LESSON 5. SYSTEMS OF INEQUALITIES LESSON 5. SYSTEMS OF INEQUALITIES
OVERVIEW Here s what ou ll learn in this lesson: Solving Linear Sstems a. Solving sstems of linear inequalities b graphing As a conscientious dog owner, ou like to make sure that our best friend gets all the vitamins and minerals he needs. When given the choice between canned dog food and fresh meat, ou consider two things: price and nutrition. You want to find combinations of fresh meat and canned food that ou can afford to bu that will give our dog all the vitamins and minerals he needs. The nutritional content of each kind of food and the amount of each ou can afford to bu describe a sstem of linear inequalities which ma have man solutions. To decide how much of each kind of food to bu, ou can graph this sstem of linear inequalities, and consider all of the possibilities. In this lesson ou will learn how to solve sstems of linear inequalities. TOPIC 5 SOLVING LINEAR SYSTEMS
EXPLAIN SOLVING LINEAR SYSTEMS Summar Solving Sstems of Linear Inequalities Just as ou can use graphing to find the solution of a sstem of linear equations, ou can also use graphing to find the solution of a sstem of linear inequalities. To find the solution of a sstem of linear inequalities b graphing:. Graph the linear equation that corresponds to the first inequalit. Use a dotted line.. Shade the region whose points satisf the inequalit.. Graph the linear equation that corresponds to the second inequalit. Use a dotted line.. Shade the region whose points satisf the inequalit. 5. The solution is the points in the region where the shading overlaps. For eample, to find the solution of this linear sstem: + > <. Graph the linear equation + =.. Shade the region above the line.. Graph the linear equation =.. Shade the region above the line. 5. The solution is the points in the region where the shading overlaps. Don't forget to use a solid line if the inequalit is or. An point that makes both inequalities true is called a solution of the sstem. + > How do ou know to shade above the line? Substitute the test point (0, 0) into the inequalit + >. Since (0) + 0 > is not true, shade on the side of the line which doesn t contain (0, 0). all the points in this region < + > LESSON 5. SYSTEMS OF INEQUALITIES EXPLAIN
+ As another eample, find the solution of this sstem: + <. Graph the linear equation = +.. Shade the region above and including the line. < + +. Graph the linear equation =.. Shade the region below the line. 5. There is no solution to this sstem because the shaded regions do not overlap. Like sstems of linear equations, it is possible for sstems of linear inequalities to have no solution onl when the corresponding equations are parallel lines (as in the previous eample). However, unlike sstems of linear equations, it is possible for a sstem of linear inequalities to have a solution even when the corresponding lines are parallel. For eample, this sstem of two parallel linear inequalities has a solution: + < The solution of this sstem is shown to the left. < TOPIC 5 SOLVING LINEAR SYSTEMS
Sample Problems. Graph the sstem below to find its solution: Answers to Sample Problems > a. Graph the line =. b. Shade the region whose points satisf the inequalit. c. Graph the line =. d. Shade the region whose points satisf the inequalit. c., d., e. e. Indicate the region whose points are the solution of the sstem.. Graph the sstem below to find its solution: all the points in this region > + a. Graph the line = +. b. Shade the region whose points satisf the inequalit. c. Graph the line =. d. Shade the region whose points satisf the inequalit. e. Indicate the region whose points are the solution of the sstem. c., d., e. all the points in this region LESSON 5. SYSTEMS OF INEQUALITIES EXPLAIN 5
Answers to Sample Problems b. Sample Problems EXPLORE On the computer ou used the Grapher to solve and analze sstems of linear inequalities. Below are some additional eploration problems.. Graph the sstem of inequalities below. Find the points of intersection of each pair of lines and the area of the region that is the solution. c. 0 0 + a. Graph the inequalit 0. b. Graph the inequalit 0. c. Graph the inequalit +. a., b., c. d. Find the intersection of = 0 + = and + = 0 + =. = intersection: (0, ) e. (, 0) e. Find the intersection of = 0 and + =. intersection: (, ) f. (0, 0) f. Find the intersection of = 0 and = 0. intersection: (, ) g. Find the area of the triangle. area = base height g., 8 a., b. = = square units. Find the coordinates of four points one in each quadrant that satisf both inequalities in the sstem below. + 5 a. Graph the inequalit. c., d., e., f. d. an point in this region c. an point in this region b. Graph the inequalit + 5. c. Find a point in Quadrant I that satisfies both inequalities. II I d. Find a point in Quadrant II that satisfies both inequalities. III e. an point in this region IV f. an point in this region e. Find a point in Quadrant III that satisfies both inequalities. f. Find a point in Quadrant IV that satisfies both inequalities. TOPIC 5 SOLVING LINEAR SYSTEMS
HOMEWORK Homework Problems Circle the homework problems assigned to ou b the computer, then complete them below. Eplain Solving Linear Sstems. Graph the sstem of inequalities below. Use the grid in Figure 5... Then write the coordinates of the plotted points that are solutions of this sstem. + < Figure 5... Graph the sstem of inequalities below to find its solution. + <. Graph the sstem of inequalities below to find its solution. 5 >. Graph the sstem of inequalities below to find its solution. > + < + 8 5. Graph the sstem of inequalities below to find its solution. + 5. Graph the sstem of inequalities below to find its solution. + < 8 7. Graph the sstem of inequalities below to find its solution. < > 7 8. Graph the sstem of inequalities below to find its solution. 5 5 + 9. Conchita is moving out of her rental house and needs to clean it up. Because she is so bus, she won t have more than 0 hours to spend cleaning. She wants to spend at least as much time cleaning the inside of the house as she spends picking up the ard. Let equal the number of hours Conchita spends picking up the ard; let equal the number of hours she spends cleaning the house. Write a sstem of two linear inequalities that represents Conchita's cleaning options. Then graph the sstem. 0. Aki is tring to decide how to divide her eercise time between running and swimming. She wants to spend at least three times as much time swimming as she spends running, but she doesn't want to spend more than 9 hours per week swimming. Let equal the number of hours per week Aki spends running; let equal the number of hours per week she spends swimming. Write a sstem of two linear inequalities that represents Aki's eercise options. Then graph the sstem.. Graph the sstem of inequalities below to find its solution.. Graph the sstem of inequalities below to find its solution. + + 5 + LESSON 5. SYSTEMS OF INEQUALITIES HOMEWORK 7
Eplore. Graph the sstem of inequalities below. Find the points of intersection of each pair of lines. 0 5 5 5 + 5. Graph the sstem of inequalities below. Then find the coordinates of four points that satisf both inequalities in the sstem. 7 5. Graph the sstem of inequalities below. Find the points of intersection of each pair of lines and the area of the region whose points are the solution of the sstem. 0 0 5. Graph the sstem of inequalities below. If the direction of both inequalit signs are reversed, what region contains the points that are the solution of the new sstem? < + + <7 7. Graph the sstem of inequalities below. Then find the coordinates of four points that satisf both inequalities in the sstem. + + 8. Graph the sstem of inequalities below. Find the points of intersection of each pair of lines and the area of the region whose points are the solution of the sstem. + + 0 + 0 8 TOPIC 5 SOLVING LINEAR SYSTEMS
APPLY Practice Problems Here are some additional practice problems for ou to tr. Solving Linear Sstems. Graph the sstem of inequalities below. > + <. Graph the sstem of inequalities below. + +. Graph the sstem of inequalities below. > < +. Graph the sstem of inequalities below. < < + 5. Graph the sstem of inequalities below. +. Graph the sstem of inequalities below. < < + 7. Graph the sstem of inequalities below. + < 8. Graph the sstem of inequalities below. > + 5 9. Graph the sstem of inequalities below. + > 0. Graph the sstem of inequalities below. > + 5 +. Graph the sstem of inequalities below. < + 5. Graph the sstem of inequalities below. <. Graph the sstem of inequalities below. > + <. Graph the sstem of inequalities below. + 5. Graph the sstem of inequalities below. + > + 5. Graph the sstem of inequalities below. + > < LESSON 5. SYSTEMS OF INEQUALITIES APPLY 9
7. Graph the sstem of inequalities below. + < 8. Graph the sstem of inequalities below. + 9. Graph the sstem of inequalities below. + 0. Graph the sstem of inequalities below. > + 5 +. Graph the sstem of inequalities below. 5 > + <. Graph the sstem of inequalities below. > <. Graph the sstem of inequalities below. + 5. Graph the sstem of inequalities below. < 5. Graph the sstem of inequalities below. >. Graph the sstem of inequalities below. < > 7. Graph the sstem of inequalities below. > + < 8. Graph the sstem of inequalities below. + 0 TOPIC 5 SOLVING LINEAR SYSTEMS
Practice Test EVALUATE Take this practice test to be sure that ou are prepared for the final quiz in Evaluate.. Graph the sstem of inequalities below. Use the grid in Figure 5... Then write the coordinates of the plotted points that are solutions of this sstem. < + + Figure 5... Graph the sstem of inequalities below to find its solution. + >. Graph the sstem of inequalities below. Use the grid in Figure 5... Then write the coordinates of the plotted points that are not solutions of either inequalit in this sstem. 5 < 5 + 5. Graph the sstem of inequalities below to find its solution. + 5. Graph the sstem of inequalities below to find its solution. +. Graph the sstem of inequalities below. Use the grid in Figure 5... Then write the coordinates of the plotted points that are solutions of this sstem. 0 + Figure 5.. Figure 5.. LESSON 5. SYSTEMS OF INEQUALITIES EVALUATE
7. The sstem of inequalities below is graphed in Figure 5..5. If the direction of the first inequalit sign is reversed, outline the region(s) that contains the points that are the solution of the new sstem. + < + > 8. Graph the sstem of inequalities below. Then write the coordinates of a point that satisfies the first two inequalities but not the third inequalit. 0 Figure 5..5 TOPIC 5 SOLVING LINEAR SYSTEMS
TOPIC 5 CUMULATIVE ACTIVITIES CUMULATIVE REVIEW PROBLEMS These problems combine all of the material ou have covered so far in this course. You ma want to test our understanding of this material before ou move on to the net topic, or ou ma wish to do these problems to review for a test.. Evaluate the epression 7 + 8 + when = and =. a. Find the GCF of 75, 8, and 80. b. Find the LCM of 75, 8, and 80.. The sum of two numbers is 5. The difference of these numbers is 5. What are the two numbers?. Write the equation of the line through the point (, ) with slope : a. in point-slope form. b. in slope-intercept form. c. in standard form. 5. Solve this sstem: + = = 7. Solve + < 5 for, then graph its solution on the number line below. 8 0 8 7. Find the slope of the line through the points (0, 5) and (8, ). 8. Solve 5 = ( ) for. 9. Graph the sstem of inequalities below to find its solution. + 0. Reduce to lowest terms: 080 590. Solve this sstem: + = 7 =. The point (, ) lies on a line with slope. Graph this line b finding another point that lies on the line.. Find the slope of the line that is perpendicular to the line 9 = 5.. Graph the sstem of inequalities below to find its solution. + 5a. Find the GCF of 8,, and 5. 5b. Find the LCM of 8,, and 5.. Write the equation of the line through the point (, 5) with slope : a. in point-slope form. b. in slope-intercept form. c. in standard form. 7. Solve: + 5 = 9 for. 8. Last ear Keith split $,500 between his savings account, which paid % in interest, and his checking account, which paid.5% in interest. If he earned a total of $.5 in interest, how was his mone divided between the two accounts? 9. Solve this sstem: + 5 = 5 + = TOPIC 5 CUMULATIVE REVIEW 5
0. Graph the sstem of inequalities below to find its solution. + 5. Evaluate the epression a b + ab when a = and b =. Use Figure 5. to answer questions through 5. B C D 9. Solve this sstem: 7 = 9 7 + = 0. Graph this sstem of inequalities to find its solution.. Solve 0 9 for, then graph its solution on the number line below. A 8 0 8 Figure 5.. Which two lines form a sstem that has a solution of (5, 5)?. Which two lines form a sstem that has a solution of (, )?. Which two lines form a sstem that has a solution of ( 5, 0)? 5. Which two lines form a sstem that has no solution?. Find the slope of the line through the points ( 9, ) and (, ). 7 7. Find: 5 9 7 8. Solve ( + 5) = ( + ) + for.. Write the equation of the line through the point ( 8, 7) with slope : a. in point-slope form. b. in slope-intercept form. c. in standard form.. Brad emptied the mone from a newspaper vending machine that accepted onl dimes and quarters and got 8 coins worth a total of $.0. How man dimes did he collect? 5. Find: 5. Solve 8 + 7 = for.. Solve this sstem: 9 + 8 = 58 5 8 + 8 = 5 7. Find the slope of the line 5 = 8. 5 TOPIC 5 SOLVING LINEAR SYSTEMS