Systems and inequalites review

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1 Name: Class: Date: Systems and inequalites review Multiple Choice Identify the choice that best completes the statement or answers the question, 1. The approximate solutions to the system of equations shown below are A (-0.2,-2.2) and (2.6,2.6) C (2.2,0.2) and (-2.6,-2.6) B (-2.2,-0.2) and (2.6,2.6) D (0.2,2.2) and (-2.6,-2.6) 1

2 Name: ED: A 2

3 Name: ID: A 3. How many solutions are there to the system of equations graphed below? A one solution C two solutions B three solutions D no real solution 3

4 1 The approximate solutions to the system of equations shown below are y o n - / [ 8-7 -fc I \ i * _. i,,, _3L_ I / :.7 i / / / \ i ( ' j 4 X 1 J- - 1 A (-2.5,0.7) and (-2.1,0.5) B (-0.7,2.5) and (-0.5,2.1) C (0.7,-2.5) and (0.5,-2.1) D (2.5,-0.7) and (2.1,-0.5)

5 Name: IB: A 5. The approximate solution to the system of equations shown below is A (-2.3,2.3) C (2.3,-6.3) B (-2.3,-6.3) D (2.3,6.3) 5

6 Name: ID: A 6. What system of equations is represented by the following graph? A y = -\Ax 2 + 3x + 2 C y=l.4x 2-3x + 2 y = 0.3x x + 3 y = -0.3JC 2-2.9x + 3 B y = -\Ax 2-3x + 2 D y =-0.3x 2-3x+ 2 y = 0.3x 2-2.9x + 3 y = I Ax 2-2.9x + 3 6

7 Name: 7. What system of equations is represented by the following graph? A y = 1.2*+ 3 y = -1.9x x + 3 B y = -l.9x-3 C y = -1.2x + 3 y= l.9x x + 3 D y = -l.2x + 3 y = l.2x x4-3 7= 1.9x 2-4.2x How many times does a line tangent to a parabola cross the parabola? A twice C once B three times D none of these 9. The line y = 9x - 4 intersects the quadratic function y = x 2 + 7x - 3 at one point. What are the coordinates of the point of intersection? A (0,0) C (-1,5) B (1,-5) D (1,5) 10. Find the coordinates of the point(s) of intersection of the line y = 4x + 8 and the quadratic function y = -4x 2-5x + 8. A (0, 8)and(-, 17) C (2, -34) B (0,0) D (--,-1) and (0,8) 7

8 ID: A 11. Solve the following system: y = -6x + 9 y = -Sx 2-9x C (-,^)and (0,9) B (0,y)and(, ) D (0,0) 12. Determine the y-intercept of a line with a slope of -2 that is tangent to the curve y = x 2-4x - 3. A 3 B 6 C -6 D The cross-section of a tunnel is in the shape of a parabola. The parabolic shape of the tunnel is given by the function y x 2 + 6x. What is the width of the tunnel, to the nearest hundredth of a metre, at a height of m? Diagram not to scale. A B m m C D m m 14. Solve the following system of equations: y = 4x y = 2x 2 A (0,0) and (2,-8) B (2,4) and (-2,8) C (2, 2) and (-2,-8) D (0, 0)and (2,8) 15. What are the coordinates of the point(s) of intersection of the line y = -Ix - 5 and the quadratic function y = - X 2 _ 15 X + 4? A (9,58) and (-1,-12) C (9,-58)and(l,12) B (-9,58), and (1,-12) D (9,-58) and (1,-12) 16. What are the solutions for the following system of equations? y = Sx + 7 y = -x 2-5x + 7 A (13,97) and (0,7) C (13,-97) and (0,7) B (-13,-97) and (0,7) D (13,97) and (0,-7) 8

9 Name: ID: A 17. What are the coordinates of the point(s) of intersection of the quadratic functions y = -2x 2-4x + 5 and y = 2x 2 + 4x + 5? A (-2,5) and (0,5) C (2,5) and (0,5) B (2,-5) and (0,-5) D (2,-5) and (0,5) 18. Solve the systemy = -^x 2 +2x-9 and y = ^x 2-6x + 9. Express your answers as exact values. A (-2, 4) and (18,-36) B 4"i)and(-^,^) C (2, -4) and (-18, 36) D (4'"i ) a n d ( "S'-i ) 19. The graph of -5JC - 6y < 6 is 9

10 20. The graph of -Ax + 7y > 1 is 10

11 Name: ID: A 22. Which number line represents the solution set to the inequality ~2x 2-7.9x > 3? A B i I 111 I I I h-» ( I I I 1 h -5^ D < I KB I I I OH h-» ( I I OH I I ' -5^ ' C 23. Which graph represents the solution to the inequality 2x 2-6x + 4 > 0? A B C D < i i i i i i i i i > i i i i i i i c o i i i >

12 Name: ID: A 24. The solution set to the inequality -2x 2 + Sx - 6 > 0 is A jx 1 <x < 3, x G R C jx x < 1, x> 3, x G i? B jx -3 <x <-l,x e i?} D jx x < -3,x > -l,x e R.J 25. The solution set to the inequality -3x 2 < -9x + 6 is A jx 1 <x<2,x G i?} C jx x<-2orx>-l,x e/?} B {x -2<x <-l,x G i?} D jx x < 1 orx > 2, x e i? J 26. Which graph represents the solution to the inequality y < -5(x + 3) +4? B D 12

13 Name: ID: A 27. The solution to the inequality y < -7(x + 4) + 3 is 13

14 28. Which quadratic inequality is represented by the graph shown below? A V>-3(JC + 2) -7 B J>3(JC + 2) 2-7 C y>-3(x-7) -2 D y<3(x-7)

15 Name: ID: A 29. Which point does not satisfy the inequality y > -2(x-3) + 8? A (-9,-234) C (5,16) B (1,1) D (2,0) Completion Complete each statement. 1. A linear-quadratic system that has one point of intersection has 2. The solution(s) to the system of equations y=x 2-4 and y = 2x - 4 is (are) 3. The solution(s) to the system of equations y = x 2 + \2x + 38 and y = -x 2-12x - 34 is (are) 4. The system of equations y = -5(x + 4) 2-4 and y = Sx x +124 has solution(s). 15

16 Name: ED: A 5. The most convenient test point to use to determine if the points in a region defined by y < I Ax satisfy the inequality is 16

17 Name: ID: A Matching Match each system of equations to the corresponding graphical representation below. 1. y = -l.5x x4-3 j; = )

18 Name: 2. y = L3x + 3 y=\.5(x + 2.3) y= 1.5x x4-3 j/ = -1.5(x-2.3) y = -1.3x + 3 v= 1.5(x42.3) j = -1.5x x4-3 v= 1.5(x-2.3) Short Answer 1. Solve the system graphically. y = 2x-4

19 Name; ID: A 2. Determine the coordinates of the point(s) of intersection of each linear-quadratic system algebraically. Identify whether you used substitution or elimination in your solution. a) y = x 2 - Ix +15 and y = 2x - 5 1, b) j = x -2x-3 andy = -2x + 1 x y 1 3. Graph the inequality - > y r~ ~ " ~- ~" 1 - I _i i i _a. > - A What is the solution for 2x 2 - Ix > -3? 5. Graph the inequality -x 2 < 24 - lox and state the solution set. 19

20 Name: ID: A a) Sketch the graph of the quadratic inequality y <--z (x-3) -1. b) Check your answer using a test point not in the solution region you graphed j i i i I * t t -\ t L 1 -f- - t t * < ; J -P f? -6 j5-4 -f - 2 -jl I r i ! 3 4 I! I + I I -t - j- t r r- 6- H 4-f ] ; ;. L 20

21 ID: A Systems and inequalites review Answer Section MULTIPLE CHOICE 1. ANS NAT KEY 2. ANS NAT KEY 3. ANS NAT KEY 4. ANS NAT KEY 5. ANS NAT KEY 6. ANS NAT KEY 7. ANS NAT KEY 8. ANS NAT KEY 9. ANS NAT KEY 10. ANS NAT KEY 11. ANS NAT KEY 12. ANS NAT KEY 13. ANS NAT KEY 14. ANS NAT KEY D PTS: 1 DIF: Easy OBJ: Section 8.1 RF6 TOP: Solving Systems of Equations Graphically linear-quadratic systems interpreting graphs C PTS: 1 DIF: Easy OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically linear-quadratic systems interpreting graphs tangent line C PTS: 1 DIF: Easy OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically linear-quadratic systems interpreting graphs number of solutions C PTS: 1 DIF: Average OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically quadratic-quadratic systems interpreting graphs D PTS: 1 DIF: Easy OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically quadratic-quadrat ic systems interpreting graphs C PTS: 1 DIF: Difficult OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically quadratic-quadratic systems interpreting graphs A PTS: 1 DIF: Average OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically linear-quadratic systems j interpreting graphs C PTS: 1 DIF: Easy OBJ: Section 8.1 RF 6 TOP: Solving Systems of Equations Graphically linear-quadratic systems tangent line D PTS: 1 DIF: Easy OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically linear-quadratic systems algebraic solution D PTS: 1 DIF: Difficult OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically linear-quadratic systems points of intersection algebraic solution C PTS: 1 DIF: Difficult OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically linear-quadratic systems points of intersection algebraic solution C PTS: 1 DIF: Difficult OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically tangent line quadratic function number of solutions C PTS: 1 DIF: Average OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically linear-quadratic systems algebraic solution D PTS: 1 DIF: Easy OBJ: Section 8.2 RF 6 TOP: Solving Systems of Equations Algebraically linear-quadratic systems algebraic solution 1

22 ID: A 15. ANS: B PTS: 1 DIF: Average OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: linear-quadratic systems algebraic solution 16. ANS: B PTS: 1 DIF: Average OBJ: Section 8.2 NAT: RF6 TOP: Solving Systems of Equations Algebraically KEY: linear-quadratic systems algebraic solution 17. ANS: A PTS: 1 DIF: Average OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: linear-quadratic systems algebraic solution 18. ANS: C PTS: 1 DIF: Difficult OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: quadratic-quadratic systems j algebraic solution exact values 19. ANS: C PTS: 1 DIF: Easy OBJ: Section 9.1 NAT: RF 7 TOP: Linear Inequalities in Two Variables KEY: linear inequality graphing 20. ANS: D PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF 7 TOP: Linear Inequalities in Two Variables KEY: linear inequality graphing 21. ANS: D PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF 7 TOP: Linear Inequalities in Two Variables KEY: linear inequality determine equation 22. ANS: B PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality one variable 23. ANS: A PTS: 1 DIF: Easy OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality j one variable 24. ANS: A PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality one variable solution set 25. ANS: D PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality one variable solution set 26. ANS: D PTS: 1 DIF: Easy OBJ: Section 9.3 NAT: RF 7 TOP: Quadratic Inequalities in Two Variables KEY: quadratic inequality two variables graphing a < ANS: B PTS: 1 DIF: Easy OBJ: Section 9.3 NAT: RF 7 TOP: Quadratic Inequalities in Two Variables KEY: quadratic inequality two variables graphing a < ANS: B PTS: 1 DIF: Average OBJ: Section 9.3 NAT: RF 7 TOP: Quadratic Inequalities in Two Variables KEY: quadratic inequality j two variables determine equation 29. ANS: D PTS: 1 DIF: Average OBJ: Section 9.3 NAT: RF 7 TOP: Quadratic Inequalities in Two Variables KEY: quadratic inequality two variables test point 2

23 ID: A COMPLETION 1. ANS: one solution PTS: 1 DIF: Easy OBJ: Section 8.1 NAT: RF 6 TOP: Solving Systems of Equations Graphically KEY: number of solutions 2. ANS: (2,0) and (0,-4) PTS: 1 DIF: Easy OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: linear-quadratic systems algebraic solution 3. ANS: (-6,2) PTS: 1 DIF: Easy OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: quadratic-quadratic systems algebraic solution 4. ANS: one PTS: 1 DIF: Average OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: quadratic-quadratic systems number of solutions 5. ANS: (0, 0) NB: Other answers will work as long as they satisfy the given inequality see graph below. PTS: 1 DIF: Easy OBJ: Section 9.1 NAT: RF 7 TOP: Linear Inequalities in Two Variables KEY: test point MATCHING 1. ANS: B PTS: 1 DIF: Average OBJ: Section 8.1 NAT: RF6 TOP: Solving Systems of Equations Graphically KEY: linear-quadratic systems graphical solution 3

24 IB: A 2. ANS: A PTS: 1 DIF: Average OBJ: Section 8.1 NAT: RF 6 TOP: Solving Systems of Equations Graphically KEY: linear-quadratic systems graphical solution 3. ANS: E PTS: 1 DIF: Average OBJ: Section 8.1 NAT: RF 6 TOP: Solving Systems of Equations Graphically KEY: quadratic-quadratic systems graphical solution 4. ANS: C PTS: 1 DBF: Average OBJ: Section 8.1 NAT: RF 6 TOP: Solving Systems of Equations Graphically KEY: quadratic-quadratic systems graphical solution 5. ANS: D PTS: 1 DIF: Average OBJ: Section 8.1 NAT: RF6 TOP: Solving Systems of Equations Graphically KEY: quadratic-quadratic systems graphical solution SHORT ANSWER PTS: 1 DIF: Easy OBJ: Section 8.1 NAT: RF 6 TOP: Solving Systems of Equations Graphically KEY: linear-quadratic systems interpreting graphs graphical solution 4

25 ID: A 2. ANS: Solution methods may vary. Examples: a) Substitution: y=x 2 -lx + \5 Substitute y - 2x - 5: 2r-5=x 2-7x+15 x 2-9x + 20 = 0 Solve for x by factoring: x 2-9x + 20 = (x-4)(x-5) x = 4 or x = 5 Substitute x = 4 and x = 5 into y - 2x - 5 and solve for y. v = 2(4)-5 y = 2(5)-5 = 3 =5 The points of intersection for the system are (4, 3) and (5, 5). b) Elimination: Subtract the second equation from the first: y = 4 y = -2x = jx \x 2-2x-3 Y =4 4 Solve for x: 1 2 A x l = 4 4 x 2 = 16 x = ±4 Substitute x = ±4 into y = -2x +1 and solve for v. 7 = -2(4) + 1 j = -2(-4) +1 = -1 =9 The points of intersection for the system are (4, -7) and (-4, 9). PTS: 1 DBF: Average OBJ: Section 8.2 NAT: RF 6 TOP: Solving Systems of Equations Algebraically KEY: linear-quadratic systems algebraic solution substitution elimination 5

26 ID: A 3. ANS: Rearrange the inequality to make it easier to graph. PTS: 1 DIF: Difficult OBJ: Section 9.1 NAT: RF 7 TOP: Linear Inequalities in Two Variables KEY: linear inequality graphing two variables 6

27 ID: A 4. ANS: First, rewrite the inequality as 2x 2 - Ix + 3 > 0. Next, factor the quadratic: 2x 2-7x + 3 = (2x- l)(x-3) 1 ~ x = orx = 3 i H H » ^9 _g Choose a test point in each interval, such as 0, 1, and 4: L.S. = 2(0) 2-7(0)+3 = 3 L.S. > R.S. R S - = 0 L.S. = 2(1) 2-7(1) +3 R-S. =0 = -2 L.S. < R.S. L.S. = 2(4) 2-7(4)+ 3 R-S. =0 = 7 L.S. ^ R.S. Therefore, the solution is jx x < - orx > 3, x it!j. < i i i i l i i i i i i i ) PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality one variable solution set 7

28 ID: A 5. ANS: First, rewrite the inequality as -x 2 + lox - 24 < 0. Next, factor the quadratic: ~x 2 + lox-24 = -(x 2 - lox + 24] = -(x-4)(x-6) x = 4 orx = 6 < I 1 ( O I O > 9 g Choose a test point in each interval, such as 0, 5, and 7: L.S. = -(0) (0) -24 R S - = 0 = -24 L.S. < R.S. L.S. =-(5) (5)-24 R-S.=0 = 1 L.S. > R.S. L.S. = -(7) (7) - 24 R S - = 0 = -3 L.S. < R.S. ( i i i i I I I I I I i i i o i o I I I o Therefore, the solution is jx x < 4 orx > 6, x e R\. PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF 7 TOP: Quadratic Inequalities in One Variable KEY: quadratic inequality one variable graphing solution set 8

29 ANS: a) 9- ft o 7 o c J 4 X. 1? \ "l. A :., 1 - J * *... m e. b) Test point used will vary. Example: Use the test point (0, 0). L.S.=0 R.S. = - (0-3) 2 -l i / «<\ E f Raj li = -6-1 = -7 L.S. > R.S. Since the test point is not in the shaded region, the graphical solution is correct. PTS: 1 DIF: Average OBJ: Section 9.3 NAT: RF 7 TOP: Quadratic Inequalities in Two Variables KEY: quadratic inequality two variables graphing

30 Systems and inequalites review [Answer Strip] C 2. C 3. D 1.

31 Systems and inequalites review [Answer Strip] ID: A C 6. A 7. C 11. A 17. D 20. C 18. C 12. C 13. C 19. C 8. D 9. D 14. D 10. B 15. B 16.

32 Systems and inequalites review [Answer Strip] ID: A _JL_21. A 24. B 27. B 28. j> 29. D 25. D 26. B 22. A 23.

33 Systems and inequalites review [Answer Strip] A 2. E 3. C 4. D 5. B 1.