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1 PLC Papers Created For: Year 11 Topic Practice Paper: Inequalities

2 Represent linear inequalities 1 Grade 6 Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. The graph shows the region that represents the inequalities < 3, <, and + > 12 by shading the unwanted regions. a) In the dataset listed below, circle the points that satisfy all three inequalities. { (4,8), (7,4), (5,6), (4,7), (5,5)} b) If the inequality < were to be changed to, what would the fully correct dataset be? (Total 3 marks)

3 Question 2. The dataset shown below lists the complete integer solution set to three inequalities. { (1,3), (1,4), (2,4) } Plot the points on the given axes and determine the three inequalities for which they are the complete integer solution set. (Total 4 marks)

4 Question 3. a) Represent the solution to the inequalities > + 2, + 5 and > 0.5 graphically on the grid below by shading the unwanted regions. Total /10 (Total 3 marks)

5 Represent linear inequalities 2 Grade 6 Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. Balpreet has correctly drawn the inequalities < 3, + 4, > 2 graphically and correctly shown the region that represents the solution by shading the unwanted regions. a) She then writes down the integer dataset of the solution but makes some mistakes, circle the incorrect answers in Balpreet s dataset. { (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,2), (4,3), (5,3) } b) Write down the inequalities that would make Balpreet s dataset the correct and complete solution. (Total 3 marks)

6 Question 2. The dataset shown below lists the complete integer solution set to three inequalities. { (2,1), (3,1), (3,2) } Plot the points on the given axes and determine the three inequalities for which they are the complete integer solution set. (Total 4 marks)

7 Question 3. Represent the solution to the inequalities + 3, < 2, 3.5 graphically on the grid below by shading the unwanted regions. (A3) (Total 3 marks) Total /10

8 Represent linear inequalities 3 Grade 6 Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. Teenagers should consume more than 1200 milligrams of calcium each day, a portion of cheese contains 150 milligrams of calcium whilst a portion of yoghurt contains 250 milligrams of calcium. It is recommended that you consume less than 10 portions of cheese and less than 5 portions of milk each day. By letting x represent the number of cheese portions consumed per day and y represent the number of milk portions consumed per day. a) Write down 3 inequalities that represent this situation b) Represent the solution to the three inequalities graphically on the grid below by shading the unwanted regions.

9 c) Explain why there are solution regions that do cannot be used? (Total 5 marks) Question 2. Gurmeet is planning to grow basil and parsley plants in a small herb garden but is restricted in the amount of water he can use each day and by the amount of space he has available. A basil plant requires 275ml of water whilst a parsley plant requires 550ml of water and Gurmeet must use less than 5.5l of water. There is also only enough space for less than 10 of each plant. By letting x represent the number of basil plants and y represent the number of parsley plants. a) Write down 3 inequalities that model Gurmeet s problem

10 b) Represent the solution to the three inequalities graphically on the grid below by shading the unwanted regions. c) How many possible solution combinations does Gurmeet have to choose from? Total /10 (Total 5 marks)

11 Represent linear inequalities 4 Grade 6 Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. You receive a gift certificate to the local sports centre for 25. At the sports centre it costs 4.50 to use the swimming baths and 7.50 to play squash and you definitely like to try squash as you have never played before. By letting x represent the number of times you go swimming and y represent the number of times you play squash. a) Write down two inequalities that represent this situation b) Represent the solution to the inequality graphically on the grid below by shading the unwanted regions.

12 c) Using set notation write down the possible alternatives of activities you can undertake. (Total 5 marks) Question 2. In the first 3 games of a rugby season a team scored at most 63 points. In rugby points are scored from converted tries scoring 7 points and penalties scoring 3 points. By letting x represent the number of converted tries scored and y represent the number of penalties scored. a) Write down an inequality that models this situation b) Represent the solution to your inequality graphically on the grid below.

13 c) In the third game the team scored exactly 63 points Write in set notation the possible converted try and penalty scoring combinations that may have occurred. Total /10 (Total 5 marks)

14 Represent quadratic inequalities 1 Grade 7 Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. a) Solve x Represent your solution on a number line. b) Write the integer answers for part a) in set notation. Question 2. (Total 2 marks) Solve Display your answer on a sketch of the graph of the solution (Total 2 marks)

15 Question 3. For which values of x is the expression greater than the expression ? Represent the possible values of on a number line. Question 4. Find the set(s) of all values for which < x + 5 Display your answer on a sketch of the graph of the solution (Total 3 marks) (Total 3 marks) TOTAL /10

16 Represent quadratic inequalities 2 Grade 7 Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. a) Solve x + 24 > 0 Represent your solution on a number line. b) Write the integer answers for part a) in set notation. (Total 2 marks) Question 2. Solve Display your answer on a sketch of the graph of the solution (Total 2 marks)

17 Question 3. For which values of x is the expression less than or equal to the expression ? Represent the possible values of on a number line. Question 4. Find the set(s) of all values for which > 7 Display your answer on a sketch of the graph of the solution (Total 3 marks) (Total 3 marks) TOTAL /10

18 Represent quadratic inequalities 3 Grade 7 Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. A grassed area is being designed in the shape of a right-angled triangle with two sides of length and ( 7) respectively. If the longest side must be no more than 13 metres, how long can the shortest side be? Show your answer on a number line. Question 2. (Total 4 marks) A woman is 4 years younger than her partner and the product of their ages is greater than 480. If the man is years old then set up and solve an inequality that represents this situation. Display your answer on a sketch of a graph of the solution.

19 (Total 3 marks) Question 3. The diagram below is a planner s representation of a company s new car park, if the total area cannot be more than 60 2 what are the possible integer values of. Give your answer in set notation. Total /10 (Total 3 marks)

20 Represent quadratic inequalities 4 Grade 7 Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. A rectangle has sides (3 4) m and ( 3) m. If its area is less than 68 m 2 and its perimeter is at least 22 m, find the possible values for. Represent the possible values of on a number line. (Total 5 marks)

21 Question 2. A builder has been asked to lay a patio 2 m longer than its width. Each slab is a square of side 0.5 m and costs 3. The builder has been asked to spend between 180 and 288. Find the 2 possible integer lengths of the patio, write your answer in set notation. (Total 5 marks) Total /10

22 PLC Papers Created For: Year 11 Topic Practice Paper: Inequalities

23 Represent linear inequalities 1 Grade 6 SOLUTIONS Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. The graph shows the region that represents the inequalities < 3, <, and + > 12 by shading the unwanted regions. a) In the dataset listed below, circle the points that satisfy all three inequalities. { (4,8), (7,4), (5,6), (4,7), (5,5)} (A1) b) If the inequality < were to be changed to, what would the fully correct dataset be? {(4,4), (4,5), (4,6), (4,7), (5,5), (5,6)} (A2) (Total 3 marks)

24 Question 2. The dataset shown below lists the complete integer solution set to three inequalities. { (1,3), (1,4), (2,4) } Plot the points on the given axes and determine the three inequalities for which they are the complete integer solution set. Plotting (1,3), (1,4) and (2,4) correctly (M1) (A1) (A1) (A1) (Total 4 marks)

25 Question 3. a) Represent the solution to the inequalities > + 2, + 5 and > 0.5 graphically on the grid below by shading the unwanted regions. (A3) (Total 3 marks) Total /10

26 Represent linear inequalities 2 Grade 6 SOLUTIONS Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. Balpreet has correctly drawn the inequalities < 3, + 4, > 2 graphically and correctly shown the region that represents the solution by shading the unwanted regions. a) She then writes down the integer dataset of the solution but makes some mistakes, circle the incorrect answers in Balpreet s dataset. { (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,2), (4,3), (5,3) } (A2) b) Write down the inequalities that would make Balpreet s dataset the correct and complete solution. 3, + 4, 2 (A1) (Total 3 marks)

27 Question 2. The dataset shown below lists the complete integer solution set to three inequalities. { (2,1), (3,1), (3,2) } Plot the points on the given axes and determine the three inequalities for which they are the complete integer solution set. Plotting (2,1), (3,1) and (3,2) all correctly (M1) (A1) (A1) (A1) (Total 4 marks)

28 Question 3. Represent the solution to the inequalities + 3, < 2, 3.5 graphically on the grid below by shading the unwanted regions. (A3) (Total 3 marks) Total /10

29 Represent linear inequalities 3 Grade 6 SOLUTIONS Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. Teenagers should consume more than 1200 milligrams of calcium each day, a portion of cheese contains 150 milligrams of calcium whilst a portion of yoghurt contains 250 milligrams of calcium. It is recommended that you consume less than 10 portions of cheese and less than 5 portions of milk each day. By letting x represent the number of cheese portions consumed per day and y represent the number of milk portions consumed per day. a) Write down 3 inequalities that represent this situation >1200 (simplifies to 3x + 5y > 24) <10 <5 (A1) b) Represent the solution to the three inequalities graphically on the grid below by shading the unwanted regions. (A3)

30 c) Explain why there are solution regions that do cannot be used? Cannot have negative portions of cheese or milk (B1) (Total 5 marks) Question 2. Gurmeet is planning to grow basil and parsley plants in a small herb garden but is restricted in the amount of water he can use each day and by the amount of space he has available. A basil plant requires 275ml of water whilst a parsley plant requires 550ml of water and Gurmeet must use less than 5.5l of water. There is also only enough space for less than 10 of each plant. By letting x represent the number of basil plants and y represent the number of parsley plants. a) Write down 3 inequalities that model Gurmeet s problem <5500 (simplifies to x + 2y < 20) <10 <10 (A1)

31 b) Represent the solution to the three inequalities graphically on the grid below by shading the unwanted regions. c) How many possible solution combinations does Gurmeet have to choose from? (A3) 16 (A1) (Total 5 marks) Total /10

32 Represent linear inequalities 4 Grade 6 SOLUTIONS Objective: Represent the solution of a linear inequality in two variables on a number line, using set notation and on a graph Question 1. You receive a gift certificate to the local sports centre for 25. At the sports centre it costs 4.50 to use the swimming baths and 7.50 to play squash and you definitely like to try squash as you have never played before. By letting x represent the number of times you go swimming and y represent the number of times you play squash. a) Write down two inequalities that represent this situation (A1) b) Represent the solution to the inequality graphically on the grid below by shading the unwanted regions. (A2)

33 c) Using set notation write down the possible alternatives of activities you can undertake. { (0,1), (0,2), (0,3), (1,1), (1,2), (2,2), (3,1)} (A2) (Total 5 marks) Question 2. In the first 3 games of a rugby season a team scored at most 63 points. In rugby points are scored from converted tries scoring 7 points and penalties scoring 3 points. By letting x represent the number of converted tries scored and y represent the number of penalties scored. a) Write down an inequality that models this situation (A1) b) Represent the solution to your inequality graphically on the grid below. (A2)

34 c) In the third game the team scored exactly 63 points. Write in set notation the possible converted try and penalty scoring combinations that may have occurred. { (0,21), (3,14), (6,7) (9,0) } (A2) Total /10 (Total 5 marks)

35 Represent quadratic inequalities 1 Grade 7 Solutions Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. a) Solve x Represent your solution on a number line. ( + 9)( + 4) b) Write the integer answers for part a) in set notation. { -9, -8, -7, -6, -5, -4 } (A1) (A1) Question 2. (Total 2 marks) Solve Display your answer on a sketch of the graph of the solution ( 6)( + 3) 0 (M1) (A1) (Total 2 marks)

36 Question 3. For which values of x is the expression greater than the expression ? Represent the possible values of on a number line > > 0 (M1) (2 + 5)(2 3) > 0 (M1) < 5 2, > 3 2 (A1) Question 4. (Total 3 marks) Find the set(s) of all values for which < x + 5 Display your answer on a sketch of the graph of the solution 2 14 < ( + 5)(2 ) 2 14 < (M1) < 0 ( + 8)( 3) < 0 (M1) (A1) (Total 3 marks) TOTAL /10

37 Represent quadratic inequalities 2 Grade 7 Solutions Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. a) Solve x + 24 < 0 Represent your solution on a number line. ( + 8)( + 3) < 0 8 < < 3 b) Write the integer answers for part a) in set notation. { -7, -6, -5, -4 } (A1) (A1) (Total 2 marks) Question 2. Solve Display your answer on a sketch of the graph of the solution ( 7)( + 3) (Total 2 marks)

38 Question 3. For which values of x is the expression less than or equal to the expression ? Represent the possible values of on a number line (M1) (3 + 5)( 2) 0 (M1) (A1) (Total 3 marks) Question 4. Find the set(s) of all values for which > 7 Display your answer on a sketch of the graph of the solution > (7 )( + 3) > (M1) > 0 ( + 9)( + 2) > 0 (M1) ( 7)( + 3) (Total 3 marks)

39 TOTAL /10

40 Represent quadratic inequalities 3 Grade 7 SOLUTIONS Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. A grassed area is being designed in the shape of a right-angled triangle with two sides of length and ( 7) respectively. If the longest side must be no more than 13 metres, how long can the shortest side be? Show your answer on a number line. 2 + ( 7) 2 13 (M1) (oe. M1) ( 12)( + 5) Shortest side: ( 7) > 0 Combining constraints gives 0 < ( 7) 5 (M1) (A1) (Total 4 marks) Question 2. A woman is 4 years younger than her partner and the product of their ages is greater than 480. If the man is years old then set up and solve an inequality that represents this situation. Display your answer on a sketch of a graph of the solution. ( 4) > 480 (M1) 2 4 > > 0

41 ( 24)( + 20) > 0 (M1) > 24 (A1) (Total 3 marks) Question 3. The diagram below is a planner s representation of a company s new car park, if the total area cannot be more than 60 2 what are the possible integer values of. Give your answer in set notation. 6 + (4 + 2) 60 (M1) ( + 5)( 3) (M1) h 0 < 3 {1,2,3} (A1) (Total 3 marks) Total /10

42 Represent quadratic inequalities 4 Grade 7 SOLUTIONS Objective: Represent the solution to a quadratic inequality on a number line, using set notation and on a graph Question 1. A rectangle has sides (3 4) m and ( 3) m. If its area is less than 68 m 2 and its perimeter is at least 22 m, find the possible values for. Represent the possible values of on a number line. Perimeter: 2(3 4) + 2( 3) (M1) (M1) Area: (3 4)( 3) < < < 0 (M1) (3 + 8)( 7) < < < 7 (M1) Satisfying both gives: < 7 (A1) (Total 5 marks)

43 Question 2. A builder has been asked to lay a patio 2 m longer than its width. Each slab is a square of side 0.5 m and costs 3. The builder has been asked to spend between 180 and 288. Find the 2 possible integer lengths of the patio, write your answer in set notation. Area of patio: ( + 2) = ( ) 2 Area of 1 slab: = 0.25 = No. of slabs: = 4( ) Cost of slabs: 3 4( ) = 12( ) Cost constraint: ( 2 + 2) 288 (M1) Min cost: 12( ) ( 3)( + 5) 0 (M1) 3 3 Max cost: 12( ) ( + 6)( 4) 0 (M1) 6 4 Possible widths: 3 4 so = 3 4 (A1) Possible lengths: 5m or 6m Set notation: {5,6} (A1) (Total 5 marks) Total /10