Mathematics
Solving Systems of Inequalities
Solving systems of inequalities involves finding the overlapping regions that satisfy multiple inequalities. This is typically done graphically by shading the areas that satisfy each individual inequality and then identifying the overlapping shaded region. The solution to the system of inequalities is the set of points within the overlapping region.
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12 Key excerpts on "Solving Systems of Inequalities"
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
478 CHAPTER 5 ■ Systems of Equations and Inequalities Graphing Inequalities (pp. 467–468) To graph an inequality: 1. Graph the equation that corresponds to the inequality. This “boundary curve” divides the coordinate plane into separate regions. 2. Use test points to determine which region(s) satisfy the inequality. 3. Shade the region(s) that satisfy the inequality, and use a solid line for the boundary curve if it satisfies the inequality ( or ) and a dashed line if it does not ( or ). Graphing Systems of Inequalities (p. 469) To graph the solution of a system of inequalities (or feasible region determined by the inequalities): 1. Graph all the inequalities on the same coordinate plane. 2. The solution is the intersection of the solutions of all the inequalities, so shade the region that satisfies all the inequalities. 3. Determine the coordinates of the intersection points of all the boundary curves that touch the solution set of the system. These points are the vertices of the solution. 1. (a) What is a system of equations in the variables x , y , and z ? (b) What are the three methods we use to solve a system of equations? 2. Consider the following system of equations: e x y 3 3 x y 1 (a) Describe the steps you would use to solve a system by the substitution method. Use the substitution method to solve the given system. (b) Describe the steps you would use to solve a system by the elimination method. Use the elimination method to solve the given system. (c) Describe the steps you would use to solve a system by the graphical method. Use the graph shown below to solve the system. y x+y=3 3x-y=1 x 1 1 0 3. What is a system of linear equations in the variables x , y , and z ? 4. For a system of two linear equations in two variables, (a) How many solutions are possible? (b) What is meant by an inconsistent system? (c) What is meant by a dependent system? 5. What operations can be performed on a linear system to arrive at an equivalent system? 6.
eBook - PDFCollege Algebra
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
The solution set of such a linear inequality is the set of all points ( x , y ) that satisfy the inequality. To find the solution set of a linear inequality, we first graph the corresponding equality. This graph divides the xy -plane into two regions. One of these regions will satisfy the inequality. Example 4 illustrates how an inequality in two variables is solved. Graphing a Linear Inequality Graph the solution set of each of the following linear inequalities. a. x ≥ 3 b. y < 2 x Solution a. First graph the line x = 3 as a solid line, because it is included in the inequality. This line separates the xy -plane into two half-planes. Choose a point not on the line and see if it satisfies the inequality x ≥ 3. We choose (0, 0) because it is easy to check. x ? ≥ 3 ⇒ 0 ? ≥ 3 Because the inequality 0 ≥ 3 is false, the half-plane that contains the point (0, 0) does not satisfy the inequality. Thus all the points in the other half-plane do satisfy the inequality, and so this region is shaded. See Figure 2. b. Graph the line y = 2 x as a dashed line, because it is not included in the inequality. Choose a point not on the line and see if it satisfies the inequality y < 2 x. We choose (2, 0) because it is easy to check. Note that we cannot use (0, 0) because it lies on the line y = 2 x. y ? < 2 x ⇒ 0 ? < 2(2) Example 4 x x = 3 Graph of x ≥ 3 y 5 4 3 2 1 1 (0, 0) –1 –1 –2 –3 –4 –5 –2 –3 2 3 4 5 6 7 8 Figure 2 5.1 Systems of Linear Equations and Inequalities in Two Variables 407 Because the inequality 0 < 4 is true, the half-plane that contains the point (2, 0) is shaded. See Figure 3. Check It Out 4 Graph the inequality y ≥ x + 5. Solving Systems of Linear Inequalities in Two Variables The solution set of a system of linear inequalities in the variables x and y consists of the set of all points ( x , y ) in the intersection of the solution sets of the individual inequalities of the system.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.5 Solving Systems of Linear Inequalities 331 EXAMPLE 3 Graph the solutions of the system: Strategy We will graph the solutions of in one color and the solutions of in another color on the same coordinate system to see where the graphs of the two inequalities intersect. Why The solution set of the system is the set of all points in the intersection of the two graphs. Solution The boundary of the graph of is the line . Since the inequality contains the symbol , we draw the boundary as a solid line. The test point makes true, so we shade the side of the boundary that contains . See part (a) of the figure below. Graph the boundary: A table of solutions Shading: Check the test point (0, 0) (0, 0) x 2 (0, 0) x 2 x 2 y 3 x 2 e x 2 y 3 x 2 2 0 2 2 2 4 (2, 4) (2, 2) (2, 0) ( x , y ) y x True Since is true, is a solution of . x 2 (0, 0) 0 2 0 2 x 2 In part (b) of the figure, the graph of is superimposed over the graph of . The boundary of the graph of is the line . Since the inequality contains the symbol , we draw the boundary as a dashed line. The test point makes false, so we shade the side of the boundary that does not contain . Graph the boundary: A table of solutions Shading: Check the test point (0, 0) (0, 0) y 3 (0, 0) y 3 y 3 x 2 y 3 y 3 0 3 1 3 4 3 (4, 3) (1, 3) (0, 3) ( x , y ) y x False Since is false, is not a solution of . y 3 (0, 0) 0 3 0 3 y 3 (a) (b) The area that is shaded twice represents the solutions of the system of inequalities. Any point in the doubly shaded region in purple has coordinates that satisfy both inequalities, including the purple portion of the boundary.
eBook - PDFPrecalculus
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
To find the solution set of a linear inequality, we first graph the corresponding equality. This graph divides the xy -plane into two regions. One of these regions will satisfy the inequality. Example 3 illustrates how an inequality in two variables is solved. Solving a Linear Inequality Graph the solution set of each of the following linear inequalities. a. x ≥ 3 b. y < 2 x Solution a. First graph the line x = 3 as a solid line, because it is included in the inequality. This line separates the xy -plane into two half-planes. Choose a point not on the line and see if it satisfies the inequality x ≥ 3. We choose (0, 0) because it is easy to check. x ? ≥ 3 ⇒ 0 ? ≥ 3 Because the inequality 0 ≥ 3 is false, the half-plane that contains the point (0, 0) does not satisfy the inequality. Thus all the points in the other half-plane do satisfy the inequality, and so this region is shaded. See Figure 2. b. Graph the line y = 2 x as a dashed line, because it is not included in the inequality. Choose a point not on the line and see if it satisfies the inequality y < 2 x. We choose (2, 0) because it is easy to check. Note that we cannot use (0, 0) because it lies on the line y = 2 x. y ? < 2 x ⇒ 0 ? < 2(2) Example 3 x x = 3 Graph of x ≥ 3 y 5 4 3 2 1 1 (0, 0) –1 –1 –2 –3 –4 –5 –2 –3 2 3 4 5 6 7 8 Figure 2 8.1 Systems of Linear Equations and Inequalities in Two Variables 647 Because the inequality 0 < 4 is true, the half-plane that contains the point (2, 0) is shaded. See Figure 3. Check It Out 3 Graph the inequality y ≥ x + 5. Solving Systems of Linear Inequalities in Two Variables The solution set of a system of linear inequalities in the variables x and y consists of the set of all points ( x , y ) in the intersection of the solution sets of the individual inequalities of the system. To find the graph of the solution set of the system, graph the solution set of each inequality and then find the intersection of the shaded regions.
eBook - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
In systems of linear equations, we sought the points that satisfied all of the equations. The solution set of a system of inequalities contains the points that satisfy all of the inequalities. The graph of a system of inequalities can be obtained by simultaneously graphing each individual inequality and finding where the shaded regions intersect (or overlap), if at all. STEP 4 Shade the region containing x y (5, 0) 2x –3y ≥ 6 or y ≤ x – 2 2 3 the point (5, 0). Your Turn Graph the inequality x − 2y ≤ 6. Answer x y x – 2y ≤ 6 or y ≥ x – 3 1 2 Video EXAMPLE 3 Graphing a System of Two Linear Inequalities Graph the system of inequalities: x + y ≥ −2 x + y ≤ 2 Solution STEP 1 Change the inequality signs to equal signs. x + y = −2 x + y = 2 780 CHAPTER 8 Systems of Linear Equations and Inequalities STEP 2 Draw the two lines. Because the inequality signs are x y y = –x + 2 y = –x – 2 not strict, use solid lines. STEP 3 Test points for each inequality. x + y ≥ −2 Substitute (−4, 0) into x + y ≥ −2. −4 ≥ −2 The point (−4, 0) does not satisfy the inequality. Substitute (0, 0) into x + y ≥ −2. 0 ≥ −2 The point (0, 0) does satisfy the inequality. x + y ≤ 2 Substitute (0, 0) into x + y ≤ 2. 0 ≤ 2 The point (0, 0) does satisfy the inequality. Substitute (4, 0) into x + y ≤ 2. 4 ≤ 2 The point (4, 0) does not satisfy the inequality. STEP 4 For x + y ≥ −2, shade the region For x + y ≤ 2, shade the region above that includes (0, 0). below that includes (0, 0). x y y ≥ –x – 2 x y y ≤ –x + 2 STEP 5 The overlapping region is the x y y ≤ –x + 2 y ≥ –x – 2 solution. Notice that the points (0, 0), (−1, 1), and (1, −1) all lie in the shaded region and all three satisfy both inequalities. Video EXAMPLE 4 Graphing a System of Two Linear Inequalities with No Solution Graph the system of inequalities: x + y ≤ −2 x + y ≥ 2 8.7 Systems of Linear Inequalities in Two Variables 781 Solution STEP 1 Change the inequality signs to equal signs.
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
1. 1 0, 0 2 2. 1 3, 0 2 3. 1 2, 2 2 4. 1 0, 4 2 Getting Ready 4.5 x y x y Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 270 CHAPTER 4 Inequalities We now use our knowledge of Solving Systems of Inequalities in two variables to solve linear programming applications. Find the maximum and minimum value of an equation in the form P 5 ax 1 by , subject to specific constraints. Linear programming is a mathematical technique used to find the optimal allocation of resources in the military, business, telecommunications, and other fields. It got its start during World War II when it became necessary to move huge quantities of people, materi-als, and supplies as efficiently and economically as possible. To solve a linear program, we maximize (or minimize) a function (called the objective function ) subject to given conditions on its variables. These conditions (called constraints ) are usually given as a system of linear inequalities. For example, suppose that the annual profit (in millions of dollars) earned by a business is given by the equation P 5 y 1 2 x , where the profit is determined by the sale of two different items, x and y , and are subject to the following constraints: μ 3 x 1 y # 120 x 1 y # 60 x $ 0 y $ 0 To find the maximum profit P that can be earned by the business, we solve the system of inequalities as shown in Figure 4-35(a) and find the coordinates of each corner point of the region R , called a feasibility region .
eBook - PDFPractical Algebra
A Self-Teaching Guide
- Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
5 LINEAR INEQUALITIES So far, we’ve worked extensively with equations. However, we often deal with situa-tions where we find values of the variable in which one expression is greater than or less than another. Fortunately, despite some important differences, the methods we use here are similar to the methods we used to solve equations. 5.1 Basic Principles of Solving Inequalities First, let’s start with a basic definition: an inequality is a statement that says that one expression is greater than or less than another expression. Like equations, inequalities must be either true or false—they can’t be both at the same time. As with equations, we often want to find the values of the variables that make the inequality true. We call these the solutions to the inequality . In Chapter 1, we used number lines to represent positive and negative numbers. Recall that positive numbers appear to the right of 0, and negative numbers appear to the left of 0. Also, any number that appears to the right of another is the greater number. We use the following inequality symbols: • > (pronounced “is greater than”) • ≥ (pronounced “is greater than or equal to,” combines the > and = symbols) • < (pronounced “is less than”) • ≤ (pronounced “is less than or equal to,” combines the < and = symbols) Here are some important points about the direction of inequalities: • The inequality symbols in statements like 6 > 2 and 5 > 3 have the same direction. • The inequality symbols in statements like 6 > 2 and 4 < 5 have the opposite direction. • If we reverse the order of the numbers, then we reverse the direction of the inequality symbol, so 3 > 1 is equivalent to 1 < 3. 111 112 PRACTICAL ALGEBRA Reading and Writing Tip The > and < symbols are easily confused. Figure 5.1 can help you remember that the inequal-ity symbols “open” in the direction of the larger number: 5 3 Figure 5.1 5 is greater than 3. Example 5.1 Determine if the inequality + 1 > − 2 is true.
eBook - PDFBeginning Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The graph of the solution set of the system of inequalities is the intersection of the two solution sets and has been shaded in the following graph. 12 6 9 3 –3 –6 y x –10 –20 10 20 –15 –5 5 15 10 x – 16 y $ –80 y , 3 ? ? ? ? ? Connecting the Concepts What’s true about points in the solution set? Remember that a point in the solution set must make both inequalities true. Skill Connection Horizontal Inequalities In Section 4.4, we learned that since the value of y gives the vertical coordinate, when graphing the solution set to a y $ b problem, we shade above the line. Likewise, when graphing the solution set to a y # b problem, we shade below the line. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. S E C T I O N 4 . 5 S y s t e m s o f L i n e a r I n e q u a l i t i e s 439 To check our work, we see that the point (0, 0) is in the solution set. Substituting x 5 0 and y 5 0 into both original inequalities yields Inequality 1 Inequality 2 10( 0 ) 2 16( 0 ) $ 2 80 0 , 3 0 $ 2 80 True 0 , 3 True PRACTICE PROBLEM FOR EXAMPLE 3 Graph the solution set for each system of inequalities. a. y , 2 5 2 x 1 8 b. 2 x 2 3 y $ 21 y $ 4 x 2 6 y , 2 3 Some of the applications that we previously looked at in systems of linear equations may be more realistic when stated as a system of inequalities. One such type of application comes from investment problems. ? Example 4 Applying constraints to an investment Mitsuko recently won a lottery game and has a maximum of $50,000.00 to invest.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 642 Chapter 9 Systems of Equations and Inequalities GO DIGITAL Graphical Interpretation of Solutions It is possible for a system of equations to have exactly one solution, two or more solutions, or no solution. In a system of linear equations, however, if the system has two different solutions, then it must have an infinite number of solutions. To see why this is true, consider the following graphical interpretation of a system of two linear equations in two variables. A system of linear equations is consistent when it has at least one solution. A system is inconsistent when it has no solution. EXAMPLE 5 Recognizing Graphs of Linear Systems See LarsonPrecalculus.com for an interactive version of this type of example. Match each system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. a. { 2x -4x - + 3y 6y = = 3 6 b. { 2x x - + 3y 2y = = 3 5 c. { 2x -4x - + 3y 6y = = 3 -6 i. x - 2 2 4 - 2 - 4 2 4 y ii. x 4 2 4 - 2 - 4 2 y iii. x - 2 2 4 - 2 - 4 2 4 y Solution a. The graph of system (a) is a pair of parallel lines (ii). The lines have no point of intersection, so the system has no solution. The system is inconsistent. b. The graph of system (b) is a pair of intersecting lines (iii). The lines have one point of intersection, so the system has exactly one solution. The system is consistent. c. The graph of system (c) is a pair of lines that coincide (i). The lines have infinitely many points of intersection, so the system has infinitely many solutions.
eBook - PDFBeginning and Intermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Solve an application by setting up and solving a system of three linear equations in three variables. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 580 CHAPTER 8 Solving Systems of Linear Equations and Inequalities Unless otherwise noted, all content on this page is © Cengage Learning. We now extend the definition of a linear equation to include equations of the form ax 1 by 1 cz 5 d where a , b , c , and d are numerical values ( d is a constant). The solution of a system of three linear equations with three variables is an ordered triple of numbers if the equations are independent and the system consistent. For example, the solution of the system • 2 x 1 3 y 1 4 z 5 20 3 x 1 4 y 1 2 z 5 17 3 x 1 2 y 1 3 z 5 16 1 x , y , z 2 , which equals 1 1, 2, 3 2 is the ordered triple 1 1, 2, 3 2 . Each equation is satisfied if x 5 1 , y 5 2 , and z 5 3 . 2 x 1 3 y 1 4 z 5 20 3 x 1 4 y 1 2 z 5 17 3 x 1 2 y 1 3 z 5 16 2 1 1 2 1 3 1 2 2 1 4 1 3 2 0 20 3 1 1 2 1 4 1 2 2 1 2 1 3 2 0 17 3 1 1 2 1 2 1 2 2 1 3 1 3 2 0 16 2 1 6 1 12 0 20 3 1 8 1 6 0 17 3 1 4 1 9 0 16 20 5 20 17 5 17 16 5 16 The graph of an equation of the form ax 1 by 1 cz 5 d is a flat surface called a plane . A system of three linear equations in three variables is consistent or inconsis-tent, depending on how the three planes corresponding to the three equations intersect. Figure 8-15 illustrates some of the possibilities. Determine whether the equation x 1 2 y 1 3 z 5 6 is satisfied by the following values.
eBook - PDFIntermediate Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Find equations for models of these data. b. Determine the years for which the number of females enrolled in U.S. colleges was greater than or equal to the number of males enrolled in U.S. colleges. Solving Inequalities Numerically and Graphically We can use a table to estimate the solutions to inequalities by looking for the input value(s) that make the left side of the inequality equal to the right side of the inequality. Once we have the input value(s) that make both sides equal, we compare the output value(s) on either side of those inputs to see when one is less than or greater than the other. Start by putting the left side into Y1 and the right side into Y2 of the Y 5 screen of your calculator. When looking for the input value(s) that make the two sides equal, notice when one column changes from being smaller than the other column to being larger. The value where these two sides are equal is between these input values. Now look at the inequality relationship in the problem to find the solution set. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. S E C T I O N 2 . 4 S o l v i n g L i n e a r I n e q u a l i t i e s 197 Example 5 Solving inequalities numerically Solve the following inequalities numerically using the calculator table. a. 5x 1 7 . 2x 1 31 b. 4x 1 9 # 6x 2 1 SOLUTION a. Enter both sides of the inequality into the calculator and then use the table to estimate a solution.
eBook - PDFCollege Algebra
Building Skills and Modeling Situations
- Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
The technique we will use to solve inequalities of this type involves graphing. Suppose, for example, we want to find the solution set for the inequality x 2 − x − 6 > 0. We begin by factoring the left side to obtain (x − 3)(x + 2) > 0 We have two real numbers x − 3 and x + 2 whose product (x − 3)(x + 2) is greater than zero. That is, their product is positive. The only way the product can be positive is either if both factors, (x − 3) and (x + 2), are positive or if they are both negative. To help visualize where x − 3 is positive and where it is negative, we draw a real number line and label it accordingly: Here is a similar diagram showing where the factor x + 2 is positive and where it is negative: Drawing the two number lines together and eliminating the unnecessary num- bers, we have We can see from the preceding diagram that the graph of the solution to x 2 − x − 6 > 0 is x < −2 or x > 3 3 - - - - - - - - - - - - + + + + + Sign of x - 3 x - 3 is negative when x < 3 x - 3 is positive when x > 3 + + + + + + + + + + + + - - - - - Sign of x + 2 x + 2 is negative when x < -2 x + 2 is positive when x > -2 -2 - - - - - - - - - - + + + + + - - - - - + + + + + + + + + + -2 3 Both factors negative, their product is positive Both factors positive, their product is positive Sign of x - 3 Sign of x + 2 3 -2 144 Chapter 2 Solving Equations and Inequalities Solve for x: x 2 − 2x − 8 ≤ 0. SOLUTION We begin by factoring: x 2 − 2x − 8 ≤ 0 (x − 4)(x + 2) ≤ 0 The product (x − 4)(x + 2) is negative or zero. The factors must have opposite signs. We draw a diagram showing where each factor is positive and where each factor is negative: From the diagram, we have the graph of the solution set: −2 ≤ x ≤ 4 Solve x 2 − 6x + 9 ≥ 0. SOLUTION x 2 − 6x + 9 ≥ 0 (x − 3) 2 ≥ 0 This is a special case in which both factors are the same. Because (x − 3) 2 is always positive or zero, the solution set is all real numbers.
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