Date:
Class:
Name:
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
B
(-0.2,-2.2) and (2.6,2.6)
(-2.2,-0.2) and (2.6,2.6)
C
D
1
(2.2,0.2) and (-2.6,-2.6)
(0.2,2.2) and (-2.6,-2.6)
Name:
ED: A
2
ID: A
Name:
3. How many solutions are there to the system of equations graphed below?
A
B
one solution
three solutions
C
D
3
two solutions
no real solution
The approximate solutions to the system of equations shown below are
y-
9. o
n/
I
/
... .1 .
:
.7
i
/
/
[
/
8
-7
-fc
-5
4
3
-2
-
I \
\
i
'
(
j
i—
1
*
_.
i
__,
,
, _3L_
J-
A
B
(-2.5,0.7) and (-2.1,0.5)
(-0.7,2.5) and (-0.5,2.1)
C
D
4
X
1
- 1 —
(0.7,-2.5) and (0.5,-2.1)
(2.5,-0.7) and (2.1,-0.5)
Name:
IB: A
5. The approximate solution to the system of equations shown below is
A
B
(-2.3,2.3)
(-2.3,-6.3)
C
D
5
(2.3,-6.3)
(2.3,6.3)
Name:
ID: A
6. What system of equations is represented by the following graph?
A
2
y = -\Ax
+ 3x + 2
C
2
2
y = -\Ax -3x
2
y = 0.3x -2.9x
+2
+2
2
y = 0.3x +2.9x + 3
B
2
y=l.4x -3x
y = -0.3JC -2.9x + 3
D
2
y =-0.3x
2
+3
-3x+ 2
y = I Ax - 2.9x + 3
6
Name:
7. What system of equations is represented by the following graph?
A
y = 1.2*+ 3
B
y = -1.9x +4.2x + 3
y = -l.9x-3
C
y = -1.2x + 3
D
y= l.9x +4.2x + 3
y = -l.2x + 3
2
2
2
y = l.2x +4.2x4-3
2
7 = 1.9x -4.2x + 3
8. How many times does a line tangent to a parabola cross the parabola?
A twice
C once
B three times
D none of these
2
9. The line y = 9x - 4 intersects the quadratic function y = x + 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
= -4x -5x + 8.
2
y
A
(0, 8)and(-, 17)
C
(2, -34)
B
(0,0)
D
( - - , - 1 ) and (0,8)
7
ID: A
11. Solve the following system:
y = -6x + 9
y = -Sx -9x + 9
3 27
2
B
27
3 9
(0,y)and(|,|)
C
( - | , ^ ) a n d (0,9)
D
(0,0)
2
12. Determine the y-intercept of a line with a slope of -2 that is tangent to the curve y = —x - 4x - 3.
A
3
B
6
C
-6
D
-3
13. 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 —
2
x + 6x. What is the width of the tunnel, to the nearest hundredth of a metre, at a height of
47.25 m?
Diagram not to scale.
A
B
63.00 m
47.25 m
14. Solve the following system of equations:
y = 4x
y = 2x
A (0,0) and (2,-8)
B (2,4) and (-2,8)
C
D
21.00 m
31.50 m
C
D
(2, 2) and (-2,-8)
(0, 0)and (2,8)
2
15. What are the coordinates of the point(s) of intersection of the line y = -Ix - 5 and the quadratic function
y = - 2 _ 15 + 4?
X
X
A
B
(9,58) and (-1,-12)
(-9,58), and (1,-12)
C (9,-58)and(l,12)
D (9,-58) and (1,-12)
16. What are the solutions for the following system of equations?
y = Sx + 7
= -x - 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)
2
y
8
ID: A
Name:
2
17. What are the coordinates of the point(s) of intersection of the quadratic functions y = -2x - 4x + 5 and
y = 2x + 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)
2
2
18. Solve the systemy = -^x +2x-9
A
(-2, 4) and (18,-36)
B
4"i)and(-^,^)
C
(2, -4) and (-18, 36)
D
(
4'"i
) a n d (
"S'-i
and y = ^ x
2
)
19. The graph of -5JC - 6y < 6 is
9
-6x + 9. Express your answers as exact values.
20. The graph of -Ax + 7y > 1 is
10
ID: A
Name:
2
22. Which number line represents the solution set to the inequality ~2x - 7.9x > 3?
A
C
i—I
111
I
I • I
1
1
1
1
h-»
(
- 5 ^ - 3 -2 -1 0 1 2 3 4 5
B
I
I •
I
1
h
-5 -4 -3 -2 -1 0 1 2 3 4 5
D
<—I
KB I
I
I
OH
1
1
1
1
h-»
(
- 5 ^ - 3 -2 -1 0 1 2 3 4 5
I
I OH
1
1-0 I
I
|
|
|
| '
-5 -4 -3 -2 -1 0 1 2 3 4 5'
2
23. Which graph represents the solution to the inequality 2x - 6x + 4 > 0?
A
C
-5 -4 -3 -2 -1 0 1 2 3 4 5
B
-5 -4 -3 -2 -1 0 1 2 3 4 5
D
<—i—i—i—i—i—i—•—i—i—i—>
-5 -4 -3 -2 -1 0 1 2 3 4 5
i—i—i—i—i—i—i—c—o—i—i—i—>
-5 -4 -3 -2 -1 0 1 2 3 4 5
11
ID: A
Name:
2
24. The solution set to the inequality -2x + Sx - 6 > 0 is
A
j x | 1 <x < 3, x
B
j x | - 3 <x < - l , x e i?}
G
C
R|
D
jx| x < 1, x> 3, x
G
i?|
jx| x < -3,x > -l,x e R.J
2
25. The solution set to the inequality -3x < -9x + 6 is
A
jx| 1 < x < 2 , x G i?}
C
j x | x < - 2 o r x > - 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
ID: A
Name:
27. The solution to the inequality y < -7(x + 4) + 3 is
13
28. Which quadratic inequality is represented by the graph shown below?
A
B
V>-3(JC + 2) - 7
2
J>3(JC + 2 ) - 7
C
D
14
y>-3(x-7)
-2
2
y<3(x-7) -2
ID: A
Name:
29. Which point does not satisfy the inequality y > -2(x-3)
A
B
(-9,-234)
(1,1)
C
D
+ 8?
(5,16)
(2,0)
Completion
Complete each statement.
1. A linear-quadratic system that has one point of intersection has
2
2. The solution(s) to the system of equations y=x
-4 and y = 2x - 4 is (are)
2
2
3. The solution(s) to the system of equations y = x + \2x + 38 and y = -x - 12x - 34 is (are)
2
2
4. The system of equations y = -5(x + 4 ) - 4 and y = Sx + 64x +124 has
solution(s).
15
ED: A
Name:
5. The most convenient test point to use to determine if the points in a region defined by y < I Ax + 0.6 satisfy
the inequality is
•
16
ID: A
Name:
Matching
Match each system of equations to the corresponding graphical representation below.
2
1. y = -l.5x
+ 1.5x4-3
2
j ; = 1.50 + 2.3) +2.5
17
Name:
2. y = L3x + 3
2
y=\.5(x
+ 2.3) +2.5
2
3. y= 1.5x 4-1.5x4-3
2
j / = -1.5(x-2.3) -2.5
4. y = -1.3x + 3
2
v = 1.5(x42.3) 42.5
2
5. j = -1.5x 4-1.5x4-3
2
v = 1.5(x-2.3) -2.5
Short Answer
1. Solve the system graphically.
y = 2x-4
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) = x - Ix +15 and y = 2x - 5
1 ,
b) j = — x - 2 x - 3 andy = -2x + 1
2
y
x y
1
3. Graph the inequality — - — > —
y
——r~
~—"-—5---
™
~
-~
"
—
1I -
7
„5
-5
„
3
-2
i
-1
i
—_i—
—_a—
...•> -
—A —0
.1—
2
4. What is the solution for 2x - Ix > -3?
2
5. Graph the inequality -x < 24 - lOx and state the solution set.
19
ID: A
Name:
2
2
6. 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.
9 +
8
i
j
t
i
I *
7
i
t
-\
5
t
L
1
4
- 3t
-f- - t - -
f?
-P
*
<
;
1
-6
j5 -4 -f -|2 -jl
3
J
4
I
I
r
•3+
i
- -|
t
r—
- !
—4
r-
]
—6I
I
-t -
+
I
j-
•H 4 - f
;
_
_;. L
20
!
ID: A
Systems and inequalites review
Answer Section
MULTIPLE CHOICE
PTS: 1
DIF: Easy
OBJ: Section 8.1
1. ANS D
TOP: Solving Systems of Equations Graphically
NAT RF6
K E Y linear-quadratic systems | interpreting graphs
PTS: 1
DIF: Easy
OBJ: Section 8.1
2. ANS C
TOP: Solving Systems of Equations Graphically
NAT RF 6
K E Y linear-quadratic systems | interpreting graphs | tangent line
PTS: 1
DIF: Easy
OBJ: Section 8.1
3. ANS C
RF
6
TOP:
Solving
Systems
of
Equations
Graphically
NAT
K E Y linear-quadratic systems | interpreting graphs | number of solutions
PTS: 1
DIF: Average
OBJ: Section 8.1
4. ANS C
RF
6
TOP:
Solving
Systems
of
Equations
Graphically
NAT
K E Y quadratic-quadratic systems | interpreting graphs
PTS: 1
DIF: Easy
OBJ: Section 8.1
5. ANS D
TOP: Solving Systems of Equations Graphically
NAT RF 6
K E Y quadratic-quadrat ic systems | interpreting graphs
PTS: 1
DIF: Difficult
OBJ: Section 8.1
6. ANS C
TOP: Solving Systems of Equations Graphically
NAT RF 6
K E Y quadratic-quadratic systems | interpreting graphs
PTS: 1
DIF: Average
OBJ: Section 8.1
7. ANS A
NAT RF 6
TOP: Solving Systems of Equations Graphically
K E Y linear-quadratic systems j interpreting graphs
PTS: 1
DIF: Easy
OBJ: Section 8.1
8. ANS C
TOP: Solving Systems of Equations Graphically
NAT RF 6
K E Y linear-quadratic systems | tangent line
PTS: 1
DIF: Easy
OBJ: Section 8.2
9. ANS D
TOP: Solving Systems of Equations Algebraically
NAT RF 6
K E Y linear-quadratic systems | algebraic solution
PTS: 1
DIF: Difficult
OBJ: Section 8.2
10. ANS D
TOP: Solving Systems of Equations Algebraically
NAT RF 6
K E Y linear-quadratic systems | points of intersection | algebraic solution
PTS: 1
DIF: Difficult
OBJ: Section 8.2
11. ANS C
TOP: Solving Systems of Equations Algebraically
NAT RF 6
K E Y linear-quadratic systems | points of intersection | algebraic solution
PTS: 1
DIF: Difficult
OBJ: Section 8.2
12. ANS C
TOP: Solving Systems of Equations Algebraically
NAT RF 6
K E Y tangent line | quadratic function | number of solutions
PTS: 1
DIF: Average
OBJ: Section 8.2
13. ANS C
NAT RF 6
TOP: Solving Systems of Equations Algebraically
K E Y linear-quadratic systems | algebraic solution
PTS: 1
DIF: Easy
OBJ: Section 8.2
14. ANS D
NAT RF 6
TOP: Solving Systems of Equations Algebraically
K E Y linear-quadratic systems | algebraic solution
1
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 < 0
27. 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 < 0
28. 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
ID: A
COMPLETION
1. ANS: one solution
PTS: 1
DIF: Easy
OBJ: Section 8.1
TOP: Solving Systems of Equations Graphically
2. ANS: (2,0) and (0,-4)
NAT: RF 6
KEY: number of solutions
PTS:
TOP:
KEY:
3. ANS:
1
DIF: Easy
OBJ: Section 8.2
Solving Systems of Equations Algebraically
linear-quadratic systems | algebraic solution
(-6,2)
NAT: RF 6
PTS:
TOP:
KEY:
4. ANS:
1
DIF: Easy
OBJ: Section 8.2
Solving Systems of Equations Algebraically
quadratic-quadratic systems | algebraic solution
one
NAT: RF 6
PTS: 1
DIF: Average
OBJ: Section 8.2
TOP: Solving Systems of Equations Algebraically
KEY: quadratic-quadratic systems | number of solutions
5. ANS:
(0, 0)
NAT: RF 6
NB: Other answers will work as long as they satisfy the given inequality—see graph below.
PTS: 1
DIF: Easy
OBJ: Section 9.1
TOP: Linear Inequalities in Two Variables
NAT: RF 7
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
IB: A
2. ANS:
NAT:
KEY:
3. ANS:
NAT:
KEY:
4. ANS:
NAT:
KEY:
5. ANS:
NAT:
KEY:
A
PTS: 1
DIF: Average
OBJ:
RF 6
TOP: Solving Systems of Equations Graphically
linear-quadratic systems | graphical solution
E
PTS: 1
DIF: Average
OBJ:
RF 6
TOP: Solving Systems of Equations Graphically
quadratic-quadratic systems | graphical solution
C
PTS: 1
DBF: Average
OBJ:
RF 6
TOP: Solving Systems of Equations Graphically
quadratic-quadratic systems | graphical solution
D
PTS: 1
DIF: Average
OBJ:
RF6
TOP: Solving Systems of Equations Graphically
quadratic-quadratic systems | graphical solution
Section 8.1
Section 8.1
Section 8.1
Section 8.1
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
ID: A
2. ANS:
Solution methods may vary. Examples:
a) Substitution:
y=x -lx
+ \5
Substitute y - 2x - 5:
2r-5=x -7x+15
2
2
2
x - 9 x + 20 = 0
Solve for x by factoring:
x - 9 x + 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
2
=3
=5
The points of intersection for the system are (4, 3) and (5, 5).
b) Elimination:
Subtract the second equation from the first:
2
y=
\x -2x-3
4
y = -2x + 1
2
0= jx -4
4
=4
— Y
4
Solve for x:
2 A
—x =4
4
x = 16
x = ±4
Substitute x = ±4 into y = -2x +1 and solve for v.
7 = -2(4) + 1
j = -2(-4) +1
1
l
2
= -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
ID: A
3. ANS:
Rearrange the inequality to make it easier to graph.
PTS: 1
DIF: Difficult
OBJ: Section 9.1
TOP: Linear Inequalities in Two Variables
KEY: linear inequality | graphing | two variables
6
NAT: RF 7
ID: A
4. ANS:
First, rewrite the inequality as 2x - Ix + 3 > 0.
Next, factor the quadratic:
2x -7x + 3 = (2x- l ) ( x - 3 )
2
2
1
~
x = — orx = 3
i—|
^9
1
_g
1
-7
1
-6
1
-5
1
-4
1
-3
1
-2
1
-1
H H
0
1
1
2
•
3
1
4
1
5
1
6
1
7
1
8
1—»
9
i
8
i
9
Choose a test point in each interval, such as 0, 1, and 4:
2
L.S. = 2(0) -7(0)+3
R S
- =
0
=3
L.S. > R.S.
2
L.S. = 2(1) -7(1) +3
R-S. =0
= -2
L.S. < R.S.
2
L.S. = 2(4) -7(4)+ 3 R-S. =0
=7
L.S. ^ R.S.
Therefore, the solution is j x | x < -| orx > 3, x € it!J.
<
- 9 - 8 - 7 - 6 - 5 - 4
i
-3
i
-2
i
-1
i •
0
l — i — •
1
2
3
i
4
i
5
i
6
i
7
PTS: 1
DIF: Average
OBJ: Section 9.2
TOP: Quadratic Inequalities in One Variable
KEY: quadratic inequality | one variable | solution set
7
)
NAT: RF 7
ID: A
5. ANS:
First, rewrite the inequality as - x + lOx - 24 < 0.
Next, factor the quadratic:
2
2
2
~x + lOx-24 = - ( x - lOx + 24]
= -(x-4)(x-6)
x = 4 orx = 6
<—I
9
|
g
1
-7
(
1
-5
1
-4
1
-3
1
-2
1
1
1
0
1
1
1
2
1
3
O—I
4
5
O
6
1
7
1
8
1—•>
9
Choose a test point in each interval, such as 0, 5, and 7:
2
R S
L.S. = - ( 0 ) + 10(0) -24
- =
0
= -24
L.S. < R.S.
2
L.S. = - ( 5 ) + 10(5)-24
R-S.=0
=1
L.S. > R.S.
2
R S
L.S. = - ( 7 ) + 10(7) - 24
- =0
= -3
L.S. < R.S.
(
i
9
i
i
8
i
7
o
I
I
.5
4
I
I
3
I
2
I
1 0
i
i
i
o—i—o
I
I
I
1
2
3
4
7
8
9
5
6
Therefore, the solution is j x | 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
ANS:
a)
—9-
ft
o
7
o
c
J
4
X.
1
1
?
-8
-7
-5
-5
-\
-5
-2
-1
1
1
"l ..
, 1 -
—J *
*| ...
A :.
m
e .
« <\
/
i
•Raj E
f
li
b) Test point used will vary.
Example: Use the test point (0, 0).
L.S.=0
2
R.S. = - | ( 0 - 3 ) - l
= -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
TOP: Quadratic Inequalities in Two Variables
KEY: quadratic inequality | two variables | graphing
NAT: RF 7
Systems and inequalites review [Answer Strip]
C
D
1.
2.
C
3.
Systems and inequalites review [Answer Strip]
C
6.
A
C
7.
C
11.
C
12.
C
13.
8.
D
9.
D
10.
D
14.
B
15.
B
16.
ID: A
A
17.
C
18.
C
19.
D
20.
Systems and inequalites review [Answer Strip]
_JL_21.
B
22.
A
23.
A
24.
D
25.
D
26.
B
27.
ID: A
B
28.
j>
29.
Systems and inequalites review [Answer Strip]
B
1.
A
2.
E
3.
C
4.
D
5.