MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

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1 FOM 11 T13 APPLICATIONS OF LINEAR INEQUALITIES 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) MATHEMATICAL MODEL = a representation of information given in a word problem using equations, inequalities and or graphs. 2) MATHEMATICAL RESTRICTIONS = limits to the types of numbers that can be used to solve the problems. e.g. x!, x W, x I, or x " MODELING REAL LIFE USING LINEAR INEQUALITIES I) Many real life situations can be represented using linear inequalities. The questions asked in word problems describing real life situations are solved by creating mathematical models. A MATHEMATICAL MODEL IS A REPRESENTATION OF INFORMATIONS GIVEN IN A WORD PROBLEM USING EQUATIONS, INEQUALITIES AND GRAPHS. The mathematical model created includes one or more linear inequalities, which are graphed on a grid. The graph is then used to answer the question posed in the problem. A) USE THESE STEPS TO SOLVE REAL LIFE SITUATIONS INVOLVING LINEAR INEQUALITIES e.g. Let x = the number of cars crossing the bridge each hour Let y = the number of trucks crossing the bridge each hour 6: Answer the question(s) found in the original word problem. WRITE A SENTENCE!! II) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Read the problem described in EXAMPLE 3 on page 300 of your text. Let x = the number of snowboards sold Let y = the number of pairs of skis sold Because the store sells snowboards and pairs of skis, the possible solutions must be non-negative entire numbers. Because it is possible to meet the minimum daily sales objective by selling only snowboards or pairs of skis, the mathematical restrictions are whole numbers. Thus the restrictions to the domain and range are: x W, y W The problem indicates that the store earns a profit of $120 for each snowboard sold and $100 for each pair of skis sold. This means that the profit earned for snowboards is 120x and pairs of skis sold is 100y. Since the manager wants to earn at least $600 each day, the inequality sign is. This information is used to create this inequality.

2 FOM 11 T13 APPLICATIONS OF LINEAR INEQUALITIES 2 120x 120x +100y 120x y 120x y x ( 6) 100 y 6 5 x + 6 b = 6 m = 6 5 6: Answer the question(s) found in the original word problem. WRITE A SENTENCE!! The question in the problem is asking you to determine two combinations of snowboard and pair of skis sales that make sense. Thus, you can choose any two stippled points found in the shaded area and check them in the inequality created in step 4. The two points below are just two of many possible combinations that the store can sell to meet their daily minimum sales of $600. P 1 (6, 1) P 2 (2, ) x = 6 y = 1 x = 2 y = 120( 6) +100 ( 1 ) ( 2) +100 ( ) Ans: The store can sell 6 snowboards and 1 pair of skis or 2 snowboards and pairs of skis. 2) Read the problem described in INVESTIGATE the Math on page 294 of your text. Let n = the amount of nuts purchased Let r = the amount of raisins purchased The problem doesn t clearly state in what quantities the nuts and raisins are sold, the possible solutions can be non-negative entire numbers or decimal numbers. Thus the restrictions to the domain and range: n!, r! where n 0, r 0 NOTE: n 0, r 0 means n and r must be 0 or positive numbers, they cannot be negative.

3 FOM 11 T13 APPLICATIONS OF LINEAR INEQUALITIES 3 The problem indicates that Amir pays $ for each kilogram of nuts and $ for each kilogram of raisins. This means that Amis pays n for nuts r for raising. Since he wants to spend less than $200 to make the mixture, the inequality sign is <. This information is used to create this inequality. The inequality must be written in slope y- intercept form. You can choose to isolate either of the REMEMBER: The isolated variable represents the vertical axis. n + r r < r n < r n < r ( ) n < r + b = m = Because the slope of the boundary line contains relatively large numbers, it is advisable to draw the boundary line using the horizontal (r) and vertical (n) intercepts. The vertical, n-intercept is. We must calculate the horizontal intercept. Because the horizontal intercept is a specific point on the grid, the inequality sign < is changed to an = sign. REMEMBER: The horizontal intercept happens when the vertical variable equals 0, n = 0. n + r = 200 ( 0) + r = r = 200 r = 200 ( ) r - int =

4 FOM 11 T13 APPLICATIONS OF LINEAR INEQUALITIES 4 6: Answer the question(s) found in the original word problem. The question in the problem is asking you to determine several combinations of quantities of nuts and raisins that make sense. Thus, you can choose any two ordered pairs found in the shaded area and check their coordinates in the inequality created in step 4. The two points below are just two of many possible combinations that Amir can purchase. P 1 (2, 6) P 2 (10.5, 4) r = 2 n = 6 r = 10.5 n = 4 ( 6) + ( 3 ) < < 200 ( 4) + ( 10.5 ) < < < < 200 Ans: Amir can purchase 2 kg of raisins and 6 kg of nuts or 10.5 kg of raisins and 4 kg of nuts and will spend less than $200. III) REQUIRED PRACTICE 1: Complete these problems in the order listed. Page 304: Questions, 9, 10, 11, 12 & 7. SHOW THE PROCESS!! {Ans. Page }

5 FOM 11 T13 APPLICATIONS OF LINEAR INEQUALITIES 5 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T13 APPLICATIONS OF LINEAR INEQUALITIES Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! GRAPH EACH INEQUALITY ON ITS OWN GRID!!! Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Copy the sentences numbered 1-7 then match it with the correct term listed below. (2) Domain Coefficient Constant Term Range Natural numbers Whole numbers Integers Real numbers 1) The set of all possible x values. 3) The set of entire numbers. 2) The set of all possible y values. 4) The set of positive entire numbers. Answer these problems. ANSWER THESE QUESTIONS FOR PROBLEMS 5 & 6 a) Define the variables needed to describe the situation. (1 each question) b) List restrictions to the domain and range. (1 each inequality) c) Write an inequality to model the situation. (1 each inequality) d) Draw a graph to model the situation. (6 each question) e) Answer the question given in the problem. (2 each question) 5) Rob s Entertainment sells CDs for $10 and video games for $20. Cara can spend no more than $160. List two combinations of CDs and video games that Cara can purchase. (11) 6) The Vancouver Canucks need 90 points to make the playoffs. A win is worth 2 points and a tie is worth 1 point. List two combinations of wins and ties that will get the team into the playoffs. (11) Following the instructions. (1) /