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Chapter 1
Real Numbers
No problems assigned
1
Chapter 2
Algebraic Methods of Solving
Equations
See book for problems:
Section Assigned Problems
2.3 p124 5, 11, 13, 17, 21, 25, 29, 31, 33, 37, 41, 47, 55, 57
2.4 p131 3, 7, 13, 21, 25, 33, 37, 51, 61
2.5 p139 3, 7, 11, 13, 21, 25, 33, 37, 41, 51, 55, 57
2
Chapter 3
Formulas and Inequalities
No problems assigned.
3
Chapter 4
Working with 2 Variables
4.1
Introduction to 2 Variables
1. Is the ordered pair (0, 8) a solution of the equation x + y = 8?
2. Is the ordered pair (3, -1) a solution of the equation 2x + y = 5?
3. Is the ordered pair (5, 2)] a solution of the equation 5x − 3y = 15?
4. Complete the ordered pair (5, ?) as a solution of the equation y = 2x + 7.
5. Complete the ordered pair (?, -3) as a solution of the equation y = 2x + 7.
6. Complete the table of values for the equation 4x − 9y = −36.
x y
0
0
8
7. Complete the table of values for the equation x = 12.
x y
3
8
0
8. Complete the table of values for the equation y + 2 = 0.
x y
9
2
0
4
9. Plot the following points on a rectangular coordinate system. Label the points.
(a) A = (6, 2)
(b) B = (-4, 2)
(c) C = (0, 4)
(d) D = (4, 0)
10. Complete the table of values for 2x − y = 4, then plot the ordered pairs.
x y
0
0
4
4.2
Graphing Linear Equations
When graphing an equation,
1. Make a table of values.
2. Plot the points in your table of values.
3. Connect the points with a line.
Remember to use arrowheads, since the line extends forever in both directions.
1. Graph the equation y = 32 x + 1.
5
2. Find the x-intercept and y-intercept of: 2x − 3y = 24.
3. Find the x-intercept and y-intercept of: 5x − 2y = 20.
4. Graph the equation: x = y + 2.
5. Graph the equation: x − y = 4.
6. Graph the equation: 2x + y = 6.
7. Graph the equation: y − 2x = 0.
8. Graph the equation: y = −1.
9. Graph the equation: x + 2 = 0.
10. The expected height y (in centimeters) of a woman is related to the length
x of her radius bone (from the elbow to the thumb-side of the wrist), and is
approximated by the linear equation: y = 3.9x + 73.5. Use this information
for the following questions.
(a) Find the expected height of a woman whose radius bone is 22cm long.
(b) How long would the radius bone be for a woman who is 167 cm tall?
11. The expected weight y (in pounds) of a man taller than 60 in. can be approximated by the linear equation: y = 5.5x − 220, where x is the height of the
man in inches. Use this equation for the following questions.
(a) Find the expected weight of a man whose height is 66 in.
(b) How tall would a man be who weighs 155 lb?
6
4.3
Slope of a Line
1. Use the indicated points to nd the slope for this line:
2. Use the indicated points to nd the slope for this line:
7
3. Use the indicated points to nd the slope for this line:
4. Find the slope of a line containing the points (1, -2) and (-3, -7).
5. Find the slope of a line containing the points (-2, 4) and (-3, 7).
6. Find the slope of a line containing the points (-12, 3) and (-12, -7).
7. Find the slope of the line y = 5x + 12.
8. Find the slope of the line 4y = x + 1.
9. Find the slope of the line y = −5.
10. Find the slope of the line x = 6.
11. For the line 3x + y = 7:
(a) What is the slope of a line parallel to it?
(b) What is the slope of a line perpendicular to it?
12. Find the slopes of these
( two lines and determine if they are parallel, perpendicular, or neither:
2x + 5y = 4
4x + 10y = 1
13. Find the slopes of these
( two lines and determine if they are parallel, perpendicular, or neither:
8x − 9y = 6
8x + 6y = −5
8
14. Find the slopes of these
( two lines and determine if they are parallel, perpendicular, or neither:
3x − 2y = 6
2x + 3y = 3
15. Find the slopes of these
( two lines and determine if they are parallel, perpendicular, or neither:
4.4
5x − y = 1
x − 5y = −10
Equations of Lines
1. Determine the slope and y-intercept from this graph. Then write the equation
of the line in slope-intercept form.
2. Determine the slope and y-intercept from this graph. Then write the equation
of the line in slope-intercept form.
9
3. Find an equation of a line with slope m = 4 and with y-intercept (0, -3). Give
the answer in slope-intercept form.
4. Write the equation of a line with slope m = 0 and containing the point (0, 3).
5. Write the equation of a line with undened slope m and containing the point
(0, -2).
6. Graph the line y = 3x + 2 using the slope and y-intercept.
7. Graph the line passing through the point (1, -5) and having slope m = − 25 .
8. Graph the line passing through the point (-2, 3) and having slope m = 0.
9. Find an equation for the line containing the point (-2, 5) and having slope
m = 32 . Give the answer in slope-intercept form.
10. Find an equation for the line containing the points (4, 10) and (6, 12). Give
the answer in slope-intercept form.
11. Find an equation for the line containing the points (-4, 0) and (0, 2). Give the
answer in slope-intercept form.
12. Find an equation for the line containing the point (2, -3) and parallel to 3x =
4y + 5. Give the answer in slope-intercept form.
13. Find an equation for the line perpendicular to x − 2y = 7 and with y-intercept
(0, -3). Give the answer in slope-intercept form.
10
4.5
Functions
1. For the relation described by {(1, 1), (1, -1), (0, 0), (2, 4), (2, -4)}:
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
2. For the relation described by {(2, 5), (3, 7), (4, 9), (5, 11)}:
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
3. For the relation shown in this graph:
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
4. For the relation shown in this graph:
11
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
12
5. For the relation shown in this graph:
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
6. For the relation described by y = x2 :
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
7. For the relation described by y =
2
x−4 :
(a) Is the relation a function?
(b) Find the domain:
(c) Find the range:
f (x) = −3x + 4 and g (x) = −x2 + 4x + 1 for the following problems.
8. Find f (0)
9. Find g (−2)
10. Find f (p)
11. Find g (−x)
12. Find f (x + 2)
13
4.6
Operations on Functions
No problems assigned
14
Chapter 5
Exponents and Polynomials
See book for problems:
Section Assigned Problems
5.1 p364 1, 3, 11, 17, 19, 27, 31, 37, 39, 43, 51, 55, 63, 69, 73, 81
5.2 p377 3, 9, 19, 23, 31, 39, 41, 45, 47, 49, 53, 55, 59, 67, 73, 77
5.3 p384 1, 3, 5, 7, 9, 11, 15, 19, 25, 29, 31, 35, 39
5.4 p390 3, 9, 13, 17, 21, 25, 29, 31, 41, 49, 55
5.5 p397 3, 5, 7, 11, 15, 19, 21, 25, 29, 31, 33, 57, 59, 63, 69, 71, 77
5.6 p404 1, 3, 7, 11, 17, 23, 29, 31, 39, 43, 51
5.7 p413 1, 5, 7, 11, 13, 17, 19, 23, 25, 41, 53
15
Chapter 6
Factoring Polynomials
6.1
White and Yellow Belts
See book for problems:
Section Assigned Problems
6.1 p438 White Belt: 3, 5, 7, 11, 23, 27, 31, 35, 41, 45;
Yellow Belt: 51, 57, 61, 65, 71, 75, 77
6.2 p445 Orange Belt: 27, 29, 31, 33, 41, 51, 61
6.3 p456 Purple Belt: 5, 13, 19, 25, 37, 55
6.4 p464 Green Belt: 1, 3, 5, 9, 15;
Blue Belt: 25, 35, 39;
Red Belt: 19, 23, 41, 61, 65
6.2
Orange Belt
See table above for problems.
6.3
Purple Belt
See table above for problems.
6.4
Green, Blue, and Red Belts
See table above for problems.
16
6.5
Factoring Review
1. Factor completely: a2 − 4a − 12
2. Factor completely: 6a + 12b + 18c
3. Factor completely: 17x3 y 2 + 51xy
4. Factor completely: 6n2 − 19n + 10
5. Factor completely: 54m2 − 24z 2
6. Factor completely: z 3 − 8
7. Factor completely: 16r2 + 24rm + 9m2
8. Factor completely: 3k 3 − 12k 2 − 15k
9. Factor completely: 6 + 3m + 2p + mp
10. Factor completely: 6a2 + 10a − 4
11. Factor completely: 2x3 + 128
6.6
Solving Equations by Factoring
1. Solve: (2m − 7)(m − 3) = 0
2. Solve: t(6t + 5) = 0
3. Solve: y 2 + 3y + 2 = 0
4. Solve: x2 = 24 − 5x
5. Solve: 3x2 + 5x − 2 = 0
6. Solve: y 2 − 9 = 0
7. Solve: (2x + 7)(x2 + 2x − 3) = 0
8. Solve: r3 − 2r2 − 8r = 0
17
6.7
Applications
1. The length of a CD jewel case is 2 cm more than its width. The area of the
rectangular top of the case is 168 cm2 . Find the length and width of the jewel
case.
2. A 10-gallon aquarium is 3 in. higher than it is wide. Its length is 21 in. and
its volume is 2730 in3 . What are the height and width of the aquarium?
3. Find three consecutive odd integers such that 3 times the sum of all three is
18 more than the product of the rst and second integers.
4. Find three consecutive even integers such that the sum of the squares of the
rst and second integers is equal to the square of the third integer.
5. Hercomer works due north of home. Mergatroid works due east. They leave
for work at the same time. By the time Hercomer is 5 mi from home, the
distance between them is 1 mi more than Mergatroid's distance from home.
How far from home is Mergatroid?
6. An ball is propelled from a height of 48 ft with an initial upward velocity of 32 ft
per sec. Its height h after t seconds is given by the formula h = −16t2 +32t+48.
(a) After how many seconds is the height 64 ft?
(b) After how many seconds is the height 60 ft?
(c) After how many seconds does the ball hit the ground?
18
Chapter 7
Rational Expressions and
Equations
7.1
Rational Expressions
1. Find the undened values for:
12
5y
2. Find the undened values for:
x+1
x−6
3. Find the undened values for:
4x2
3x + 5
4. Find the undened values for:
5m + 2
m2 + m − 6
5. Reduce:
4 (y − 2)
10 (y − 2)
m2 − n2
6. Reduce:
m+n
7. Reduce:
x2 + 2x − 15
x2 + 6x + 5
z 3 + 27
8. Reduce: 3
z − 3z 2 + 9z
9. Which of the following are equal to −1 (more than one answer is possible):
19
2x + 3
2x − 3
2x − 3
(b)
3 − 2x
2x + 3
(c)
3 + 2x
2x + 3
(d)
−2x − 3
(a)
10. Reduce:
m2 − 1
1−m
15a2 7
·
11. Multiply and reduce:
14 5a
12. Multiply and reduce:
2(c + d)
18
·
3
6(c + d)2
3x (x + 3)2
13. Multiply and reduce:
·
x+3
6x2
2x
x2
14. Divide and reduce:
÷
x−1 x+2
(x − 3)2 x − 3
÷
15. Divide and reduce:
6x
x2
16. Simplify:
27 − 3z
12
·
4
2z − 18
p2 + 4p − 5
p−1
17. Simplify: 2
÷
p + 7p + 10 p + 4
2k 2 − k − 1
4k 2 − 1
18. Simplify: 2
÷
2k + 5k + 3 2k 2 + k − 3
m2 + 3m + 2 m2 + 10m + 24
·
19. Simplify: 2
m + 5m + 4 m2 + 5m + 6
7.2
Adding and Subtracting Rational Expressions
1. Find the LCD for
7
15
and
21
20
20
2. Find the LCD for
15
−2
and
5p
6p
3. Find the LCD for
7
15
and
6p
4p − 8
4. Find the LCD for
25
37
and
6r − 12
9r − 18
5. Find the LCD for
29
18
and
p−q
q−p
6. Find the LCD for:
5
3
2
,
and
p2 + 8p + 15 p2 − 3p − 18
p2 − p − 30
7. Rewrite with new denominator (i.e., expand):
2a2
?
a + 2b
= 3
2
+ ab − b
2a b + a2 b2 − ab3
7
4
+
m m
5m
1 + 4m
9. Simplify:
−
m+1
m+1
8. Simplify:
z 1
+
5 3
x + 1 3x + 3
11. Simplify:
+
6
9
7
2
12. Simplify: 2 −
3p
p
10. Simplify:
13. Simplify:
x
−8
+ 2
x−2 x −4
14. Simplify:
−1
4y − 3
−
1−y
y−1
15. Simplify:
3
9
+
4p − 5 5 − 4p
16. Simplify:
5
x+2
−
x2 − 9 x2 + 4x + 3
17. Simplify:
x + 3y
x−y
+
x2 + 2xy + y 2 x2 + 4xy + 3y 2
21
7.3
Complex Fractions
No problems assigned.
7.4
Solving Equations with Rational Expressions
1. Find all undened values for x:
3
5
− =1
x+2 x
2. Solve:
5
3
−
=8
m m
3. Solve:
z−1 z+3
=
4
3
4. Solve:
k
4
−5=
k−4
k−4
5. Solve:
a+7 a−2 4
−
=
8
3
3
6. Solve:
3
2
7
+
=
x − 1 4x − 4 4
7. Solve:
5x
2
3x
=
−
x2 + 5x + 6 x2 + 2x − 3 x2 + x − 2
8. Solve for F : m =
kF
a
9. Solve for A: h =
2A
B+b
10. Solve for z : 9x +
3
5
=
z
y
7.5
Applications of Rational Expressions
1. In a certain fraction, the denominator is 6 more than the numerator. If 3 is
added to both the numerator and the denomnator, the resulting fraction is
equivalent to 57 . What was the original fraction?
22
2. In the 2002 Winter Olympics, Catriona LeMay Doan of Canada won the 500-m
skating event for women. Her rate was 6.6889 m per sec. What was her time
(to the nearest hundredth of a second)?
3. A boat can go 20 mi against a current in the same time that it can go 60 mi
with the current. The speed of the current is 4 mph. Find the speed of the
boat in still water.
4. Working alone, Hercomer can paint a large room in 8 hr. Mergatroid can paint
the same size room working alone in 6 hr. How long will it take them if they
work together?
5. One pipe can ll a pool in 6 hr, and another can do it in 9 hr. How long would
it take the two pipes working together to ll the pool?
6. If the water in Hercomer's bathtub is turned on full force, it can ll the tub in
24 minutes if the drain is plugged completely. The plug is leaky, though, and
it will drain a full tub in 30 minutes. How long would it take to ll the tub
with the water on full force with the leaky plug?
7.6
Variation
1. If a varies directly as c, and a = 7.2 when c = 2.4, nd a when c = 5.
2. If z varies inversely as w, and z = 10 when w = 0.5, nd z when w = 8.
3. If p varies jointly as q and r2 , and p = 200 when q = 2 and r = 3, nd p when
q = 5 and r = 2.
4. For an object falling from rest in a vacuum, the distance the object falls varies
directly as the square of the time. If an object falls 576 ft in 6 sec, how far
would it fall in 4 sec?
23
Chapter 8
Systems of Equations
8.1
1.
2.
3.
4.
5.
6.
8.2
Solving Linear Systems by Graphing
(
4x = 26 − y
Is the ordered pair (7, -2) a solution of the system:
3x = 29 + 4y
(
−2y = x + 10
Is the ordered pair (6, -8) a solution of the system:
3y = 2x + 30
(
x−y =2
Solve this system by graphing:
x+y =6
(
2x − 3y = −6
Solve this system by graphing:
y = −3x + 2
(
2x − y = 6
Solve this system by graphing:
4x − 2y = 8
(
3x + y = 5
Solve this system by graphing:
6x + 2y = 10
Solving Linear Systems by Substitution
(
x + y = 12
1. Solve by substitution:
y = 3x
24
2. Solve by substitution:
3. Solve by substitution:
4. Solve by substitution:
5. Solve by substitution:
6. Solve by substitution:
7. Solve by substitution:
8.3
(
3x + 2y = 27
x=y+4
(
3x + 5y = 25
x − 2y = −10
(
7x + 4y = 13
x+y =1
(
2x + y = 0
4x − 2y = 2
(
2x + 8y = 3
x = 8 − 4y
(
x
8
5 + 2y = 5
y
7
3x
5 + 2 = − 10
Solving Linear Systems by Elimination
1. Solve by elimination:
2. Solve by elimination:
3. Solve by elimination:
4. Solve by elimination:
5. Solve by elimination:
(
x − y = −2
x + y = 10
(
2x + y = −5
x−y =2
(
4x − 3y = 1
8x = 3 + 6y
(
2x − y = 12
3x + 2y = −3
(
5x + 4y = 12
3x + 5y = 15
25
(
−x + 3y = 4
6. Solve by elimination:
−2x + 6y = 8
(
6x − 2y = −22
7. Solve by elimination:
−3x + 4y = 17
26
Chapter 9
Working with Radicals
9.1
Radical Expressions
1. Find all square roots of 64.
2. Find all square roots of
25
196 .
√
3. Find the square root: − 121
4. Find the square of:
5. Find the square of:
√
√
19
3x2 + 4
6. Classify each root as rational, irrational, or irreal. If rational, give the exact
value. If irrational, give the decimal approximation to 3 decimal places.
(a)
√
25
√
(b) − 300
√
(c) −29
7. Find the real root:
8. Find the real root:
9. Find the real root:
10. Find the real root:
11. Simplify:
√
√
3
√
3
1
−64
√
6
64
q
3
8
27
122
27
12. Simplify:
p
5
(−9)5
√
13. Simplify: 3 x15
14. Find the decimal approximation to 3 decmal places for this root:
9.2
Simplifying Radical Expressions
Assume all variables represent positive numbers.
1. Multiply:
√
14 ·
√
x
√
√
2. Multiply: 3 7x · 3 2y
q
3. Simplify:
3
25
r
4. Simplify:
5. Simplify:
6. Simplify:
7. Simplify:
8. Simplify:
9. Simplify:
10. Simplify:
11. Simplify:
12. Simplify:
13. Simplify:
14. Simplify:
p6
81
r
81
−4 4
x
√
12
√
3
128
√
3
−16
p
144x3 y 9
√
− 100m8 z 4
p
13x7 y 8
p
4
81x12 y 16
r
y 11
36
√
10
x25
15. Rewrite radicals with the same index, then simplify:
28
√
3
4·
√
3
√
3
423
16. Find the unknown length of this right triangle: (3,4,c)
17. Find the distance between the points (6, 13) and (1, 1).
18. Find the distance between the points (−8, 2) and (−4, 1).
9.3
Rational Exponents
1. Simplify: 1691/2
2. Simplify:
64
81
1/2
3. Simplify: 64−3/2
4. Write with radicals: 121/2
5. Write with radicals: (9q) /8 − (2x) /3
5
2
6. Write with radicals: (2y + x) /3
2
7. Simplify by converting to rational exponents:
√
212
√
x· x
√
3 4
t
9. Simplify by converting to rational exponents: √
5 4
t
2
2
x /3
8. Simplify by converting to rational exponents:
10. Simplify and write using positive exponents:
√
3
(x2 ) /3
7
p /5 p /10 p /2
and write using positive exponents:
(p3 )−1/5
−1/4 −3/2 −2
p q
and write using positive exponents:
3−1 p−2 q −2/3
and write using positive exponents: k 1/4 k 3/2 − k 1/2
√
x5
in exponential form: √
x8
p√
in exponential form: 4 3 m
1
11. Simplify
12. Simplify
13. Simplify
14. Simplify
15. Simplify
29
7
1
9.4
Opertions with Radicals
Assume all variables represent positive numbers.
1. Simplify:
2. Simplify:
3. Simplify:
4. Simplify:
5. Simplify:
6. Simplify:
7. Simplify:
√
36 −
√
100
√
√
4
32 + 3 4 2
√
√
5 6 + 2 10
√
√
√
8 2x − 8x + 72x
√
√
2 3 27x − 2 3 8x
√
√
64
8− √
16
r
√
50
2
3
+8· √
9
8
8. Find the perimeter
√
√ of a√triangle with sides of these lengths (dimensions are in
inches): 3 20, 2 45, 75
√
√
9. Multiply, then simplify: 5( 72 − 8)
√
√
√
√
10. Multiply, then simplify: ( 2 − 3)( 2 + 3)
√
√
11. Multiply, then simplify: ( 2 + 1)( 3 − 1)
√
12. Multiply, then simplify: ( 5 + 2)2
√
√
√
√
√
13. Multiply & simplify: (2 + 3 2)(4 − 2 3 2 + 3 4)
√
√
14. Multiply & simplify: (2 x + y)(2 x − y)
15
3
r
7
16. Rationalize the denominator:
2
15. Rationalize the denominator: √
s
17. Rationalize the denominator and simplify:
30
288x7
y9
18. Simplify:
19. Simplify:
r
3
r
4
2
3
2y
z
3
√
4+ 5
√
√
2− 3
√
21. Rationalize the denominator: √
6− 5
√
√
x− y
22. Rationalize the denominator: √
√
x+ y
20. Rationalize the denominator:
23. Rationalize the denominator: √
9.5
Solving Equations with Radicals
1. Solve:
2. Solve:
3.
4.
5.
6.
7.
√
r−2=3
√
3k + 1 − 4 = 0
√
√
Solve: 9a − 4 = 8a − 1
√
Solve: k = k 2 + 4k − 20
√
√
Solve: 3 2x + 5 = 3 6x + 1
√
√
Solve: k + 2 − k − 3 = 1
r
2K
Solve for K : V =
m
8. Solve for L: f =
9.6
1
x+y
1
√
2π LC
Functions with Radicals
No problems assigned.
31
9.7
Complex Numbers
1. Write as a product of i and a simplied radical:
2. Write as a product of i and a simplied radical:
√
√
−169
−48
√
−7 · −15
√
−75
4. Simplify: √
3
3. Simplify:
√
5. Simplify and write in a + bi form: (3 + 2i) + (−4 + 5i)
6. Simplify and write in a + bi form: (5 − i) + (−5 + i)
7. Simplify and write in a + bi form: (−4 + 11i) + (−2 − 4i) + (7 + 6i)
9.8
Multiplication and Division with Complex Numbers
1. Simplify: (3i)(27i)
2. Simplify: (−8i)(−2i)
3. Simplify and write in a + bi form: 5i(−6 + 2i)
4. Simplify and write in a + bi form: (4 + 5i)2
5. Simplify and write in a + bi form: (4 + 3i)(1 − 2i)
6. Simplify and write in a + bi form: 2i(−4 − i)2
7. Simplify and write in a + bi form:
2
1−i
8. Simplify and write in a + bi form:
8i
2 + 2i
9. Simplify and write in a + bi form:
2 − 3i
2 + 3i
10. Simplify and write in a + bi form:
3+i
i
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Chapter 10
Solving Quadratic Equations
10.1
Square Root Property, Completing the Square
Solve the following equations using the square root property.
1. x2 = 81
2. m2 = 32
3. 3n2 − 72 = 0
4. (x − 3)2 = 25
5. (3k + 1)2 = 18
6. Find the complex roots: (r − 5)2 = −3
Solve the following equations by completing the square.
7. x2 − 2x − 24 = 0
8. x2 + 3x − 2 = 0
9. 3w2 − w = 24
10. (x + 2)(x + 1) = 10
11. Find the complex roots: m2 + 4m + 13 = 0
33
10.2
Quadratic Formula
Solve these equations using the quadratic formula: x =
1. m2 − 8m + 15 = 0
2. 4k 2 + 4k − 1 = 0
3. (r − 3)(r + 5) = 2
4. Find the complex roots: x2 − 3x + 6 = 0
5. Find the complex roots: x(3x + 4) = −2
34
−b ±
√
b2 − 4ac
2a