (, ) is a solution to the sstem if it satisfies both inequalities. Let and in each inequalit: 6 ( ) () 6 ( ) Because both inequalities are satisfied, the point (, ) is a solution to the sstem. b) Let 4 and in each inequalit: 6 (4) ( ) 6 (4) 6 7 Because onl one inequalit is satisfied, the point (4, ) is not a solution to the sstem. c) Let 5 and in each inequalit: 6 (5) () 6 (5) 6 9 Because neither inequalit is satisfied, the point (5, ) is not a solution to the sstem.
GRAPHING SYSTEMS OF LINEAR INEQUALITIES
444 (8 5) Chapter 8 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student
More information3.4. section. Definition. A linear inequality is a linear equation with the equal sign replaced by an inequality symbol.
. Linear Inequalities and Their Graphs () a) Write the equation of the line through (8., 0.) and (7.6, 77.) and epress w as a linear function of d. b) What is the flow when the depth is 7.8 feet? c) Is
More informationSection 7.1 Graphing Linear Inequalities in Two Variables
Section 7.1 Graphing Linear Inequalities in Two Variables Eamples of linear inequalities in two variables include + 6, and 1 A solution of a linear inequalit is an ordered pair that satisfies the
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More information12.2 Graphing Systems of Linear Inequalities
Name Class Date 1. Graphing Sstems of Linear Inequalities Essential Question: How do ou solve a sstem of linear inequalities? Resource Locker Eplore Determining Solutions of Sstems of Linear Inequalities
More informationLinear Inequalities, Systems, and Linear Programming
8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed
More information12.2 Graphing Systems of Linear Inequalities
  0   Locker LESSON 1. Graphing Sstems of Linear Inequalities Common Core Math Standards The student is epected to: AREI.1 Graph the solutions to a linear inequalit in two variables as a halfplane
More information3.3. section. 140 (320) Chapter 3 Graphs and Functions in the Cartesian Coordinate System FIGURE FOR EXERCISE 52 MISCELLANEOUS
0 (0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Selling price (in thousands of dollars) 0 a) Use the graph on the net page to estimate the average retail price of a earold car in
More informationChapter 5 Graphing Linear Equations and Inequalities
.1 The Rectangular Coordinate Sstem (Page 1 of 28) Chapter Graphing Linear Equations and Inequalities.1 The Rectangular Coordinate Sstem The rectangular coordinate sstem (figure 1) has four quadrants created
More informationInequalities and Linear Programming
4CH_PHCalter_TMSETE_949118 3//007 1:38 PM Page 1 Inequalities and Linear Programming OBJECTIVES When ou have completed this chapter, ou should be able to: Graph linear inequalities on the number line.
More informationEssential Question How can you graph a linear inequality in two variables?
5.6 Graphing Linear Inequalities in Two Variables Essential Question How can ou graph a linear inequalit in two variables? A solution of a linear inequalit in two variables is an ordered pair (, ) that
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationSolving Systems of Linear Inequalities
  0   Locker LESSON 5. Solving Sstems of Linear Inequalities Teas Math Standards The student is epected to: A.3.F Solve sstems of two or more linear inequalities in two variables. Also A.3.E, A.3.G
More informationGraphs and Functions in the Cartesian Coordinate System
C H A P T E R Graphs and Functions in the Cartesian Coordinate Sstem List price (in thousands of dollars) 0 0 0 (99, 0,) (98,,67) (00, 7,) he first selfpropelled automobile to carr passengers was built
More informationLINEAR PROGRAMMING J
9. Sstems of Linear Inequalities 9. Linear Programming Involving Two Variables 9. The Simple Method: Maimization 9.4 The Simple Method: Minimization 9.5 The Simple Method: Mied Constraints John von Neumann
More informationSECTION 84 Systems of Linear Inequalities in Two Variables
84 Sstems of Linear Inequalities in Two Variables 627 at the same time and follow the same route on the 7mile trip across the English Channel to Cherbourg, France. The average speed of boat A is miles
More informationINTRODUCTION TO FUNCTIONS
4.6 Introduction to Functions (4 47) 0 with a height of 60 cm, B is a linear function of the person s weight w (in kilograms). For a weight of 45 kg, B is 00 calories. For a weight of 50 kg, B is 65 calories.
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 informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationSECTION 26 Inverse Functions
182 2 Graphs and Functions SECTION 26 Inverse Functions OnetoOne Functions Inverse Functions Man important mathematical relationships can be epressed in terms of functions. For eample, C d f(d) V s
More informationLesson 5.2 Exercises, pages
Lesson 5. Eercises, pages 6 68 A. Determine whether each point is a solution of the given inequalit. a)  16 A(, ) In the inequalit, substitute:, L.S.: ( ) () 17 R.S. 16 Since the L.S.
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationEssential Question: How do you write and graph linear inequalities in two variables?
5     Locker LESSON 7.3 Linear Inequalities in Two Variables Teas Math Standards The student is epected to: A1.3.D Graph the solution set of linear inequalities in two variables on the coordinate plane.
More informationThe Slope of a Line 4.2. On the coordinate system below, plot a point, any point.
.2 The Slope of a Line.2 OBJECTIVES 1. Find the slope of a line 2. Find the slopes of parallel and perpendicular lines 3. Find the slope of a line given an equation. Find the slope given a graph 5. Graph
More informationAnnual rate GRAPHS OF FUNCTIONS. Linear and Constant Functions. Linear Function
.6 Graphs of Functions () 7 80. Printing costs. To determine the cost of printing a book, a printer uses a linear function of the number of pages. If the cost is $8.60 for a 00page book and $.0 for a
More informationInequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,
Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 7577 Es. 420. 2328, 2939 odd, 4043, 4952, 5973 odd
More informationGraphing Linear Inequalities
7.4 Graphing Linear Inequalities 7.4 OBJECTIVE 1. Graph a linear inequalit in two variables In Section 2.7 ou learned to graph inequalities in one variable on a number line. We now want to etend our work
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 informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationThe slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationLines and Linear Equations. Slopes
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationchangeeofyy slope = m =
LLEVADA S ALGEBRA Section The Slope In section, linear equations and the use of the slope were introduced Now the question will be how to come up with the slope and its corresponding linear equation without
More informationChapter 2 Section 5: Linear Inequalities
Chapter Section : Linear Inequalities Introduction Now we ll see what happens in the coordinate plane when we replace the equal sign in a linear equation with an inequality symbol. A line with equation
More informationSOLVING SYSTEMS BY GRAPHING AND SUBSTITUTION
94 (8 ) Chapter 8 Sstems of Linear Equations and Inequalities In this Solving a Sstem b Graphing section Independent, Inconsistent, and Dependent Equations Solving b Substitution Applications E X A M P
More informationon the left graph below.
3.1 Graphing Linear Inequalities Graphing linear inequalities in two variables: The solution set for an inequality in two variables is shown on the Cartesian coordinate system. Boundary lines divide the
More informationTwoVariable Inequalities
7 What You ll Learn To graph linear inequalities To graph absolute value inequalities...and Wh To solve problems involving combinations, as in Eample TwoVariable Inequalities Check Skills You ll Need
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information2.4 Linear Inequalities
6360_ch0pp07668.qd 0/6/08 4:3 PM Page 3 3 CHAPTER Linear Functions and Equations.4 Linear Inequalities Understand basic terminolog related to inequalities Solve linear inequalities smbolicall Solve linear
More informationSECONDDEGREE INEQUALITIES
74 ( 6) Chapter Nonlinear Sstems and the Conic Sections Working in groups, simplif this equation. First get the radicals on opposite sides of the equation, then square both sides twice to eliminate the
More informationUnit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system
Unit Stud Guide Sstems of Linear Equations and Inequalities 6 Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and  in for in
More informationCollege Prep Math Notes Quadratics Unit Quadratics. Math Background
Quadratics Math Background Previousl, ou Identified and graphed quadratic functions in Algebra II Applied transformations to parent functions Solved quadratic functions in Algebra II Worked with comple
More informationEssential Question How can you graph a system of linear inequalities?
5.7 Sstems of Linear Inequalities Essential Question How can ou graph a sstem of linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain
More informationSystems of Equations
Sstems of Equations Sstem of equations two or more equations where ou want to find a sloution that makes all of them true simultaneousle (the same point makes them all true). Solution of a sstem of equations
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line () 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationGraphing Inequalities in Two Variables
Graphing Inequalities in Two Variables Then You graphed linear equations. (Lesson 31) Now Graph linear inequalities on the coordinate plane. Solve inequalities b graphing. New Vocabular boundar halfplane
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving sstems of linear inequalities, including realworld problems through graphing two and three variables
More informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationLINEAR PROGRAMMING: THE GRAPHICAL METHOD
r Chapter LINEAR PROGRAMMING: THE GRAPHICAL METHOD. Graphing Linear Inequalities Your Turn 8 6 + 8 6 8 Your Turn 6 +. Eercises. + First graph the boundar line + = using the points (, ) and (, ). Since
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More information1.2. Graphs of Equations. The Graph of an Equation. What you should learn. Why you should learn it
3330_010.qd 1 1/7/05 Chapter 1 1. 8:31 AM Page 1 Function and Their Graphs Graphs of Equations What ou should learn Sketch graphs of equations. Find  and intercepts of graphs of equations. Use smmetr
More informationSystems of Linear Equations in Two Variables
5.1 Sstems of Linear Equations in Two Variables 5.1 OBJECTIVES 1. Find ordered pairs associated with two equations 2. Solve a sstem b graphing 3. Solve a sstem b the addition method 4. Solve a sstem b
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More informationSolving x < a. Section 4.4 Absolute Value Inequalities 391
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationWarmUp. What is the solution for this equation? 2x 3 = 5. What is the solution for this equation? x +10 =12
CST/CAHSEE:Algebra WarmUp Review: Algebra What is the solution for this equation? x = What is the solution for this equation? x + = A. x = or x = B. x = or x = C. x = or x = D. x = or x = Show two was
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More information2.5 Absolute Value Equations and Inequalities
660_ch0pp07668.qd 0/6/08 4: PM Page 46 46 CHAPTER Linear Functions and Equations Writing about Mathematics 0. Suppose the solution to the equation a + b = 0 with a 7 0 is = k. Discuss how the value of
More informationLesson 8.3 Exercises, pages
Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5  b) = 5 c) =  d) =. Sketch broken lines to represent the vertical and horizontal
More information1.3 LINES IN THE PLANE AND SLOPE
000_00.qd //0 : AM Page CHAPTER Functions, Graphs, and Limits. LINES IN THE PLANE AND SLOPE Use the slopeintercept form of a linear equation to sketch graphs. Find slopes of lines passing through two
More information( 7, 3) means x = 7 and y = 3
3 A: Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two linear equations that each represents the graph of a line. Eamples
More informationChapter 4: Linear Systems of Equations
HOSP 1107 (Business Math) Learning Centre Chapter 4: Linear Sstems of Equations An pair of linear equations (with two variables) can be solved b using algebra or graphing. To solve sstems of equations
More informationLinear Programming: A Geometric Approach
3 Linear Programming: A Geometric Approach Graphing Sstems of Linear Inequalities in Two Variables Linear Programming Problems Graphical Solutions of Linear Programming Problems Sensitivit Analsis (skip)
More informationDo NOT use a calculator. ( i ) x + 11 = 57 ( ii ) x  13 = 14. ( iii ) 5x = 115 ( iv ) 5x + 8 = 33. ( v ) 4 x  7 = 33 ( vi ) 8x + 3 = 7
INEQUALITIES These lesson notes are available from www.pilean.com The ma be freel duplicated and distributed but copright remains with the author. Martin Hansen Chapter.. Solving Simple Equations & Inequalities
More informationF6.1 Inequalities on a Number Line. F6.2 Solution of Linear Inequalities (Inequations) F6.3 Inequalities Involving Quadratic Terms
Mathematics SKE: STRAND F UNIT F Solving Inequalities: Tet STRAND F: ALGEBRA Unit Solving Inequalities Tet Contents Section * * * F. Inequalities on a Number Line F. of Linear Inequalities (Inequations)
More informationPre Calculus Math 40S: Explained!
Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph
More informationIn Lesson 7.1, you learned that you can write rules for some of the coding grids. y x 1
LEON 7. PLANNING LEON OUTLINE One da: 0 min Investigation min haring min Eample min min ATERIAL Closing Eercises Function or Not? (T), optional Calculator Note 1J LEON 7. Functions and Graphs In Lesson
More information2.5. Direct and Inverse Variation Stacking Boxes. My Notes ACTIVITY
Direct and Inverse Variation SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Think/Pair/Share, Look for a Pattern You work for a packaging and shipping compan. As part of our job there,
More informationGRAPHS OF POLYNOMIAL FUNCTIONS
(0) Chapter 0 Polnomial and Rational Functions. A bo of frozen specimens measures inches b inches b inches. It is wrapped in an insulating material of uniform thickness for shipment. The volume of the
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationInequalities. After completing this chapter you should be able to:
After completing this chapter ou should be able to: Manipulate inequalities Determine the critical values of an inequalit Find solutions of algebraic inequalities Inequalities 1 Most applications of mathematics
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationGraph Linear Inequalities in Two Variables
.8 Graph Linear Inequalities in Two Variables Before You solved linear inequalities in one variable. Now You will graph linear inequalities in two variables. Wh? So ou can model data encoding, as in Eample
More informationAnswers may vary. Sample: The set of all positive numbers is one example.
61 Solving Sstems b Graphing Vocabular Review Write I if the amount described is infinite. Write F if the amount is finite. I 1. the rational numbers greater than 6 F. the number of seats in a movie theater
More informationAlgebra 2 / Trigonometry Summer Math Packet Summer 2015
Algebra / Trigonometr Summer Math Packet Summer 015 Covering Prerequisite Concepts for Incoming Algebra / Trigonometr Students This summer packet contains eciting math problems designed to ensure our readiness
More informationExample 2 Finding the Domain and Range of a Function
7_00.qd /7/06 0:9 AM Page 5 Section. Graphs of Functions 5. Graphs of Functions The Graph of a Function In Section., functions were represented graphicall b points on a graph in a coordinate plane in which
More informationSection 0.2 Set notation and solving inequalities
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationLearning Objectives for Section 1.2 Graphs and Lines. Cartesian coordinate system. Linear Equations in Two Variables
Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to identif and work with the Cartesian coordinate sstem. calculate the slope of a line.
More informationDomain, Range, and End Behavior
COMMON CORE Locker LESSON Domain, Range, and End Behavior Common Core Math Standards The student is epected to: COMMON CORE FIF.B.5 Relate the domain of a function to its graph and, where applicable,
More informationAlgebra 1 Unit 3. Review Worksheet Review Worksheet Review Algebra 1 Unit 3 1
Algebra 1 Unit 3 1. Students will be able to determine whether an ordered pair is a solution of an equation or a point on a line. The will be able to graph a line b making a table of values. Worksheet
More informationStudent Lesson: Inverses of Functions
TEKS: Objectives: A.1 Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. A.1A The student is epected to identif the mathematical
More informationSolving Systems Using Tables and Graphs
31 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slopeintercept 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 informationWork with a partner. The following steps show a method of solving ax 2 + bx + c = 0. Explain what was done in each step.
9.5 Solving Quadratic Equations Using the Essential Question How can ou derive a formula that can be used to write the solutions of an quadratic equation in standard form? Deriving the Work with a partner.
More informationRates of Change in Rational Functions. LEARN ABOUT the Math
. Rates of Change in Rational Functions YOU WILL NEED graphing calculator or graphing software GOAL Determine average rates of change, and estimate instantaneous rates of change for rational functions.
More information