Ch. 1 Graphs, Equations, and Inequalities
1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equation
s
1 Use the Distance Formula
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the distance d(P1, P2) between the points P1and P2.
1)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A) 13 B) 5 C) 6 D) 4
2)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
A) 194 B) 12 C) 65 D) 8
Page 1
3)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
A) 2 5 B) 12 3 C) 12 D) 6
4)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
A) 2 13 B) 20 5 C) 20 D) 2
5) P1 = (–4
,
–4); P2 = (–4
,
4)
A) 8 B) 2 2 C) 9 D) 7
6) P1 = (1
,
–3); P2 = (–5
,
–11)
A) 10 B) 100 C) 11 D) 20
7) P1 = (0, 10); P2 = (3
,
10)
A) 3 B) 10 C) 109 D) 9
8) P1 = (0, 0); P2 = (10
,
7)
A) 149 B) 149 C) 17 D) 70
9) P1 = (5
,
3); P2 = (–5
,
–4)
A) 149 B) 51 C) 70 D) 3
10) P1 = (4
,
–7); P2 = (6
,
–1)
A) 2 10 B) 32 2 C) 32 D) 4
Page 2
11) P1 = (–4
,
–7); P2 = (2
,
–3)
A) 2 13 B) 20 5 C) 20 D) 2
12) P1 = (0.2
,
0.5); P2 = (–1.5
,
–1.6) Round to three decimal places, if necessary.
A) 2.702 B) 19 C) 8.544 D) 2.802
Decide whether or not the points are the vertices of a right triangle.
13) (–6
,
–7), (1
,
–7), (1
,
1)
A) Yes B) No
14) (9
,
7), (11
,
11), (13
,
10)
A) Yes B) No
15) (–6
,
–3), (0
,
–1), (–1
,
–6)
A) Yes B) No
16) (–3
,
11), (3
,
13), (9
,
6)
A) Yes B) No
Solve the problem.
17) Find all values of k so that the given points are 29 units apart.
(–5, 5), (k, 0)
A) –3, –7B)
–7 C) 3, 7 D) 7
18) Find the area of the right triangle ABC with A =(–2, 7), B =(7, –1), C =(3, 9).
A) 29 square units B) 58 square units C) 58
2 square units D) 29
2 square units
19) Find all the points having an x–coordinate of 9 whose distance from the point (3, –2) is 10.
A) (9, 6), (9, –10) B) (9, 2), (9, –4) C) (9, –12), (9, 8) D) (9, 13), (9, –7)
20) A middle school’s baseball playing field is a square, 55 feet on a side. How far is it directly from home
plate to second base (the diagonal of the square)? If necessary, round to the nearest foot.
A) 78 feet B) 79 feet C) 77 feet D) 85 feet
21) A motorcycle and a car leave an intersection at the same time. The motorcycle heads north at an average
speed of 20 miles per hour, while the car heads east at an average speed of 48 miles per hour. Find an
expression for their distance apart in miles at the end of t hours.
A) 52t miles B) t 68 miles C) 52 t miles D) 2t 13 miles
22) A rectangular city park has a jogging loop that goes along a length, width, and diagonal of the park. To the
nearest yard, find the length of the jogging loop, if the length of the park is 125 yards and its width is 75
yards.
A) 346 yards B) 146 yards C) 345 yards D) 145 yards
Page 3
23) Find the length of each side of the triangle determined by the three points P1
,
P2
,
and P3. State whether
the triangle is an isosceles triangle, a right triangle, neither of these, or both.
P1 = (–5, –4), P2 = (–3, 4), P3 = (0, –1)
A) d(P1, P2) = 217
; d(P2, P3) = 34; d(P1, P3) = 34
both
B) d(P1, P2) = 217
; d(P2, P3) = 34; d(P1, P3) = 34
isosceles triangle
C) d(P1, P2) = 217
; d(P2, P3) = 34; d(P1, P3) = 52
right triangle
D) d(P1, P2) = 217
; d(P2, P3) = 34; d(P1, P3) = 52
neither
2 Use the Midpoint Formula
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the midpoint of the line segment joining the points P1and P2.
1) P1 = (7
,
8); P2 = (9
,
2)
A) (8
,
5) B) (16
,
10) C) (–2
,
6) D) (5
,
8)
2) P1 = (2
,
–4); P2 = (–8
,
1)
A) – 3, – 3
2B) 5, – 5
2C) 10
,
–5 D) –6
,
–3
3) P1 = (7, 1); P2 = (–16, –16)
A) – 9
2, – 15
2B) 23
2, 17
2C) –9, –15 D) 9, 15
4) P1 = (–0.5
,
0.6); P2 = (–1.1
,
–1.9)
A) (–0.8
,
–0.65) B) (–0.65
,
–0.8) C) (–0.3
,
–1.25) D) (–1.25
,
–0.3)
5) P1 = (a
,
6); P2 = (0, 4)
A) a
2, 5 B) a
,
10 C) – a
2, 2 D) a
,
5
6) P1 = (6y
,
2); P2 = (7y
,
9)
A) 13y
2, 11
2B) 13y, 11 C) y, 7 D) 11y
2, 13
2
Solve the problem.
7) If (2
,
–5) is the endpoint of a line segment, and (3
,
–3) is its midpoint, find the other endpoint.
A) (4
,
–1) B) (4
,
–7) C) (0
,
–9) D) (6
,
–3)
8) If (4
,
3) is the endpoint of a line segment, and (1
,
–1) is its midpoint, find the other endpoint.
A) (–2
,
–5) B) (–2
,
7) C) (10
,
11) D) (–4
,
–3)
9) If (–1
,
2) is the endpoint of a line segment, and (0
,
–1) is its midpoint, find the other endpoint.
A) (1
,
–4) B) (1
,
5) C) (–3
,
8) D) (–7
,
4)
Page 4
10) If (10
,
–10) is the endpoint of a line segment, and (8
,
–7) is its midpoint, find the other endpoint.
A) (6
,
–4) B) (6
,
–13) C) (14
,
–16) D) (16
,
–14)
11) The medians of a triangle intersect at a point. The distance from the vertex to the point is exactl
y
two–thirds of the distance from the vertex to the midpoint of the opposite side. Find the exact distance of
that point from the vertex A(3, 4) of a triangle, given that the other two vertices are at (0, 0) and (8, 0).
A) 217
3B) 17
3C) 2 D) 8
3
Page 5
3 Graph Equations by Plotting Points
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Plot the point in the xy–plane. Tell in which quadrant or on what axis the point lies.
1) (4
,
1)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant I
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant I
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant IV
Page 6
2) (–3
,
5)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant IV
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant I
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant III
Page 7
3) (6
,
–5)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant IV
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant III
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant I
Page 8
4) (–5
,
–1)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant III
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant III
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant IV
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
Page 9
5) (0, 2)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
y–axis
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
x–axis
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
y–axis
Page 10
6) (–6
,
0)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
x–axis
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
y–axis
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Quadrant II
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
x–axis
Determine whether the given point is on the graph of the equation.
7) Equation: y = x4 – x
Point: (0, 0)
A) Yes B) No
8) Equation: x2 + y2 = 4
Point: (2, 0)
A) Yes B) No
Page 11
Graph the equation by plotting points.
9) y = x – 5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 12
10) y = 2x – 8
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 13
11) y = –x2+ 9
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 14
12) 3x + 4y = 12
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 15
13) 4x2+ 9y = 36
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page 16
4 Graph Equations Using a Graphing Utility
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine the coordinates of the point shown. Tell in which quadrant the point lies. Assume the coordinates are
integers.
1)
A) (–4, 6); quadrant II B) (–4, 3); quadrant II
C) (–4, 6); quadrant I D) (–4, 3); quadrant I
Select a setting so that each of the given points will lie in the viewing window.
2) (8, 3), (5, 1), (9, 19)
A) B)
C) D)
Page 17
Determine the viewing window used.
3)
A) B)
C) D)
Page 18
Graph the equation using a graphing utility.
4) y = 2x + 1
A) B)
C) D)
5) 2x – 3y = 3
A) B)
C) D)
Page 19
6) y = –2x2 + 9
A) B)
C) D)
7) 3x2 – 2y = 56
A) B)
C) D)
Page 20
5 Use a Graphing Utility to Create Tables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a graphing utility to create a table that displays the points on the graph of the given equation for x = –3, –2, –1,
0, 1, 2, and 3.
1) y = 3x + 5
A) B)
C) D)
2) 6x + 2y = 28
A) B)
C) D)
Page 21
3) y = –3x2 + 10
A) B)
C) D)
4) 4x2 – 2y = 16
A) B)
C) D)
Page 22
6 Find Intercepts from a Graph
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
List the intercepts of the graph.
1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (–2
,
0), (2
,
0) B) (0, –2), (2
,
0) C) (0, –2), (0, 2) D) (–2
,
0), (0, 2)
2)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) (0, –1) B) (0, 0) C) (–1
,
–1) D) (–1
,
0)
Page 23
3)
x
––
2
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
––
2
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) – π
2, 0 , (0, –2), π
2, 0 B) – π
2, 0 , (–2, 0), π
2, 0
C) 0, – π
2, (–2, 0), 0, π
2D) 0, – π
2, (0, –2), 0, π
2
4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (–4
,
0), (0, 8), (2
,
0) B) (–4
,
0), (0, 8), (0, 2)
C) (0, –4), (8
,
0), (0, 2) D) (0, –4), (0, 8), (2
,
0)
5)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (–1
,
0) B) (0, –1) C) (1
,
0) D) (0, 1)
Page 24
6)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (–9
,
0), (0, –9), (0, 9), (9
,
0) B) (–9
,
0), (0, 9)
C) (–9
,
0), (0, –9), (0, 0), (0, 9), (9
,
0) D) (0, 9), (9
,
0)
7)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) (4
,
0), (1, 0) (–5
,
0), (0, 4) B) (4
,
0), (0, 4), (0, 1), (0, –5)
C) (–4
,
0), (1, 0), (5
,
0), (0, 4) D) (4
,
0), (0, –4), (0, 1), (0, 5)
8)
A) (–4, 0), (0, 4), (4, 0) B) (–2, 0), (0, 2), (2, 0) C) (–2, 0), (0, 4), (2, 0) D) (–2, 0), (2, 0)
Page 25
7 Use a Graphing Utility to Approximate Intercepts
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the equation using a graphing utility. Use a graphing utility to approximate the intercepts rounded to two
decimal places, if necessary. Use the TABLE feature to help establish the viewing window.
1) y = –2x + 15
A) (0, 15), (7.5, 0) B) (0, 15), (–7.5, 0)
C) (0, –15), (7.5, 0) D) (0, 15), (–7.5, 0), (7.5, 0)
2) y = –4x + 15
A) (0, 15), (3.75, 0) B) (0, 3.75), (15, 0) C) (0, 15), (–3.75, 0) D) (0, –3.75), (15, 0)
3) y = 3x2 – 19
A) (0, –19), (–2.52, 0), (2.52, 0) B) (0, –19), (2.52, 0)
C) (0, 19), (–2.52, 0), (2.52, 0) D) (0, –19), (–2.51, 0), (2.51, 0)
4) y = 5x2 – 13
A) (0, –13), (1.61, 0), (–1.61, 0) B) (0, 1.61), (0, –1.61), (–13, 0)
C) (0, –13), (2.60, 0), (–2.60, 0) D) (0, 2.60), (0, –2.60), (–13, 0)
5) 3x – 4y = 56
A) (0, –14), (18.67, 0) B) (0, –14), (18.66, 0)
C) (0, –14), (–18.67, 0), (18.67, 0) D) (0, 14), (18.67, 0)
6) 6x – 5y = 67
A) (0, –13.40), (11.17, 0) B) (0, 11.17), (–13.40, 0)
C) (0, 13.40), (–11.17,0) D) (0, –13.41), (11.18, 0)
7) 3x2 – 5y = 34
A) (0, –6.8), (–3.37, 0), (3.37, 0) B) (0, –6.8), (–3.36, 0), (3.36, 0)
C) (0, 6.8), (–3.37, 0), (3.37, 0) D) (0, –6.8), (3.37, 0)
8) 4x2 – 5y = 68
A) (0, –13.60), (4.12, 0), (–4.12, 0) B) (0, 4.12), (0, –4.12), (–13.60, 0)
C) (0, –13.59), (4.13, 0), (–4.13, 0) D) (0, 13.60), (4.12, 0), (–4.12, 0)
1.2 Solving Equations Using a Graphing Utility; Linear and Rational Equation
s
1 Solve Equations Using a Graphing Utility
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places.
1) x3 – 6x + 3 = 0
A) {2.15, 0.52, –2.67} B) {–0.48} C) {2.67, –0.52, –2.15} D) no solution
2) x4 – 3x2 + 4x + 15 = 0
A) {–0.84, –1.93} B) {2.11, –2.60} C) {3.94, –1.27} D) no solution
3) 2x4– 5x2 + 7x = 14
A) {–2.31, 1.69} B) {–2.31, 1.70} C) {–2.30, 1.69} D) {–2.32, 1.70}
Page 26
4) x4 – 5x3 + 6x – 2 = 0
A) {4.75, 1, 0.38, –1.13} B) {0.31, –5.23}
C) {4.71, 1.44, –0.38, –0.77} D) no solution
5) –x4 + 3x3 + 4
3x2 = 9
2x + 2
A) {2.82, 1.61, –0.46, –0.97} B) {–3.34}
C) {1.09, –0.44} D) no solution
2 Solve Linear Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
1) 5x = –10
A) {–2} B) {2} C) {5} D) {–5}
2) 10x = 4
A) 2
5B) – 2
5C) 5
2D) – 5
2
3) 2x + 6 = 0
A) {–3} B) {3} C) {2} D) {–2}
4) 3x – 1 = 0
A) 1
3B) – 1
3C) 3 D) – 3
5) 10x + 16 = 0
A) – 8
5B) 8
5C) 5
8D) – 5
8
6) 2
3x = – 5
9
A) – 5
6B) – 6
5C) 5
6D) – 5
7) 16x = 3 + 15x
A) {3} B) {–3} C) {4} D) {18}
8) 8x – 4 = 4x + 12
A) {4} B) {–4} C) {3} D) {–3}
9) –5x – 24 = –2x + 3
A) {–9} B) {9} C) {–8} D) {8}
10) 12 – 7x = –4 – 5x
A) {8} B) {–8} C) {6} D) {–6}
11) 5x – (4x – 1) = 2
A) 1 B) 1
9C) –1 D) – 1
9
Page 27
12) 3(3x – 1) = 12
A) 5
3B) 1 C) 13
9D) 11
9
13) 12(3x – 5) = 9x – 2
A) 58
27 B) 62
27 C) – 58
27 D) 58
45
14) 6(x + 5) = 7x – (3 – x)
A) 51
8B) – 15
4C) 15
4D) – 51
8
15) 4(x + 6) = 5(x – 7)
A) {59} B) {–59} C) {–11} D) {5}
16) 5(2x – 2) = 9(x + 4)
A) {46} B) {–26} C) {–46} D) {31}
17) 5(x + 6) = (5x + 30)
A) {60} B) {0} C) all real numbers D) no solution
18) –6x + 7 + 7(x + 1) = 7x + 4
A) 5
3B) 1 C) 2
5D) {7}
19) –8x + 3 + 6x = –2x + 8
A) {5} B) {–3} C) all real numbers D) no solution
20) x
4 – 4 = 1
A) {20} B) {–20} C) {12} D) {–12}
21) x
3 – 1
3 = –4
A) {–11} B) {11} C) {–13} D) {13}
22) 2x
5 = 3 + x
3
A) {45} B) {–45} C) {90} D) {–90}
23) x
2 + 4 = x
3 + 1
A) – 18 B) 18 C) – 1
2D) 1
2
24) 1
4 + x
2 = 25
8
A) 23
4B) – 23
4C) 23
2D) – 23
2
Page 28
25) –4.9x + 1.8 = –16.8 – 1.8x
A) {6} B) {–22} C) {3.8} D) {4.2}
26) 5x + 8
4 + 1
2 = – 5x
3
A) – 6
7B) – 18
35 C) 18
35 D) 6
27) (x + 8)(x – 1) = (x + 1)2
A) 9
5B) 8
5C) 9
8D) {9}
28) x(6x – 2) = (6x + 1)(x – 2)
A) – 2
9B) – 1
5C) – 1
2D) {3}
29) x(1 + 3x) = (3x – 1)(x – 3)
A) 3
11 B) – 3
11 C) 3
121 D) – 3
121
30) x(x2 + 4) = 6 + x3
A) 3
2B) {4} C) {6} D) 2
3
Solve the problem.
31) If (a, 3) is a point on the graph of y = 2x –5, what is a?
A) 4 B) 1 C) –1D)
–4
32) If (3, b) is a point on the graph of 3x –2y =17, what is b?
A) –4B)4 C)
23
3D) 11
3
Solve the equation. The letters a, b, and c are constants.
33) ax – b = c, a ≠ 0
A) x = b + c
aB) x = b –c
aC) x = c –b
aD) x = – b +c
a
34) x
a + x
b = c, a ≠ 0, b ≠ 0, a ≠ –b
A) x = abc
a + b B) x = abc C) x = c
ab D) x = a +b
abc
3 Solve Rational Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
1) 1 – 3
4x = 5
9
A) 27
16 B) – 27
16 C) – 27
4D) 1
Page 29
2) 5
x + 2
5 = 7
x
A) {5} B) {–5} C) {2} D) {–2}
3) 9
x + 4
x = –3
A) – 13
3B) – 3
13 C) 13
3D) 3
13
4) 3
x + 5 + 2
2x + 1 = 4
x – 2
A) – 46
47 B) 46
47 C) – 47
46 D) 47
46
5) 5 – x
x + 3
4 = 7
x
A) {–8} B) – 8
7C) {8} D) {–4}
6) x
x – 7 + 3 = 7
x – 7
A) {7
,
–7} B) {7} C) {–7} D) no solution
7) x
2x + 2 = 2x – 3
x + 1 – 2x
4x + 4
A) {3} B) 3
2C) {–3} D) no solution
8) 1
x + 1
x + 3 = x + 4
x + 3
A) {1} B) {–3} C) {–1} D) {3}
9) 6
2x – 3 = 4
2x + 5
A) – 21
2B) 21
2C) – 2
21 D) 2
21
10) 2x
x2 – 16
= 3
x2 – 16
– 1
x + 4
A) 7
3B) – 1
3C) –1 D) 1
11) x – 2
x – 5 = x + 6
x – 7
A) 22
5B) – 15
7C) {–4} D) 8
5
Page 30
12) 4x + 8
2x – 3 = 6x + 4
3x – 2
A) 2
13 B) 2
3C) – 14
13 D) – 14
3
13) 5
4x – 1
x+1 = 1
x(3x + 3)
A) – 11
3B) {–11} C) – 11
12 D) no solution
14) 5
x + 5 – 3
x – 5 = 4
x2 – 25
A) {22} B) {–22} C) {6} D) {44}
15) –1
x – 1 = –1
x – 2 – –1
x2 – 3x + 2
A) {1
,
2} B) {1} C) {2} D) no solution
Solve the equation. The letters a, b, and c are constants.
16) a
x + b
x = c, c ≠ 0
A) x = a + b
cB) x = c
a + b C) x = ab
cD) x = c
ab
4 Solve Problems That Can Be Modeled by Linear Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the formula for the indicated variable.
1) PV = nRT for n
A) n = PV
RT B) n = PV
TC) n = RPV
TD) n = PVT
R
2) I = PRT for R
A) R = I
PT B) R = IP
TC) R = IT
PD) R =IPT
3) S = 2πrh + 2πr2 for h
A) h = S – 2πr2
2πrB) h = S –rC)h
= S
2πr – 1D)h =2π(S –r)
4) F = 9
5C + 32 for C
A) C = 5
9(F – 32) B) C = 9
5(F – 32) C) C = F –32
9D) C = 5
F – 32
5) A = P(1 + rt) for t
A) t = A – P
rP B) t = A +P
rP C) t = – A +P
rP D) t = P –A
rP
Page 31
6) A = 1
2h(b1 + b2) for b1
A) b1 = 2A
h – b2B) b1 = 2A
h + b2C) b1 = 2A –b2
hD) b1 = 2A +b2
h
7) P – 5Q
3 = P + 7
2 + 1 for P
A) P = 27 + 10Q
3B) P = 27 –10Q
3C) P = 15 –10Q
3D) P = 15 +10Q
3
Solve the problem.
8) Mary and her brother John collect foreign coins. Mary has three times the number of coins that John has.
Together they have 140 foreign coins. Find how many coins Mary has.
A) 105 coins B) 35 coins C) 21 coins D) 98 coins
9) Center City East Parking Garage has a capacity of 254 cars more than Center City West Parking Garage. If
the combined capacity for the two garages is 1224 cars, find the capacity for each garage.
A) Center City East: 739 cars
Center City West: 485 cars
B) Center City East: 485 cars
Center City West: 739 cars
C) Center City East: 749 cars
Center City West: 475 cars
D) Center City East: 475 cars
Center City West: 749 cars
10) During an intramural basketball game, Team A scored 13 fewer points than Team B. Together, both teams
scored a total of 145 points. How many points did Team A score during the game?
A) 66 points B) 79 points C) 67 points D) 72 points
11) An inheritance of $620,000 is to be divided among Chris, Kelly, and Julie in the following manner: Kelly is
to receive 2
5 of what Chris gets, while Julie gets 2
3 of what Chris gets. How much does Kelly receive?
A) $120,000 B) $300,000 C) $200,000 D) $60,000
12) An auto repair shop charged a customer $459 to repair a car. The bill listed $59 for parts and the remainder
for labor. If the cost of labor is $40 per hour, how many hours of labor did it take to repair the car?
A) 10 hr B) 9 hr C) 11 hr D) 10.5 hr
13) Going into the final exam, which will count as three tests, Jerome has test scores of 61, 72, 59, 75, and 77.
What score does Jerome need on the final in order to earn a C, which requires an average of 70?
A) 72 B) 82 C) 74 D) 76
14) Going into the final exam, which will count as two tests, Emily has test scores of 79
,
85
,
68
,
61
,
and 93.
What score does Emily need on the final in order to have an average score of 80?
A) 87 B) 47 C) 94 D) 67
15) After a 16% price reduction, a boat sold for $24,360. What was the boat’s price before the reduction?
(Round to the nearest cent, if necessary.)
A) $29,000 B) $3897.60 C) $152,250.00 D) $28,257.60
16) Inclusive of a 7.8% sales tax, a diamond ring sold for $2802.80. Find the price of the ring before the tax was
added. (Round to the nearest cent, if necessary.)
A) $2600 B) $3021.42 C) $2584.18 D) $218.62
Page 32
17) It costs $33 per hour plus a flat fee of $27 for a plumber to make a house call. After writing an equation for
this situation, suppose the total cost to have a plumber come to a house is $258. How many hours did the
plumber work?
A) 7 hr B) 6 hr C) 17 hr D) 16 hr
18) A rectangular carpet has a perimeter of 216 inches. The length of the carpet is 76 inches more than the
width. What are the dimensions of the carpet?
A) 92 by 16 in. B) 92 by 108 in. C) 62 by 78 in. D) 100 by 108 in.
19) The perimeter of a triangle is 65 centimeters. Find the lengths of its sides, if the longest side is 7 centimeter
s
longer than the shorter side, and the remaining side is 4 centimeters longer than the shorter side.
A) 18 cm, 22 cm, 25 cm B) 20 cm, 20 cm, 25 cm
C) 18 cm, 23 cm, 25 cm D) 20 cm, 21 cm, 24 cm
1.3 Quadratic Equations
1 Solve Quadratic Equations by Factoring
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation by factoring.
1) x2 + 3x = 0
A) {0, – 3} B) {0, 3} C) {–3} D) {3}
2) 15x2 – 9x = 0
A) 3
5, 0 B) 3
5, – 3
5C) {0} D) – 3
5, 0
3) 6x2 + 15x = 0
A) – 5
2, 0 B) 5
2, – 5
2C) {0} D) 5
2, 0
4) x2 – 9 = 0
A) {3
,
–3} B) {3} C) {–3} D) {9}
5) x2 + x – 12 = 0
A) {–4
,
3} B) {3
,
4} C) {–3
,
4} D) {–4
,
–3}
6) x2 – 9x + 18 = 0
A) {6
,
3} B) {6
,
–3} C) {–6
,
3} D) {–6
,
–3}
7) x2 – 2x – 24 = 0
A) {–4
,
6} B) {–4
,
–6} C) {4
,
–6} D) {4
,
6}
8) 3x2 + 7x – 20 = 0
A) 5
3, – 4 B) 5
3, 4 C) – 5
3, – 4 D) – 5
3, 4
9) 2x2 – 18 = 0
A) {–3
,
3} B) {10} C) {–9
,
9} D) {9}
Page 33
10) x(x – 6) + 5 = 0
A) {1
,
5} B) {1
,
–5} C) {–1
,
5} D) {–1
,
–5}
11) x(x + 9) = 22
A) {–11
,
2} B) {11
,
–2} C) {–11
,
–2} D) {11
,
2}
12) 25x2 – 40x + 16 = 0
A) 4
5B) 5
4C) – 4
5D) – 5
4
13) 12x2 – 5x – 25 = 0
A) – 5
4, 5
3B) 5
4, 5
3C) 5
4, – 5
3D) – 5
4, – 5
3
14) 4x – 11 = 3
x
A) – 1
4, 3 B) – 1
4, 4 C) 1
11, – 1
4D) {–4
,
3}
15) 30x + 25
x = – 55
A) – 1, – 5
6B) 1, 5
6C) – 1, 5
6D) 30, 6
5
16) x – 4
x = 15
x + 4
A) {16
,
–1} B) {16
,
1} C) {4
,
–1} D) {4
,
1}
2 Solve Quadratic Equations Using the Square Root Method
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation by the Square Root Method.
1) x2 = 36
A) {6
,
–6} B) {6} C) {7
,
–7} D) {18}
2) x2 = 3
A) { 3, –3}B){3
,
–3} C) { 3} D) no real solution
3) x2 – 7 = 0
A) { 7, –7}B){
–7
,
7} C) { 7}D){7}
4) (x – 2)2 = 9
A) {5
,
–1} B) {11} C) {3
,
–3} D) {–1
,
–5}
5) (x + 5)2 = 25
A) {–10
,
0} B) {0} C) {–5
,
5} D) {–10}
6) (2x – 1)2 = 121
A) {6
,
–5} B) {5
,
–6} C) {12
,
–10} D) {10
,
–12}
Page 34
7) (2x + 5)2 = 49
A) {–6
,
1} B) {1, 6} C) {–27
,
27} D) {0, 1}
8) (x + 2)2 = 11
A) {–2 + 11, –2 – 11}B){2
+ 11, 2 – 11}
C) { 11, –11}D){9}
3 Solve Quadratic Equations by Completing the Square
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
What number should be added to complete the square of the expression?
1) x2 + 6x
A) 9 B) 3 C) 18 D) 5
2) x2 – 6x
A) 9 B) –3C)18D)5
3) x2 + 3
5x
A) 9
100 B) 9
50 C) 9
25 D) 1
5
4) x2 – 2
3x
A) 1
9B) – 2
9C) 4
9D) – 1
6
Solve the equation by completing the square.
5) x2 – x = 9
A) 1 – 37
2, 1 + 37
2B) –1 – 37
2, –1 + 37
2
C) 1 – 37
4, 1 + 37
4D) – 5
2, 7
2
6) x2 + 4x = 9
A) {–2 – 13, –2 + 13}B){ 2 + 13}
C) {–1 – 13, –1 + 13}D){
–2 – 213, –2 + 213}
7) x2 + 4x – 3 = 0
A) {–2 – 7
, –2 + 7}B){2
+ 7}C){
–1 – 7, –1 + 7}D){2
– 7, 2 + 7}
8) x2 + 6x – 40 = 0
A) {4
,
–10} B) {–4
,
10} C) { 7, –1} D) {–30
,
–10}
9) x2 + 10x + 15 = 0
A) {–5 – 10, –5 + 10}B){5
+ 10}
C) {5 – 15, 5 + 15}D){
–10 + 15}
Page 35
10) x2 – 4x – 7 = 0
A) {2 – 11, 2 + 11}B){2
– 7, 2 + 7}
C) {–2 – 11, –2 + 11}D){4
– 23, 4 + 23 }
11) x2 + 5x – 5 = 0
A) –5 – 35
2, –5 + 35
2B) 5 + 35
2
C) –5 – 35
2D) {–5 – 35, –5 + 35}
12) x2 + 1
7x – 2
49 = 0
A) – 2
7, 1
7B) 2
7, 1
7C) – 2
7, – 1
7D) 2
7, – 1
7
13) 1
3x2 + 1
12x – 1
6 = 0
A) 33 – 1
8, – 33 + 1
8B) 33
8, – 33
8
C) 1
8, – 1
8D) 33 – 1
8, 33 + 1
8
14) 49x2 + 98x + 40 = 0
A) – 4
7, – 10
7B) – 4
49, – 10
49 C) 4
7, 10
7D) – 10
49, 50
49
15) 7x2 – 2x – 4 = 0
A) 1 – 29
7, 1 + 29
7B) 7 – 29
49 , 7 + 29
49
C) –4, 30
7D) –1 – 29
7, –1 + 29
7
4 Solve Quadratic Equations Using the Quadratic Formula
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the real solutions, if any, of the equation. Use the quadratic formula.
1) x2 + 4x – 7 = 0
A) {–2 – 11, –2 + 11}B){2
+ 11}
C) {–1 – 11, –1 + 11}D){
–2 – 111, –2 + 111}
2) x2 – 4x – 7 = 0
A) {2 + 11, 2 – 11}B){2
+ 7, 2 – 7}
C) {–2 + 11, –2 – 11}D){4
+ 11, 4 – 11}
Page 36
3) x2 + x + 8 = 0
A) –1 – 31
2, –1 + 31
2B) 1 – 31
2, 1 + 31
2
C) –1 – 31
2, 1 + 31
2D) no real solution
4) 5x2 + 5x – 10 = 0
A) 1
,
–2 B) 1
,
2 C) –1
,
–2 D) – 1
,
2
5) 8x2 – x + 4 = 0
A) –1 – 129
16 , –1 + 129
16 B) –1 + 129
16 , 1 + 129
16
C) –1 – 129
16 , 1 + 129
16 D) no real solution
6) x2 + 7x + 4 = 0
A) –7 – 33
2, –7 + 33
2B) 7 – 33
2, 7 + 33
2
C) –7 – 33
14 , –7 + 33
14 D) –7 – 65
2, –7 + 65
2
7) 2x2 + x – 5 = 0
A) –1 – 41
4, –1 + 41
4B) –1 – 41
2, –1 + 41
2
C) 1 – 41
4, 1 + 41
4D) no real solution
8) 3x2 + 10x + 6 = 0
A) –5 – 7
3, –5 + 7
3B) –5 – 7
6, –5 + 7
6
C) –10 – 7
3, –10 + 7
3D) –5 – 43
3, –5 + 43
3
9) 16x2 + 8x + 1 = 0
A) – 1
4B) 1
4C) – 1
4, 4 D) no real solution
10) 9x2 – 7 = 18x
A) 7
3, – 1
3B) 7
9, – 1
9C) – 7
3, 1
3D) – 1
9, – 2
3
11) 3x2 + 10x = – 6
A) –5 – 7
3, –5 + 7
3B) –5 – 7
6, –5 + 7
6
C) –10 – 7
3, –10 + 7
3D) –5 – 43
3, –5 + 43
3
Page 37
12) 5x2 = –12x – 6
A) –6 – 6
5, –6 + 6
5B) –6 – 6
10 , –6 + 6
10
C) –12 – 6
5, –12 + 6
5D) –6 – 66
5, –6 + 66
5
13) 7x = 1 + –9
x
A) 1 – 253
14 , 1 + 253
14 B) – 1 + 253
14 , 1 – 253
14
C) 1 – 253
14 D) no real solution
14) 3 + 10
x + 2
x2 = 0
A) –5 – 19
3, –5 + 19
3B) –5 – 19
6, –5 + 19
6
C) 5 – 19
3, 5 + 19
3D) no real solution
Use the discriminant to determine whether the quadratic equation has two unequal real solutions, a repeated real
solution, or no real solution without solving the equation.
15) x2 + 2x – 3 = 0
A) repeated real solutio
n
B) two unequal real solutions C) no real solution
16) x2 + 12x + 36 = 0
A) repeated real solutio
n
B) two unequal real solutions C) no real solution
17) x2 + 2x + 4 = 0
A) repeated real solutio
n
B) two unequal real solutions C) no real solution
18) 5x2 + 3x + 8 = 0
A) repeated real solutio
n
B) two unequal real solutions C) no real solution
19) 3x2 + 6x – 8 = 0
A) repeated real solutio
n
B) two unequal real solutions C) no real solution
5 Solve Problems That Can Be Modeled by Quadratic Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The length of a vegetable garden is 3 feet longer than its width. If the area of the garden is 88 square feet,
find its dimensions.
A) 8 ft by 11 ft B) 7 ft by 12 ft C) 9 ft by 12 ft D) 7 ft by 10 ft
2) Find the dimensions of a rectangle whose perimeter is 46 meters and whose area is 126 square meters.
A) 9 m by 14 m B) 8 m by 15 m C) 10 m by 13 m D) 8 m by 13 m
Page 38
3) The area of a circle is found by the equation A = πr2. If the area A of a certain circle is 121π square
centimeters, find its radius r.
A) 11 cm B) 11πcm C) 11 π cm D) {11 cm, –11 cm}
4) A 41–inch–square TV is on sale at the local electronics store. If 41 inches is the measure of the diagonal of
the screen, use the Pythagorean theorem to find the length of the side of the screen.
A) 41 2
2 in. B) 41 in. C) 41
2 in. D) 1681
2 in.
5) An open box is to be constructed from a square sheet of plastic by removing a square of side 2 inches from
each corner and turning up the sides. If the box must have a volume of 242 cubic inches, what is the length
of one side of the open box?
A) 11 in. B) 13 in. C) 15 in. D) 10 in.
6) An open box is to be made from a square sheet of cardboard by cutting out 6–inch squares from each
corner and turning up the sides. If the box is to have a volume of 294 cubic inches, what are the original
dimensions of the sheet of cardboard?
A) 19 in. by 19 in. B) 7 6 in. by 7 6 in. C) 7 in. by 42 in. D) 7 in. by 7 in.
7) A ball is thrown vertically upward from the top of a building 128 feet tall with an initial velocity of 112 feet
per second. The distance s (in feet) of the ball from the ground after t seconds is s = 128 + 112t – 16t2. After
how many seconds will the ball pass the top of the building on its way down?
A) 7 sec B) 128 sec C) 6 sec D) 9 sec
8) As part of a physics experiment, Ming drops a baseball from the top of a 340–foot building. To the nearest
tenth of a second, for how many seconds will the baseball fall? (Hint: Use the formula h = 16t2, which
gives the distance h, in feet, that a free–falling object travels in t seconds.)
A) 4.6 sec B) 85 sec C) 21.3 sec D) 1.2 sec
9) The net income y (in millions of dollars) of Pet Products Unlimited from 1997 to 1999 is given by the
equation y = 9x2 + 15x + 52, where x represents the number of years after 1997. Assume this trend
continues and predict the year in which Pet Products Unlimited’s net income will be $916 million.
A) 2006 B) 2008 C) 2005 D) 2007
10) The formula A = P(1 + r)2 is used to find the amount of money, A, in an account after P dollars have been
invested in the account paying an annual interest rate, r, for 2 years. Find the interest rate r if $500 grows
to $845 in 2 years.
A) 30% B) 69% C) 3% D) 230%
Page 39
11) A circular pool measures 8 feet across. One cubic yard of concrete is to be used to create a circular border
of uniform width around the pool. If the border is to have a depth of 4 inches, how wide will the border
be? Use 3.14 to approximate π. Express your solution rounded to two decimal places. (1 cubic yard = 27
cubic feet)
8
A) 2.46 ft B) 7.74 ft C) 2.84 ft D) 5.39 ft
12) If a polygon of n sides has 1
2n(n – 3) diagonals, how many sides will a polygon with 152 diagonals have?
A) 19 sides B) 20 sides C) 18 sides D) 21 sides
13) Find a positive value of k such that the equation x2 + kx + 9 = 0 has a repeated real solution.
A) 6 B) 5 C) 4 D) 7
1.4 Complex Numbers; Quadratic Equations in the Complex Number System
1 Add, Subtract, Multiply, and Divide Complex Numbers
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the expression in the standard form a + bi.
1) (2 – 9i) + (6 + 2i)
A) 8 – 7i B) 8 + 7i C) –4+11i D) –8 +7i
2) (8 + 7i) – (–6 + i)
A) 14 + 6i B) 14 –6i C) 2 +8i D) –14 –6i
3) 9(9 – 2i)
A) 81 – 18i B) 81 +18i C) 81i –2i D) 9 – 18i
4) 3i(2 – 6i)
A) 18+ 6i B) 6i– 18 C) 6i – 18i2D) 6i + 18i2
5) –9i(–2 + 8i)
A) 72 + 18i B) –72 +18i C) 18i – 72i2D) 18i + 72i2
6) (–2 – 5i)(5 + i)
A) –5 – 27i B) –15 –27i C) –5+23i D) –15 +23i
7) (6 + 7i)(8 + 3i)
A) 27 + 74i B) 27 –74i C) 69 +38i D) 21i2 + 74i + 48
8) (9 + 4i)(5 – 3i)
A) 57 – 7i B) 57 +7i C) 33 +47i D) –12i2 – 7i + 45
Page 40
9) (8 + 2i)(8 – 2i)
A) 68 B) 64 – 4i2C) 60 D) 64 –4i
10) (–9 + i)(–9 – i)
A) 82 B) –9C)81D)
–80
11) (5 + 3i)(5 – 3i)
A) 34 B) 25 – 9i2C) 16 D) 25 –9i
12) 4
7 + i
A) 14
25 – 2
25iB)
14
25 + 2
25iC)
7
12 + 1
12iD)
7
12 – 1
12i
13) 9
2 – i
A) 18
5 + 9
5iB)
18
5 – 9
5iC)6
+3i D) 6 – 3i
14) 6
3 + 6i
A) 2
5 – 4
5iB)
2
5 + 4
5iC)
– 2
3 – 4
3iD)
– 2
3 + 4
3i
15) 10i
3 – i
A) –1 + 3i B) 1 + 3i C) –1+10i D) –1 –3i
16) 3 + 5i
5 – 3i
A) i B) –iC)1 D)
–1
17) 33 – 21i
5 + 3i
A) 3 – 6i B) 3 + 6i C) –6+3i D) –6 –3i
18) 7 + 6i
7 – 3i
A) 31
58 + 63
58iB)
31
40 – 63
40iC)
67
58 – 21
58iD)
67
40 – 63
40i
19) (4 – 8i)2
A) –48 – 64i B) 80 –64i C) –48 D) 16 – 64i + 64i2
20) 2
2 + 2
2 i 2
A) i B) –iC)
i
2D) – i
2
Page 41
21) i8
A) 1 B) –1C)i D)
–i
22) i19
A) –i B) i C) 1 D) –1
23) i9
A) i B) –iC)1 D)
–1
24) i18
A) –1 B) 1 C) i D) –i
25) 2i15 – i7
A) –iB)i C)
–1D)1
26) 5i5(1 + i3)
A) 5 + 5i B) 5 – 5i C) –5 –5i D) –5 +5i
27) (1 + i)5
A) –4 – 4i B) 4 – 4i C) –4+4i D) –4 +i
28) i18 + i16 + i14 + 1
A) 0 B) 1 C) i D) –1
Perform the indicated operations and express your answer in the form a +bi.
29) –16
A) 4i B) –i4 C) –4i D) ±4
30) –81
A) 9i B) i 9 C) –9i D) ±9
31) (5 + 12i)(12i – 5)
A) 13i B) –13i C) –13 D) 13
Write the expression in the standard form a + bi.
32) Given z = 8 – 3i, evaluate z + z.
A) 16 B) 16 –6i C) –6i D) 16 +6i
33) Given w = 8 + 6i, evaluate w – w.
A) 12i B) –16 +12i C) 16 D) 0
34) Given z = 8 – 7i, evaluate zz.
A) 113 B) 64 – 49i2C) 15 D) 64 –49i
35) Given z = 5 + 9i and w = –6 + i, evaluate z – w.
A) 11 – 8i B) 11 +8i C) –1+10i D) –11 –8i
36) Given z = 4 – 7i and w = 6 + 5i, evaluate z + w.
A) 10 – 2i B) 10 +2i C) –2+12i D) –10 +2i
Page 42
Solve the problem.
37) The impedance Z, in ohms, of a circuit element is defined as the ratio of the phasor voltage V, in volts,
across the element to the phasor current I, in amperes, through the element. That is, Z = V
I. If the voltage
across a circuit element is 7 + i volts and the current through the element is 3 – 4i amperes, what is the
impedance?
A) 17
25 + 31
25i ohms B) 48
25 ohms C) 1 + 31
25i ohms D) 56
25 ohms
38) In an ac circuit with two parallel pathways, the total impedance Z, in ohms, satisfies the formula
1
Z = 1
Z1 + 1
Z2, where Z1 is the impedance of the first pathway and Z2 is the impedance of the second
pathway. Determine the total impedance if the impedances of the two pathways are Z1 = 2 + i ohms and
Z2 = 5 – i ohms.
A) 11
7 + 3
7i ohms B) 9
7 + 3
7i ohms C) 8
7 ohms D) 6
7 ohms
2 Solve Quadratic Equations in the Complex Number System
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation in the complex number system.
1) x2 + 100 = 0
A) {–10i, 10i} B) {10i} C) {10} D) {–10
,
10}
2) x2 – 10x + 50 = 0
A) {5 + 5i, 5 – 5i} B) {5 – 25i, 5 +25i} C) {5 +5i} D) {10
,
0}
3) 16x2 – 7x + 1 = 0
A) 7
32 – 15
32 i, 7
32 + 15
32 i B) – 7
32 – 15
32 i, 7
32 + 15
32 i
C) 7
32 – 15
32 i, – 7
32 + 15
32 i D) – 7
32 – 15
32 i, – 7
32 + 15
32 i
4) x2 + x + 6 = 0
A) – 1
2 – 23
2i, – 1
2 + 23
2i B) 1 – 23
2, 1 + 23
2
C) 1
2 – 23
2i, 1
2 + 23
2i D) –1 – 23
2, –1 + 23
2
5) x3 – 343 = 0
A) 7, – 7
2 – 73
2i, – 7
2 + 73
2i B) 7, – 7
2 – 73
2, – 7
2 + 73
2
C) {7
,
–7i, 7i} D) {7}
6) x4 – 256 = 0
A) {–4
,
4
,
–4i, 4i} B) {–4
,
4} C) {4} D) {–4
,
4
,
4i}
Page 43
7) x4 – 2x2 – 3 = 0
A) {– 3
, 3, i, –i} B) { 3i, i} C) {–3i, –i} D) { 3, 3}
Without solving, determine the character of the solutions of the equation in the complex number system.
8) x2 + 6x + 8 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
9) x2 + 8x + 16 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
10) x2 – 2x + 6 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
11) 4x2 – 4x + 3 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
12) 2x2 – 6x + 3 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
13) 3x2 – 6x + 3 = 0
A) a repeated real solutio
n
B) two unequal real solutions
C) two complex solutions that are conjugates of each other
Solve the problem.
14) 3 – i is a solution of a quadratic equation with real coefficients. Find the other solution.
A) 3 + iB)3 – iC)
–3+iD)
–3 –i
15) 3 + 9i is a solution of a quadratic equation with real coefficients. Find the other solution.
A) 3 – 9i B) –3 –9i C) –3+9i D) 3 + 9i
1.5 Radical Equations; Equations Quadratic in Form; Absolute Value Equations; Factorable
Equations
1 Solve Radical Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the real solutions of the equation.
1) x + 4 = 4
A) {12} B) {16} C) {20} D) {64}
Page 44
2) 3x – 2 = 2
A) {2} B) {4} C) 2
3D) 4
3
3) 4x + 5 = 5
A) {5} B) {25} C) 15
2D) 25
4
4) 34x + 3 = –4
A) – 67
4B) 13
4C) –16 D) – 17
5) 51 – x = –2
A) {33} B) {–32} C) {–33} D) {32}
6) 31 + x = –1
A) {–2} B) {2} C) {–1} D) {1}
7) x = 12 x
A) {0, 144} B) {0, 12} C) {–144
,
144} D) {–12
,
12}
8) 10x + 24 = x
A) {12} B) {–2
,
12} C) – 8
3D) no real solution
9) 20x – 60 = x + 2
A) {8} B) {–7} C) {–8} D) {7}
10) x + 7 + x = 3
A) 1
9B) {4} C) {1} D) no real solution
11) x2 – 3x + 16 = x + 1
A) {3} B) {–3} C) – 3
2D) {4}
12) x2 – 2x + 73 = x + 5
A) {4} B) {–4} C) {9} D) {–1}
13) x2 + 2 – 2x
+ 5 = 0
A) {3, –1} B) {–3, 1} C) {3} D) no real solution
14) 2x + 3 – x
+ 1 = 1
A) {3, –1} B) {3} C) {–3, –1} D) no real solution
15) 2x + 5 – x
– 2 = 3
A) {2, 38} B) {3, 8} C) {2} D) {–2}
Page 45
16) 3x + 10 – x
+ 2 = 2
A) {–2, 2} B) {2} C) {–3} D) {–2}
17) 2 – 3x = 6
A) 34
3B) –34
3C) 2
3D) no real solution
18) 1 + 4x
= 1 + x
A) {0, 4} B) 0, 4
3C) {0, 16} D) {0, 36}
19) (9x – 8)1/2 = 8
A) {8} B) {64} C) 56
9D) 64
9
20) (3x + 3)1/2 = 4
A) 13
3B) 16
3C) { –1}D){9}
21) (3x + 1)1/3 = 3
A) 26
3B) 8
3C) 9 D) 24
22) (x + 3)1/3 = –2
A) {–11} B) {1} C) {–9} D) no real solution
23) (x2 – 2)1/2 = 9
A) { 83, – 83}B){83
,
–83} C) { 11} D) {11}
24) x3
/
4 – 8x1
/
4 = 0
A) {0, 64} B) {0, 4096} C) {2 2}D){2}
25) x5/4 – 4x1/4 = 0
A) {0, 4} B) {–4, 0, 4} C) {0, 2} D) {0}
Solve the problem.
26) Find all points having an x–coordinate of 5 whose distance from the point (2
,
2) is 5.
A) (5
,
6); (5
,
–2) B) (5
,
6); (5
,
–6) C) (5
,
2); (5
,
–2) D) (5
,
2); (5
,
–6)
27) Find all points on the y–axis that are 13 units from the point (–12
,
–2).
A) (0, 3); (0, –7) B) (0, 10); (0, –14) C) (0, –3); (0, 7) D) (3
,
0); (–7
,
0)
28) For a cone, the formula r = 3V
πh describes the relationship between the radius r of the base, the volume V,
and the height h. Find the volume if the radius is 5 inches and the cone is 6 inches high. (Use 3.14 as an
approximation for π, and round to the nearest tenth.)
A) 157.0 cu in. B) 31.4 cu in. C) 1413.0 cu in. D) 26.2 cu in.
Page 46
29) The formula v = 2.5r can be used to estimate the maximum safe velocity v, in miles per hour, at which a
car can travel along a curved road with a radius of curvature r, in feet. To the nearest whole number, find
the radius of curvature if the maximum safe velocity is 15 miles per hour.
A) 90 ft B) 563 ft C) 36 ft D) 225 ft
30) The function f(x) = 6.75 x + 12 models the amount, f(x), in billions of dollars of new student loans x years
after 1993. According to the model, in what year is the amount loaned expected to reach $32.25 billion?
A) 2002 B) 2005 C) 2007 D) 2006
31) When an object is dropped to the ground from a height of h meters, the time it takes for the object to reach
the ground is given by the equation t = h
4.9, where t is measured in seconds. Solve the equation for h.
Use the result to determine the height from which an object was dropped if it hits the ground after falling
for 6 seconds.
A) h = 4.9t2; 176.4 m B) h =24.01t; 144.1 m
C) h = 24.01t2; 864.4 m D) h =4.9t; 29.4 m
32) The maximum number of volts, E, that can be placed across a resistor is given by the formula E = PR,
where P is the number of watts of power that the resistor can absorb and R is the resistance of the resistor
in ohms. Solve this equation for R. Use the result to determine the resistance of a resistor if P is 1
2 watts
and E is 12 volts.
A) R = E2
P; 288 ohms B) R = E2
P2; 576 ohms
C) R = E2P; 288 ohms D) R = E2P2; 576 ohms
33) The number of centimeters, d, that a spring is compressed from its natural, uncompressed position is given
by the formula d = 2W
k, where W is the number of joules of work done to move the spring and k is the
spring constant. Solve this equation for W. Use the result to determine the work needed to move a spring
2 centimeters if it has a spring constant of 0.2.
A) W = d2k
2; 0.4 joules B) W = d2k2
4; 0 joules
C) W = 2d2
k; 40 joules D) W = 2d2k; 1.6 joules
2 Solve Equations Quadratic in Form
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the real solutions of the equation.
1) x4 – 50x2 + 49 = 0
A) {–1, 1, –7
,
7} B) {–7
,
7} C) {–49
,
49} D) {–50
,
50}
2) 5x4 + 13x2 – 6 = 0
A) – 2
5, 2
5B) – 2
5, 2
5C) – 3
5, 3
5D) {– 3, 3}
Page 47
3) x6 – 26x3 – 27 = 0
A) {3
,
–1} B) {27} C) {3} D) {–3
,
1}
4) 4(x + 1)2 + 21(x + 1) + 5 = 0
A) – 5
4, –6 B) 0
,
4 C) – 1
4, –6 D) – 5
4, –5
5) (3x + 2)2 – 9(3x + 2) + 18 = 0
A) 1
3, 4
3B) – 1
3, – 4
3C) 5
3, – 8
3D) – 5
2, 8
3
6) (x – 4)2 + 3(x – 4) – 18 = 0
A) {–2, 7} B) {–7, 2} C) {–6, 1} D) {–1, 6}
7) x + x
= 90
A) {81} B) {9} C) {10} D) {100}
8) x + 7x1/2 + 6 = 0
A) {81
,
256} B) {9
,
–16} C) 625
4D) no real solution
9) x1/2 – 10x1/4 + 24 = 0
A) {256
,
1296} B) {16
,
36} C) {4
,
6} D) {–4
,
–6}
10) x2 + 5x – x
2 + 5x = 2
A) –5 + 41
2, –5 – 41
2B) {25
,
–25}
C) {1} D) {5}
11) 1
(x – 2)2 – 2
x – 2 = 3
A) 1, 7
3B) –1, 1
3C) 1, 1
3D) –1, 7
3
12) 2 + 5
2x – 1 = –2
(2x – 1)2
A) – 1
2, 1
4B) –2, – 1
2C) – 1
2, 0 D) – 1
2, – 1
4
13) 2x–2 – 7x–1 – 4 = 0
A) – 2, 1
4B) 2, 1
4C) – 1
2, –4 D) – 1
2, 4
14) x2/3 + 4x1/3 – 5 = 0
A) {–125
,
1} B) {–5
,
1} C) {–1
,
5} D) {–1
,
125}
Page 48
15) x2/3 + 2x1/3 – 15 = 0
A) {–125
,
27} B) {–5
,
3} C) {–3
,
5} D) {–27
,
125}
Find the real solutions of the equation. Use a calculator to express the solutions rounded to two decimal places.
16) π(1 + x)2 – 5 = 2(1 + x)
A) {0.62, –1.98} B) {–0.62, 0.82} C) {–1.62, –0.62} D) {–0.62, –0.18}
17) x2/5 – 3x1/5 – 4 = 0
A) {–1, 1024} B) {–1, 4} C) {–1, 1.32} D) {–1}
Solve the problem.
18) If k = x + 3
x – 1 and k2 – 5k = 6, find x.
A) 9
5, – 1 B) 6
5, – 1 C) 9
5, 1
2D) 3
2, – 1
3 Solve Absolute Value Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the equation.
1) x = 7
A) {–7
,
7} B) {7} C) {–7} D) {49}
2) x = –2
A) {2} B) {2
,
–2} C) {–2} D) no real solution
3) x – 4 = 0
A) {4} B) {–4
,
4} C) {–4} D) no real solution
4) x + 1 = 4
A) {–5
,
3} B) {5
,
3} C) {–3} D) no real solution
5) 8x + 4 = 3
A) – 1
8, – 7
8B) – 1
4, – 7
4C) 1
8, 7
8D) no real solution
6) 3 – 6x = 9
A) – 1
,
2 B) – 1
,
5 C) –2
,
1 D) – 5
,
1
7) –2x = 8
A) – 4
,
4 B) – 1
4, 1
4C) {–8
,
8} D) no real solution
8) –x = 11
A) {–11
,
11} B) {11} C) {–11} D) {121}
9) 18 – 7x = 9
A) – 9
7, 9
7B) – 7
9, 7
9C) {–9
,
9} D) no real solution
Page 49
10) 1
3x = 2
A) – 6
,
6 B) – 2
3, 2
3C) {–5
,
5} D) no real solution
11) x
3 + 1
2 = 1
A) – 9
2, 3
2B) – 3
2, 9
2C) 3
2D) no real solution
12) u – 2 = – 1
2
A) 3
2, 5
2B) 3
2C) 5
2D) no real solution
13) x2 – 36 = 0
A) {–6
,
6} B) {–1296
,
1296} C) {6} D) {–6}
14) x2 – 3x = 0
A) {0, 3} B) {–3
,
0, 3} C) {–3
,
0} D) no real solution
15) x2 + 5x – 3 = 3
A) {–6
,
–5
,
0, 1} B) {–6
,
6
,
–1, 1} C) {–6
,
–5
,
1} D) {–5
,
–1, 0, 6}
4 Solve Equations by Factoring
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the real solutions of the equation by factoring.
1) x3 – 4x = 0
A) {0, 2
,
–2} B) {0, 2} C) {0, –2} D) {0, 4}
2) 2x5 = 8x3
A) {–2
,
0, 2} B) {–22, 0, 2 2}C){
–2
,
2} D) {0}
3) x3 + 6x2 – 4x – 24 = 0
A) {–2
,
2
,
–6} B) {4
,
–6} C) {2
,
–6} D) {–2
,
2
,
6}
4) x3 + 2x2 + 9x + 18 = 0
A) {–2} B) {2} C) {–3
,
3
,
–2} D) no real solution
5) 3x4 – 108x2 = 0
A) {–6
,
0, 6} B) {–63, 0, 6 3}C){
–6
,
6} D) {0}
6) 5x4 = 320x
A) {0, 4} B) {–4
,
0, 4} C) {0, 5
,
4} D) {0}
7) x3 + 8x2 + 15x = 0
A) {0, –3
,
–5} B) {–3
,
–5} C) {0, 3
,
5} D) {3
,
5}
Page 50
8) x3 + 5x2 – x – 5 = 0
A) {–1, 1, –5} B) {1, –5
,
5} C) {–5
,
5} D) {25}
9) 12x3 + 84x2 + 120x = 0
A) {0, –5
,
–2} B) {–5
,
–2} C) {0, 5
,
2} D) {– 1
5, –2}
1.6 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Jobs
1 Translate Verbal Descriptions into Mathematical Expressions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Translate the sentence into a mathematical equation. Be sure to identify the meaning of all symbols.
1) The surface area of a sphere is 4π times the square of the radius.
A) If S represents the surface area and r the radius, then S = 4πr2.
B) If S represents the surface area and r the radius, then S =4πr.
C) If S represents the surface area and r the radius, then S = πr2.
D) If S represents the surface area and r the radius, then 4πS =r2.
2) The volume of a right prism is the area of the base times the height of the prism.
A) If V represents the volume, B the area of the base, and h the height, then V = Bh.
B) If V represents the volume, B the area of the base, and h the height, then V = B + h.
C) If V represents the volume, B the area of the base, and h the height, then V = B
h.
D) If V represents the volume, B the area of the base, and h the height, then V = 1
2Bh.
3) Speed is measured by distance divided by time.
A) If S represents speed, d distance, and t time, then S = d
t.
B) If S represents speed, d distance, and t time, then S = t
d.
C) If S represents speed, d distance, and t time, then d = S
t.
D) If S represents speed, d distance, and t time, then t = S
d.
4) Momentum is the product of the mass of an object and its velocity.
A) If M represents momentum, m mass, and v velocity, then M =mv.
B) If M represents momentum, m mass, and v velocity, then M = 1
2mv.
C) If M represents momentum, m mass, and v velocity, then M = m
v.
D) If M represents momentum, m mass, and v velocity, then M =m +v.
Page 51
5) The force of gravity between two objects is the gravitational constant times the product of their masses
divided by the square of the distance between them.
A) If F is the force of gravity, G the gravitational constant, m1the mass of one object, m2 the mass of the
second, and d the distance between them, then F = Gm1m2
d2.
B) If F is the force of gravity, G the gravitational constant, m1the mass of one object, m2 the mass of the
second, and d the distance between them, then FG = m1m2
d2.
C) If F is the force of gravity, G the gravitational constant, m1the mass of one object, m2 the mass of the
second, and d the distance between them, then F = Gm1m2
d.
D) If F is the force of gravity, G the gravitational constant, m1the mass of one object, m2 the mass of the
second, and d the distance between them, then F = Gm1 + m2
d2.
6) The total cost of producing refrigerators in one production line is $3200 plus $310 per unit produced.
A) If C is the total cost and x is the number of units produced, then C =3200 + 310x.
B) If C is the total cost and x is the number of units produced, then C =3200x + 310.
C) If C is the total cost and x is the number of units produced, then C =(3200 + 310)x.
D) If C is the total cost and x is the number of units produced, then C = 3200
310x .
7) The profit derived from the sale of x video cameras is $420 per unit less the sum of $2000 costs plus $120
per unit.
A) If P is profit and x the units sold, then P =420x –(2000 +120x) or P =300x – 2000.
B) If P is profit and x the units sold, then P =420x –(2000 –120x) or P =540x – 2000.
C) If P is profit and x the units sold, then P = 420
x – 2000
+ 120
x or P = 300
x – 2000.
D) If P is profit and x the units sold, then P =420x +2000 –120x or P =300x + 2000.
2 Solve Interest Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Don James wants to invest $57,000 to earn $5600 per year. He can invest in B–rated bonds paying 12% per
year or in a Certificate of Deposit (CD) paying 8% per year. How much money should be invested in each
to realize exactly $5600 in interest per year?
A) $26,000 in B–rated bonds and $31,000 in a CD B) $31,000 in B–rated bonds and $26,000 in a CD
C) $27,000 in B–rated bonds and $30,000 in a CD D) $30,000 in B–rated bonds and $27,000 in a CD
2) A bank loaned out $61,000
,
part of it at the rate of 11% per year and the rest at a rate of 5% per year. If the
interest received was $4970, how much was loaned at 11%?
A) $32,000 B) $29,000 C) $33,000 D) $28,000
3) A loan officer at a bank has $83,000 to lend and is required to obtain an average return of 16% per year. If
he can lend at the rate of 17% or the rate of 13%, how much can he lend at the 13% rate and still meet his
required return?
A) $20,750.00 B) $2766.67 C) $684,750.00 D) $4882.35
Page 52
4) A college student earned $8400 during summer vacation working as a waiter in a popular restaurant. The
student invested part of the money at 8% and the rest at 7%. If the student received a total of $627 in
interest at the end of the year, how much was invested at 8%?
A) $3900 B) $4500 C) $4200 D) $1200
5) Susan purchased some municipal bonds yielding 7% annually and some certificates of deposit yielding 9%
annually. If Susan’s investment amounts to $19,000 and the annual interest is $1590, how much money is
invested in bonds and how much is invested in certificates of deposit?
A) $6000 in bonds; $13,000 in certificates of deposit
B) $5500 in bonds; $13,500 in certificates of deposit
C) $13,000 in bonds; $6000 in certificates of deposit
D) $13,500 in bonds; $5500 in certificates of deposit
6) Kevin invested part of his $10,000 bonus in a certificate of deposit that paid 6% annual simple interest, and
the remainder in a mutual fund that paid 11% annual simple interest. If his total interest for that year was
$900, how much did Kevin invest in the mutual fund?
A) $6000 B) $4000 C) $7000 D) $5000
3 Solve Mixture Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The manager of a coffee shop has one type of coffee that sells for $10 per pound and another type that sells
for $14 per pound. The manager wishes to mix 80 pounds of the $14 coffee to get a mixture that will sell
for $11 per pound. How many pounds of the $10 coffee should be used?
A) 240 lb B) 120 lb C) 320 lb D) 160 lb
2) The owners of a candy store want to sell, for $6 per pound, a mixture of chocolate–covered raisins, which
usually sells for $3 per pound, and chocolate–covered macadamia nuts, which usually sells for $8 per
pound. They have a 30–pound barrel of the raisins. How many pounds of the nuts should they mix with
the barrel of raisins so that they hit their target value of $6 per pound for the mixture?
A) 45 lb B) 42 lb C) 39 lb D) 48 lb
3) The manager of a candy shop sells chocolate covered peanuts for $5 per pound and chocolate covered
cashews for $15 per pound. The manager wishes to mix 60 pounds of the cashews to get a cashew–peanut
mixture that will sell for $7 per pound. How many pounds of peanuts should be used?
A) 240 lb B) 120 lb C) 300 lb D) 150 lb
4) A chemist needs 130 milliliters of a 27% solution but has only 24% and 37% solutions available. Find how
many milliliters of each that should be mixed to get the desired solution.
A) 100 mL of 24%; 30 mL of 37% B) 110 mL of 24%; 20 mL of 37%
C) 30 mL of 24%; 100 mL of 37% D) 20 mL of 24%; 110 mL of 37%
5) How much pure acid should be mixed with 7 gallons of a 50% acid solution in order to get an 80% acid
solution?
A) 10.5 gal B) 3.5 gal C) 28 gal D) 17.5 gal
6) The radiator in a certain make of car needs to contain 40 liters of 40% antifreeze. The radiator now contains
40 liters of 20% antifreeze. How many liters of this solution must be drained and replaced with 100%
antifreeze to get the desired strength?
A) 10.0 L B) 16 L C) 20 L D) 13.3 L
Page 53
7) How many gallons of a 30% alcohol solution must be mixed with 60 gallons of a 14% solution to obtain a
solution that is 20% alcohol?
A) 36 gal B) 27 gal C) 7 gal D) 12 gal
8) How many liters of 80% hydrochloric acid must be mixed with 40% hydrochloric acid to get 15 liters of
65% hydrochloric acid? Write your answer rounded to three decimals.
A) 9.375 L B) 3.125 L C) 4.688 L D) 8 L
4 Solve Uniform Motion Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) An airplane flies 490 miles with the wind and 330 against the wind in the same length of time. If the speed
of the wind is 20, what is the speed of the airplane in still air?
A) 102.5 mph B) 92.5 mph C) 107.5 mph D) 41.25 mph
2) A boat heads upstream a distance of 30 miles on the Mississippi river, whose current is running at 5 miles
per hour. If the trip back takes an hour less, what was the speed of the boat in still water? Give the answer
rounded to two decimal places, if necessary.
A) 18.03 mph B) 16.58 mph C) 6 mph D) 15 mph
3) Two friends decide to meet in Chicago to attend a Cub’s baseball game. Rob travels 96 miles in the same
time that Carl travels 86 miles. Rob’s trip uses more interstate highways and he can average 5 mph more
than Carl. What is Rob’s average speed?
A) 48 mph B) 43 mph C) 51 mph D) 45 mph
4) Gary can hike on level ground 3 miles an hour faster than he can on uphill terrain. Yesterday, he hiked 29
miles, spending 2 hours on level ground and 5 hours on uphill terrain. Find his average speed on level
ground.
A) 6 2
7 mph B) 3 2
7 mph C) 4 1
7 mph D) 6 5
7 mph
5) Two cars start from the same point and travel in the same direction. If one car is traveling 62 miles per
hour and the other car is traveling at 55 miles per hour, how far apart will they be after 2 hours?
A) 14 mi B) 234 mi C) 124 mi D) 110 mi
6) Two trains leave a train station at the same time. One travels east at 11 miles per hour. The other train
travels west at 10 miles per hour. In how many hours will the two trains be 197.4 miles apart?
A) 9.4 hr B) 18.8 hr C) 4.7 hr D) 9.9 hr
7) Ken and Kara are 26 miles apart on a calm lake paddling toward each other. Ken paddles at 5 miles per
hour, while Kara paddles at 8 miles per hour. How long will it take them to meet?
A) 2 hr B) 1 1
10 hr C) 13 hr D) 8 2
3 hr
8) A freight train leaves a station traveling at 32 km/h. Two hours later, a passenger train leaves the same
station traveling in the same direction at 52 km/h. How long does it takes the passenger train to catch up to
the freight train?
A) 3.2 hr B) 4.2 hr C) 5.2 hr D) 2.2 hr
Page 54
9) Five friends drove at an average rate of 55 miles per hour to a weekend retreat. On the way home, they
took the same route but averaged 75 miles per hour. What was the distance between home and the retreat
if the round trip took 10 hours?
A) 317 4
13 mi B) 5 10
13 mi C) 2062 1
2 mi D) 634 8
13 mi
10) During a hurricane evacuation from the east coast of Georgia, a family traveled 260 miles west. For part of
the trip, they averaged 60 mph, but as the congestion got bad, they had to slow to 20 mph. If the total time
of travel was 7 hours, how many miles did they drive at the reduced speed?
A) 80 mi B) 85 mi C) 90 mi D) 75 mi
5 Solve Constant Rate Job Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) An experienced bank auditor can check a bank’s deposits twice as fast as a new auditor. Working together
it takes the auditors 4 hours to do the job. How long would it take the experienced auditor working alone?
A) 6 hr B) 12 hr C) 4 hr D) 8 hr
2) BJ can overhaul a boat’s diesel inboard engine in 15 hours. His apprentice takes 30 hours to do the same
job. How long would it take them working together assuming no gain or loss in efficiency?
A) 10 hr B) 4 hr C) 45 hr D) 6 hr
3) Tracy can wallpaper 5 rooms in a new house in 25 hours. Together with her trainee they can wallpaper the
5 rooms in 14 hours. How long would it take the trainee working by herself to do the job?
A) 30 hr B) 25 hr C) 55 hr D) 60 hr
4) Brandon can paint a fence in 12 hours and Elaine can paint the same fence in 11 hours. How long will they
take to paint the fence if they work together?
A) 5 17
23 hr B) 5 13
24 hr C) 5 3
4 hr D) 11 1
2 hr
5) Sue can sew a precut dress in 3 hours. Helen can sew the same dress in 2 hours. If they work together, how
long will it take them to complete sewing that dress? Give your answer rounded to one decimal place, if
necessary.
A) 1.2 hr B) 5 hr C) 1.8 hr D) 2.5 hr
1.7 Solving Inequalities
1 Use Interval Notation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Express the graph shown using interval notation. Also express it as an inequality involving x.
1)
–8–7–6–5–4–3–2–10123456789–8–7–6–5–4–3–2–10123456789
A) [–1
,
2)
–1 ≤ x < 2
B) (–1
,
2)
–1 < x < 2
C) [–1
,
2]
–1 ≤ x ≤ 2
D) (–1
,
2]
–1 < x ≤ 2
Page 55
2)
-11 -10 -9 -8 -7 -6 -5 -4 -3-11 -10 -9 -8 -7 -6 -5 -4 -3
A) (–∞
,
–7]
x ≤ –7
B) (–∞
,
–7)
x < –7
C) [–7
,
∞)
x ≥ –7
D) (–7
,
∞)
x > –7
3)
–101234567–101234567
A) (3
,
∞)
x > 3
B) (–∞
,
3]
x ≤ 3
C) [3
,
∞)
x ≥ 3
D) (–∞
,
3)
x < 3
4)
–1012345678910111213–1012345678910111213
A) (4
,
8)
4 < x < 8
B) [4
,
8]
4 ≤ x ≤ 8
C) (4
,
8]
4 < x ≤ 8
D) [4
,
8)
4 ≤ x < 8
5)
456789101112456789101112
A) [6
,
10]
6 ≤ x ≤ 10
B) (6
,
10)
6 < x < 10
C) [6
,
10)
6 ≤ x < 10
D) (6
,
10]
6 < x ≤ 10
6)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) [–4
,
5)
–4 ≤ x < 5
B) (–4
,
5]
–4 < x ≤ 5
C) (–∞
,
5)
x < 5
D) [–4
,
5]
–4 ≤ x ≤ 5
Write the inequality using interval notation, and illustrate the inequality using the real number line.
7) –10
<
x
<
–3
A) (–10
,
–3)
–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1
B) [–10
,
–3]
–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1
C) (–10
,
–3)
–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1
D) [–10
,
–3)
–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0 1
Page 56
8) 2 ≤ x ≤ 4
A) [2
,
4]
–101234567–101234567
B) (2
,
4)
–101234567–101234567
C) [2
,
4]
–101234567–101234567
D) (2
,
4]
–101234567–101234567
9) –8 ≤ x
<
2
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) [–8
,
2)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) (–8
,
2]
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) (–∞
,
2)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) [–8
,
2)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
10) t ≥ –2
A) [–2
,
∞)
-6 -5 -4 -3 -2 -1 0 1 2-6 -5 -4 -3 -2 -1 0 1 2
B) [–2
,
∞]
-6 -5 -4 -3 -2 -1 0 1 2-6 -5 -4 -3 -2 -1 0 1 2
C) (–2
,
∞)
-6 -5 -4 -3 -2 -1 0 1 2-6 -5 -4 -3 -2 -1 0 1 2
D) (–2
,
∞]
-6 -5 -4 -3 -2 -1 0 1 2-6 -5 -4 -3 -2 -1 0 1 2
11) y
<
9
A) (–∞
,
9)
56789101112135678910111213
B) (–∞
,
9]
56789101112135678910111213
C) [–∞
,
9]
56789101112135678910111213
D) [–∞
,
9)
56789101112135678910111213
Page 57
Write the interval as an inequality involving x, and illustrate the inequality using the real number line.
12) [–3
,
9)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) –3 ≤ x
<
9
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) –3
<
x ≤9
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) x
<
9
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) –3≤x
<
9
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
13) [–9
,
1]
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) –9 ≤ x ≤ 1
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) –9
<
x ≤1
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) –9
<
x
<
1
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) –9≤x
<
1
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
14) (5
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) x > 5
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) x ≥5
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) x > 5
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) x ≥5
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
15) [7
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) x ≥ 7
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) x >7
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) x > 7
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) x ≥7
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
Page 58
16) (–∞
,
–8)
A) x
<
–8
–12–11–10–9-8-7-6-5-4–12–11–10–9-8-7-6-5-4
B) x ≤–8
–12–11–10–9-8-7-6-5-4–12–11–10–9-8-7-6-5-4
C) x
<
–8
–12–11–10–9-8-7-6-5-4–12–11–10–9-8-7-6-5-4
D) x ≤–8
–12–11–10–9-8-7-6-5-4–12–11–10–9-8-7-6-5-4
2 Use Properties of Inequalities
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the inequality obtained by performing the indicated operation on the given inequality.
1) Add 3 to each side of the inequality 4 +3x > –4.
A) 7 + 3x > –1B)7
+ 3x
<
–1C)7
+6x > –1D)7
+ 6x
<
–1
2) Multiply each side of the inequality 5 –5x ≥2 by 4.
A) 20 – 20x ≥ 8 B) 20 –20x ≤8C)20
–5x ≥8 D) 20 –5x ≤8
Fill in the blank with the correct inequality symbol.
3) If x
<
3
,
then x – 3 0.
A)
<
B) >C) ≤D) ≥
4) If x
<
–3
,
then x + 3 0.
A)
<
B) >C) ≤D) ≥
5) If x > –5
,
then 2x –10.
A) >B)
<
C) ≥D) ≤
6) If x
<
4
,
then –6x –24.
A) >B)
<
C) ≥D) ≤
7) If x > –2
,
then –6x 12.
A)
<
B) >C) ≥D) ≤
8) If 3x
<
– 9
,
then x _____ –3.
A)
<
B) >C) ≥D) ≤
Page 59
3 Solve Linear Inequalities Algebraically and Graphically
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the inequality. Express your answer using interval notation. Graph the solution set.
1) x – 5
<
0
A) (–∞
,
5)
–2–10123456789101112–2–10123456789101112
B) (5
,
∞)
–2–10123456789101112–2–10123456789101112
C) (–∞
,
–5)
–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
D) (–∞
,
–5]
–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
2) x + 2
<
7
A) (–∞
,
5)
–2–10123456789101112–2–10123456789101112
B) (5
,
∞)
–2–10123456789101112–2–10123456789101112
C) (–∞
,
9)
2 3 4 5 6 7 8 9 10 11 12 13 14 15 162 3 4 5 6 7 8 9 10 11 12 13 14 15 16
D) (–∞
,
5]
–2–10123456789101112–2–10123456789101112
3) 4x + 1
<
9
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) (–∞
,
2)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) (2
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) [2
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) (–∞
,
2]
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
Page 60
4) 8x – 5 > 7x – 6
A) (–1
,
∞)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
B) (–∞
,
–1]
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
C) (–11
,
∞)
–18-17-16-15-14-13-12-11-10 –9 –8 –7 –6 –5 –4–18-17-16-15-14-13-12-11-10 –9 –8 –7 –6 –5 –4
D) [–1
,
∞)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
5) –2x – 7 ≤ –3x – 12
A) (–∞
,
–5]
–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
B) (–∞
,
–5)
–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
C) [–5
,
∞)
–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
D) [–19
,
∞)
–26-25-24-23-22-21-20-19-18-17-16-15-14-13-12–26-25-24-23-22-21-20-19-18-17-16-15-14-13-12
Page 61
6) –3x + 3 ≥ –4x + 2
A) [–1
,
∞)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
B) (–∞
,
–1)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
C) (–∞
,
–1]
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
D) (5
,
∞)
–2–10123456789101112–2–10123456789101112
7) 5x + 7 > 4x + 13
A) (6
,
∞)
–1012345678910111213–1012345678910111213
B) (–∞
,
6]
–1012345678910111213–1012345678910111213
C) (20
,
∞)
13 14 15 16 17 18 19 20 21 22 23 24 25 26 2713 14 15 16 17 18 19 20 21 22 23 24 25 26 27
D) [6
,
∞)
–1012345678910111213–1012345678910111213
8) 5 – 3(1 – x) ≤ –1
–10–8-6-4-2 0 2 4 6 8 10–10–8-6-4-2 0 2 4 6 8 10
A) (–∞
,
–1]
–10–8–6–4–20246810–10–8–6–4–20246810
B) [–1
,
∞)
–10–8–6–4–20246810–10–8–6–4–20246810
C) (–∞
,
0]
–10–8–6–4–20246810–10–8–6–4–20246810
D) (–∞
,
–1)
–10–8–6–4–20246810–10–8–6–4–20246810
Page 62
9) –28x + 4 ≤ –4(6x – 8)
A) [–7
,
∞)
–14–13–12–11–10–9-8-7–6-5-4–3-2-1 0–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0
B) (–∞
,
–7]
–14–13–12–11–10–9-8-7–6-5-4–3-2-1 0–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0
C) [–7
,
∞)
–14–13–12–11–10–9-8-7–6-5-4–3-2-1 0–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0
D) (–∞
,
–7)
–14–13–12–11–10–9-8-7–6-5-4–3-2-1 0–14–13–12–11–10–9-8-7-6-5-4-3-2-1 0
10) –4(2x + 6)
<
–12x – 28
A) (–∞
,
–1)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
B) (–1
,
∞)
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
C) (–∞
,
–1]
–8–7–6–5–4–3–2–10123456–8–7–6–5–4–3–2–10123456
D) (–∞
,
13]
6 7 8 9 10111213141516171819206 7 8 9 1011121314151617181920
11) x
2 ≥ 5 + x
12
–20 –16 -12 -8 -4 0 4 8 12 16 20–20 –16 -12 -8 -4 0 4 8 12 16 20
A) [12
,
∞)
–20 –16 –12 –8 –4 0 4 8 12 16 20–20 –16 –12 –8 –4 0 4 8 12 16 20
B) [–12
,
∞)
–20 –16 –12 –8 –4 0 4 8 12 16 20–20 –16 –12 –8 –4 0 4 8 12 16 20
C) (–∞
,
12]
–20 –16 –12 –8 –4 0 4 8 12 16 20–20 –16 –12 –8 –4 0 4 8 12 16 20
D) (12
,
∞)
–20 –16 –12 –8 –4 0 4 8 12 16 20–20 –16 –12 –8 –4 0 4 8 12 16 20
Page 63
12) x(9x + 5) ≤ (3x + 4)2
–5–4–3–2–1012345–5–4–3–2–1012345
A) – 16
19, ∞
–5–4–3–2–1012345–5–4–3–2–1012345
B) –∞, – 16
19
–5–4–3–2–1012345–5–4–3–2–1012345
C) 16
19, ∞
–5–4–3–2–1012345–5–4–3–2–1012345
D) –∞, 16
19
–5–4–3–2–1012345–5–4–3–2–1012345
Solve the problem.
13) During the first five months of the year, Len earned commissions of $3480
,
$3180
,
$2800
,
$2200
,
and $3990.
If Len must have average monthly earnings of at least $3210 in order to qualify for retirement benefits,
what must he earn in the sixth month in order to qualify for benefits?
A) at least $3610 B) at least $3143 C) at least $3210 D) at least $3130
14)
J
im has gotten scores of 67 and 66 on his first two tests. What score must he get on his third test to keep an
average of 75 or better?
A) at least 92 B) at least 66.5 C) at least 69 D) at least 90
15) At Bargain Car Rental, the cost of renting an economy car for one day is $19.95 plus 20 cents per mile. At
Best Deal Car Rental, the cost of renting a similar car for one day is $24.95 plus 15 cents per mile. Solve the
inequality 24.95 + 0.15x < 19.95 + 0.20x to find the range of miles driven such that Best Deal is a better deal
than Bargain.
A) x > 100 mi B) x
<
100 mi C) x >10 mi D) x
<
10 mi
Page 64
4 Solve Combined Inequalities Algebraically and Graphically
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the inequality. Express your answer using interval notation. Graph the solution set.
1) 12
<
4x ≤ 20
A) (3
,
5]
–3–2–101234567891011–3–2–101234567891011
B) [3
,
5)
–3–2–101234567891011–3–2–101234567891011
C) (–5
,
–3]
-11 -10 -9 -8 -7 -6 -5 -4 -3 –2 -1 0 1 2 3-11 -10 -9 -8 -7 -6 -5 -4 -3 –2 -1 0 1 2 3
D) [–5
,
–3)
-11 -10 -9 -8 -7 -6 -5 -4 -3 –2 -1 0 1 2 3-11 -10 -9 -8 -7 -6 -5 -4 -3 –2 -1 0 1 2 3
2) –7
<
x + 1 ≤ 7
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) (–8
,
6]
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) (–6
,
8]
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) [–8
,
6)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) [–6
,
8)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
3) 1 ≤ 2x – 3 ≤ 7
A) [2
,
5]
–3–2–1012345678910–3–2–1012345678910
B) (2
,
5)
–3–2–1012345678910–3–2–1012345678910
C) [–5
,
–2]
–10–9-8-7-6-5-4–3-2-1 0 1 2 3–10–9-8-7-6-5-4–3-2-1 0 1 2 3
D) (–5
,
–2)
–10–9-8-7-6-5-4–3-2-1 0 1 2 3–10–9-8-7-6-5-4–3-2-1 0 1 2 3
Page 65
4) –9 ≤ –2x + 5
<
–1
A) (3
,
7]
–2–10123456789101112–2–10123456789101112
B) [3
,
7)
–2–10123456789101112–2–10123456789101112
C) [–7
,
–3)
–12–11–10–9-8–7-6-5–4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
D) (–7
,
–3]
–12–11–10–9-8–7-6-5–4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
5) –34 ≤ –5x – 4 ≤ –24
A) [4
,
6]
–2–10123456789101112–2–10123456789101112
B) (4
,
6)
–2–10123456789101112–2–10123456789101112
C) [–6
,
–4]
–12–11–10–9-8–7-6-5–4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
D) (–6
,
–4)
–12–11–10–9-8–7-6-5–4-3-2-1 0 1 2–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2
6) 0 ≤ 2x + 1
2 < 4
–5–4–3–2–1012345–5–4–3–2–1012345
A) – 1
2, 7
2
–5–4–3–2–1012345–5–4–3–2–1012345
B) – 1
2, 7
2
–5–4–3–2–1012345–5–4–3–2–1012345
C) – 1
2, 7
2
–5–4–3–2–1012345–5–4–3–2–1012345
D) – 1
2, 7
2
–5–4–3–2–1012345–5–4–3–2–1012345
Page 66
7) 7 ≤ 5
4x + 2 < 17
–5–4–3–2–10123456789101112131415–5–4–3–2–10123456789101112131415
A) [4
,
12)
–5–4–3–2–10123456789101112131415–5–4–3–2–10123456789101112131415
B) (4
,
12]
–5–4–3–2–10123456789101112131415–5–4–3–2–10123456789101112131415
C) [4
,
5)
–5–4–3–2–10123456789101112131415–5–4–3–2–10123456789101112131415
D) (4
,
5]
–5–4–3–2–10123456789101112131415–5–4–3–2–10123456789101112131415
8) – 1
3 ≤ 7x – 1
9 < 1
3
-2 -1 0 1 2-2 -1 0 1 2
A) – 2
7, 4
7
-2 -1 0 1 2-2 -1 0 1 2
B) – 8
21, 10
21
-2 -1 0 1 2-2 -1 0 1 2
C) – 2
7, 4
7
-2 -1 0 1 2-2 -1 0 1 2
D) – 2
7, 4
7
-2 -1 0 1 2-2 -1 0 1 2
Solve the problem.
9) In one city, the local cable TV company charges $1.88 for each pay–per–view movie watched. In addition,
each monthly bill contains a basic customer charge of $29.50. If last month’s bills ranged from a low of
$40.78 to a high of $53.94, over what range did customers watch pay–per–view movies?
A) movies watched varied from 6 to 13 inclusive B) movies watched varied from 5 to 12 inclusive
C) movies watched varied from 7 to 14 inclusive D) movies watched varied from 5 to 14 inclusive
10) A real estate agent agrees to sell an office building according to the following commission schedule:
$39,000 plus 25% of the selling price in excess of $900,000. Assuming that the office building will sell at
some price between $900,000 and $1,200,000, inclusive, over what range does the agent’s commission vary?
A) The commission will vary between $39,000 and $114,000
,
inclusive.
B) The commission will vary between $40,000 and$114,000
,
inclusive.
C) The commission will vary between $39,000 and $329,000
,
inclusive.
D) The commission will vary between $264,000 and $339,000
,
inclusive.
Page 67
11) In his algebra class, Rob has scores of 79, 85, 81, and 65 on his first four tests. To get a grade of C, the
average of the first five tests must be greater than or equal to 70 and less than 80. Solve an inequality to
find the range of scores that Rob can earn on the fifth test to get a C.
A) 40 ≤ x
<
90, where x represents Bob’s score on the fifth test
B) 40
<
x
<
90, where x represents Bob’s score on the fifth test
C) 40 ≤ x ≤ 90, where x represents Bob’s score on the fifth test
D) x ≥ 40, where x represents Bob’s score on the fifth test
12) Marianne is planning a shopping trip to buy birthday gifts for her son. She estimates that the total price of
the items she plans to purchase will be between $350 and $400 inclusive. If sales are taxed at a rate of
8.375% in her area, what is the range of the amount of sales tax she should expect to pay on her purchases?
If Marianne’s budget for the shopping trip is $425, will she necessarily be able to buy all the gifts that she
has planned?
A) $29.31 ≤ x ≤ $33.50, where x represents the amount of sales tax; No.
B) $29.31 ≤ x ≤ $33.50, where x represents the amount of sales tax; Yes.
C) $29.31
<
x
<
$33.50, where x represents the amount of sales tax; No.
D) $29.31
<
x
<
$33.50, where x represents the amount of sales tax; Yes.
5 Solve Absolute Value Inequalities Algebraically and Graphically
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the inequality. Express your answer using interval notation. Graph the solution set.
1) x
<
4
A) (–4
,
4)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
B) (–∞
,
4]
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
C) (–∞
,
–4) ∪ (4
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
D) [–4
,
4]
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
Page 68
2) x > 4
A) (–∞
,
–4) ∪ (4
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
B) [4
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
C) (–4
,
4)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
D) [–4
,
4]
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
3) x > –4
A) (–4
,
4)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
B) [–4
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
C) (–∞
,
–4) ∪ (4
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
D) (–∞
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
Page 69
4) x
<
–3
A) (–3
,
3)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
B) (–∞
,
–3]
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
C) (–∞
,
–3) ∪ (3
,
∞)
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
D) ∅
–7–6–5–4–3–2–101234567–7–6–5–4–3–2–101234567
5) 9x
<
81
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) (–9
,
9)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) (–∞
,
9)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) (–∞
,
–9) ∪ (9
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) (–9
,
9)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
Page 70
6) 4x > 28
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
A) (–∞
,
–7) ∪ (7
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
B) (7
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
C) (–7
,
7)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
D) (–∞
,
–7] ∪ [7
,
∞)
–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910
7) x – 4
<
10
A) (–6
,
14)
–5 0 5 10 15 20 25 30 35–5 0 5 10 15 20 25 30 35
B) (–14
,
6)
–10–5 0 5 10152025–10–5 0 5 10152025
C) (–∞
,
14)
–5 0 5 10 15 20 25 30 35–5 0 5 10 15 20 25 30 35
D) (–∞
,
–6)
–25-20-15-10 –5 0 5 10 15–25-20-15-10 –5 0 5 10 15
8) x – 7 > 16
A) (–∞
,
–9) ∪ (23
,
∞)
–5 0 5 10 15 20 25 30–5 0 5 10 15 20 25 30
B) (–23
,
9)
-20 –15 -10 -5 0 5 10 15 20-20 –15 -10 -5 0 5 10 15 20
C) (–9
,
23)
–5 0 5 10 15 20 25 30–5 0 5 10 15 20 25 30
D) (23
,
∞)
–5 0 5 10 15 20 25 30–5 0 5 10 15 20 25 30
Page 71
9) 4k – 5 ≥ 6
A) –∞, – 1
4 ∪ 11
4, ∞
012345678910111213012345678910111213
B) – 1
4, 11
4
012345678910111213012345678910111213
C) – 1
4, 11
4
012345678910111213012345678910111213
D) 11
4, ∞
012345678910111213012345678910111213
10) 2k – 1 ≤ 2
A) – 1
2, 3
2
012345678910111213012345678910111213
B) –∞, – 1
2 ∪ 3
2, ∞
012345678910111213012345678910111213
C) – 1
2, 3
2
012345678910111213012345678910111213
D) –∞, 3
2
012345678910111213012345678910111213
Page 72
11) x + 4 – 9 ≤ –3
A) [–10
,
2]
-10 -5 0 5 10 15-10 -5 0 5 10 15
B) (–10
,
2)
-10 -5 0 5 10 15-10 -5 0 5 10 15
C) [–10
,
–3]
-10 -5 0 5 10 15-10 -5 0 5 10 15
D) ∅
-10 -5 0 5 10-10 -5 0 5 10
12) x + 2 – 6 ≥ –2
A) (–∞
,
–6] ∪ [2
,
∞)
–5 0 5 10 15 20–5 0 5 10 15 20
B) [–6
,
2]
–5 0 5 10 15 20–5 0 5 10 15 20
C) (–6
,
2)
–5 0 5 10 15 20–5 0 5 10 15 20
D) [2
,
∞)
–5 0 5 10 15 20–5 0 5 10 15 20
Page 73
13) 7k + 7 – 8 > 0
A) –∞, – 15
7 ∪ 1
7, ∞
–2–10123456789101112–2–10123456789101112
B) – 15
7, 1
7
–2–10123456789101112–2–10123456789101112
C) –∞, – 15
7 ∪ 1
7, ∞
–2–10123456789101112–2–10123456789101112
D) 1
7, ∞
–2–10123456789101112–2–10123456789101112
14) 8k + 3 – 3
<
3
A) – 9
8, 3
8
–1012345678910111213–1012345678910111213
B) –∞, – 9
8 ∪ 3
8, ∞
–1012345678910111213–1012345678910111213
C) –∞, – 9
8
–1012345678910111213–1012345678910111213
D) –∞, 3
8
–1012345678910111213–1012345678910111213
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15) x – 6 ≥ 0
A) 6
-10 -5 0 5 10-10 -5 0 5 10
B) (–∞
,
–6) ∪ (–6
,
∞)
-10 -5 0 5 10-10 -5 0 5 10
C) (–∞
,
6) ∪ (6
,
∞)
-10 -5 0 5 10-10 -5 0 5 10
D) (–∞
,
∞)
-10 -5 0 5 10-10 -5 0 5 10
Solve the problem.
16) Express the fact that x differs from –7 by more than 3 as an inequality involving absolute value. Solve for
x.
A) |x + 7| > 3; {x|x
<
–10 or x > –4} B) |x +7| >3; {x|–10
<
x
<
–4}
C) |x + 7|
<
3; {x|–10
<
x
<
–4} D) |x +7|
<
3; {x|x
<
–10 or x > –4}
17) A landscaping company sells 40–pound bags of top soil. The actual weight x of a bag, however, may differ
from the advertised weight by as much as 0.75 pound. Write an inequality involving absolute value that
expresses the relationship between the actual weight x of a bag and 40 pounds. Over what range may the
weight of a 40–pound bag of to soil vary?
A) |x – 40| ≤ 0.75; {x|39.25 ≤ x ≤ 40.75} B) |x –40|
<
0.75; {x|39.25
<
x
<
40.75}
C) |x – 40| ≥ 0.75; {x|x ≤ 39.25 or x ≥40.75} D) |x –40| ≥0.75; {x|39.25 ≤ x ≤ 40.75}
Page 75
Ch. 1 Graphs, Equations, and Inequalities
Answer Key
1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equation
s
1 Use the Distance Formula
2 Use the Midpoint Formula
3 Graph Equations by Plotting Points
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4 Graph Equations Using a Graphing Utility
5 Use a Graphing Utility to Create Tables
6 Find Intercepts from a Graph
7 Use a Graphing Utility to Approximate Intercepts
1.2 Solving Equations Using a Graphing Utility; Linear and Rational Equation
s
1 Solve Equations Using a Graphing Utility
2 Solve Linear Equations
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3 Solve Rational Equations
4 Solve Problems That Can Be Modeled by Linear Equations
Page 78
1.3 Quadratic Equations
1 Solve Quadratic Equations by Factoring
2 Solve Quadratic Equations Using the Square Root Method
3 Solve Quadratic Equations by Completing the Square
4 Solve Quadratic Equations Using the Quadratic Formula
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5 Solve Problems That Can Be Modeled by Quadratic Equations
1.4 Complex Numbers; Quadratic Equations in the Complex Number System
1 Add, Subtract, Multiply, and Divide Complex Numbers
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2 Solve Quadratic Equations in the Complex Number System
1.5 Radical Equations; Equations Quadratic in Form; Absolute Value Equations; Factorable
Equations
1 Solve Radical Equations
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2 Solve Equations Quadratic in Form
3 Solve Absolute Value Equations
Page 82
4 Solve Equations by Factoring
1.6 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Jobs
1 Translate Verbal Descriptions into Mathematical Expressions
2 Solve Interest Problems
3 Solve Mixture Problems
4 Solve Uniform Motion Problems
5 Solve Constant Rate Job Problems
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1.7 Solving Inequalities
1 Use Interval Notation
2 Use Properties of Inequalities
3 Solve Linear Inequalities Algebraically and Graphically
4 Solve Combined Inequalities Algebraically and Graphically
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5 Solve Absolute Value Inequalities Algebraically and Graphically
Page 85