Linear Inequality in Two Variables


 Nickolas Lee
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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.
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