Review for College Mathematics
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Transcript of Review for College Mathematics
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A SELECTION OF PROBLEMS INTENDED TO MAINTAIN STUDENTS ESSENTIAL ALGEBRA
SKILLS
INSTRUCTOR COPY
Review for College Mathematics
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
A moving company charges $46 to move a certain machine 10 miles and $58 to move the same machine 30 miles.
a) Find an equation that defines this relationship if it is linear.b) What will it cost to move the machine 25 miles?c) What is the minimum charge for moving the machine?d) What is the rate for each mile the machine is moved?
Solution:a)b)c)d)
C = 0.6 m + 14$55$40$0.60 per mile
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
The total cost of producing a certain item consists of paying rent for the building and paying a fixed amount per unit for material. The total cost is $250 it 10 units are produced and $330 if 30 units are produced.a) Find the equation that defines this relationship if it is linear.b) What will it cost to produce 100 units?c) How much is paid in rent?d) What is the cost of the material for each unit?
Solution:
a) b)c)d)
C = 4x + 210, where x = number of units produced.$61
0$210$4 per unit
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
If x2 + 2x = 3, demonstrate algebraically which of the following is a possible value for x.a) – 3b) –2c) –1d) 0e) 3
Solution:
a) – 3
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
How many solutions are there to the systems of equations? Solve algebraically.
Solution:Infinite: the lines coincide so the solution is
x 2y6y3x
6 1
, : 2 6x y x y
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
The sum of the square of a number and the product of the number and 3 equals 40.
a) Write an equation to model this situation.b) What are the positive answer(s)?
Solution:a)b)
2 3 40n n n =5
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Two truckers start on a trip at the same time. One heads west at 60 mph while the other heads east at 55 mph. Their CB radios work for a distance up to 500 miles apart. How long will they be able to communicate with each other? Model and solve algebraically.
Solution:4 hr 20 min.
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Solve the system of equations algebraically.
Solution:empty set: The lines are parallel.
3x4y2y3x
4 1
6
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Solve for w: p = 2 ( l + w )Give your answer in two forms, one answer with
a single fraction and the other as a sum or difference of two terms.
Solutions:
22
2
p lw
pw l
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Solve for a:
Solution:
1a
31b
1 3bab
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Solve for x:
Solution:
1x 1y1a
ayxy a
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
The Boosters Club held a spaghetti dinner as a fundraiser. They sold 300 tickets and collected $2200. If an adult’s ticket cost $8.50 and a child’s ticket cost $3.50, how many of each were sold?
Solution: 230 adult tickets 70 children’s tickets
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Admission to a school football game was $2 for students and $3 for nonstudents. How many of each group attended if there were 900 people and $1920 collected? Model and solve algebraically.
Solution:120 nonstudents.780 students.
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Solve algebraically: x2 – 5 = 3x
Solutions:
3 292
x
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MATHEMATICS for COLLEGE READINESS
Bill needs to make 1000 copies before the meeting that starts in 15 minutes. He begins using a machine that makes 30 copies per minute after laminates, a newer machine which copies at 45 pages per minutes become free . Using both machines, will Bill be able to make all the copies before the meeting? Explain algebraically.
Equations and Modeling
Solution:No, it will take almost 17 min to make the copies.
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling
Find the equation of a line with slope of 0 through (6, 10).
Solution:
y = 10
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Function Theory
MATHEMATICS for COLLEGE READINESS
Which of the following are functions?
a. x3 y = 12
b. x2 + y2 = 20
c. x3 + y = 15
d. x (x+2) = 5 + y
3 42x y x e.
f. y(y + 2) = 4 + x
g. x = 4
h. y = 5
Solution: c, d, e, h
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Function Theory
MATHEMATICS for COLLEGE READINESS
If f(x) = x2 – 5, find f(b) – f( c) .
Solution: b2 – c2
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Function Theory
MATHEMATICS for COLLEGE READINESS
If f(x) = 2x2 – x + 3, find f(x + h) .
Solution:
2 22 4 2 3x xh h x h
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Function Theory
MATHEMATICS for COLLEGE READINESS
If f(x) = x2 – 4, find f(a + 5) .
Solution:
a2 +10a+21
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Function Theory
MATHEMATICS for COLLEGE READINESS
If f(x) = x2 – 3 x + 5, find f(x + h) – f(x) .
Solution:2xh + h2 – 3h
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Function Theory
MATHEMATICS for COLLEGE READINESS
If f(x) = 3x4 – 4x3 -2x2 +x - 5, find f(-x).
Solution:
2xh + h2 – 3h
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Expressions with Exponents
MATHEMATICS for COLLEGE READINESS
Simplify:
Solution:
10 2
1
x yy
3
1
y
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MATHEMATICS for COLLEGE READINESS
Simplify. Give the answer with positive exponents only:
Expressions with Exponents
3 0
4
2 3x x
Solution: 4
8x
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MATHEMATICS for COLLEGE READINESS
Simplify. Give the answer with positive exponents only:
Expressions with Exponents
2 3 4
1 2 2
6 .4x y zx y z
5
6
3 .2yx z
Solution:
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MATHEMATICS for COLLEGE READINESS
Multiply
a.
b. c.
d.
e.
Solution:
Expressions with Exponents
8r2s 3r5s4
11r7s5
24r7s5
7 5d. 24r s
5r7s5
5r10s4
24r10s4
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MATHEMATICS for COLLEGE READINESS
Solution:
Expressions with Exponents
1/2 2/5 1/49r s r s
9/10 5/4 9r s
Multiply
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MATHEMATICS for COLLEGE READINESS
Solution:
Expressions with Exponents
1/2 5/4 016r s r s
3/4 16r s
Multiply
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MATHEMATICS for COLLEGE READINESS
If the area of a circle if what is the area of a circle if the radius is 3a?
Expressions with Exponents
2 ,r
Solution: 29A a
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MATHEMATICS for COLLEGE READINESS
If the volume of a sphere is what is the volume of a sphere with radius 3a?
Expressions with Exponents
3 ,43r
Solution: 336V a
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MATHEMATICS for COLLEGE READINESS
If the volume of a cylinder is find the volume of a cylinder with radius 3x and a height that is twice the radius.
Expressions with Exponents
2 ,r h
354V aSolution:
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MATHEMATICS for COLLEGE READINESS
If , then x =
a. 3 + ln t
b.
Exponential and Logarithmic Functions
3xt e
t 3e
t3
c.
d. ln ( t – 3 )
e. ln t – 3
Solution: e. ln t – 3
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MATHEMATICS for COLLEGE READINESS
If 7x = 3, then x =
a.
b.
37
73
log3 7c.
d.
e.
7log 3
log1073
Solution:
3c. log 7
Exponential and Logarithmic Functions
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MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
Simplify
a.b.c.d.
3 2 3 4x x x x
2
2
2
2
21 4
27 4
10 20
10 10
x x
x x
x x
x x
Solution:
a
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MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
Simplify
a.b.c.d.
22 22 2a b a b
2 2
2 2
2 2
2 2
3
5
4 5
4 3
a b
a b
a ab b
a ab b
Solution:
c
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MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
Simplify
230 7 4 5k k k
Solution:
5k2 + 250k -16
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MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
Evaluate the following expression using x = -3.1 and y = 4.23
4x-2 + y3
Solution:
76.103
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MATHEMATICS for COLLEGE READINESS
Factor completely using integer coefficients.
Algebraic Expressions
4 3 2 312 48m n m n
Solution: 12 m2n3 (m2 + 4)
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MATHEMATICS for COLLEGE READINESS
Factor completely using integer coefficients.
Algebraic Expressions
212 8 9 6m m m
Solution:(4m – 3) (3m + 2)
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Which of the following is a factor of
a.b.c.d. not factorable
212 6?x x
3 2
4 3
4 3
x
x
x
Solution:c. (4x + 3)
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Which of the following is a factor of
a.b.c.d.
3 23 18 48 ?x x x
4
2
8
2
x
x
x
x
Solution: b. (x + 2)
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Write in simplest radical form.
9 8180x y
Solution: 4 46 5x y x
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Write in simplest radical form.
4 249x y
Solution:7x2y
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Write in simplest radical form.
2 2 3
Solution: 2 3 2
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Write in simplest radical form.
3 32 7 50
Solution: 23 2
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MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
Perform the indicated operations and reduce answers to lowest terms.
a.
b.
c.
d. 2
2 2 22m n m mnm n m mn n
1
mm n m n
m nm m n
m
m
Solution: d
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MATHEMATICS for COLLEGE READINESS
Solve algebraically.
Quadratic Equations
225 144 0x
Solution:12 12,5 5
x
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MATHEMATICS for COLLEGE READINESS
Solve algebraically.
Quadratic Equations
26 150x
Solution: 5,5x
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MATHEMATICS for COLLEGE READINESS
Solve algebraically.
Quadratic Equations
66 5xx
Solution: 2 3,3 2
x
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MATHEMATICS for COLLEGE READINESS
Solve algebraically.
Quadratic Equations
2 4x x
Solution: 1 172
x
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MATHEMATICS for COLLEGE READINESS
Solve algebraically.
Quadratic Equations
23 2 5x x
Solution:1, 2
3x
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Rational Expressions
MATHEMATICS for COLLEGE READINESS
State the restricted values:
Solution:
12
2
16
xx x
x ≠ –3, 2
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MATHEMATICS for COLLEGE READINESS
Over the set of real numbers, state where the function is undefined:
Rational Expressions
2
2
49
xx
Solution: The expression is undefined over the set of real numbers where -2<x<2 or x = -3 or 3.
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MATHEMATICS for COLLEGE READINESS
Find the product:
Rational Expressions
2 24 5 63 6 4 12x x xx x
Solution: 2212x
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MATHEMATICS for COLLEGE READINESS
Calculate and write the answer in scientific notation.
a.b.c.d.
Rational Expressions
4 2
1 8
85 10 0.03 10
0.5 10 3.2 10
11
11
1
9
1.59375 10
1.59375 10
1.59375 10
1.59375 10
a Solution:
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MATHEMATICS for COLLEGE READINESS
Find the area of a rectangle if the length isand the width is
Rational Expressions
2 2
,x yy
.x yx y
Solution:2( )x y
y
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MATHEMATICS for COLLEGE READINESS
Add and simplify
Rational Expressions
2 2
5 5 .1 1x
x x
Solution: 51x
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MATHEMATICS for COLLEGE READINESS
Simplify
Rational Expressions
32 .25
abba
Solution: 35
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MATHEMATICS for COLLEGE READINESS
Simplify
Rational Expressions
2
3 4.1
xxx
Solution:23 4
1x xx
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MATHEMATICS for COLLEGE READINESS
Simplify:
Rational Expressions
3 3 2 2
2 2x y x yx x
Solution:2 2
2
2 2 2x xy yx xy
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MATHEMATICS for COLLEGE READINESS
Simplify:
a.
b.
c.
d.
Rational Expressions
63
x x yx y
63x
18
x x y x y
2x
2x x yx y
Solution: b
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MATHEMATICS for COLLEGE READINESS
Simplify:
a.
b.
c.
d.
Rational Expressions
2
2
3 2
2 4 1
x x
x x
3
2 1x
3
2 1x
3 22 2 1
xx x
32
Solution: a
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MATHEMATICS for COLLEGE READINESS
Simplify:
a. c.
b. d.
Rational Expressions
2
2
5 5 12 4 4
v vv v v
5
2 1v v
5 1 21 1
v vv v
5 5 21 1
v vv v
5 21
vv
Solution: d
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MATHEMATICS for COLLEGE READINESS
What values for x are not allowed?
a. c.
b. d.
Rational Expressions
2
3 4 52 8
x xx x
50,4
4,2
54,0, ,24
54, ,24
Solution: b
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Miscellaneous
Which of the following sets of numbers could NOT be the measures of the sides of a right triangle?
a. c.
b. d.
3,7, 10
8,5 3, 11
6,8,10
10,24,26
Solution: a
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Miscellaneous
Find the length of the hypotenuse of the right triangle whose legs are 5 and 11.
Solution: 6
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MATHEMATICS for COLLEGE READINESS
Miscellaneous
Find the distance between (-3,5) and (-7,8).
Solution: 5
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Miscellaneous
MATHEMATICS for COLLEGE READINESS
Find the coordinates of the point midway between (-5,2) and (3,-10).
a. (-4,6)b. (-1,-4)c. (1,6)d. (-4,-1)
Solution:
b. (-1, -4)
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Miscellaneous
MATHEMATICS for COLLEGE READINESS
Find y in the diagram below.
a. 24b. 18c. 4d. 8
Solution:
12
69
y
a
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MATHEMATICS for COLLEGE READINESS
Miscellaneous
MATHEMATICS for COLLEGE READINESS
Find y in the diagram below.
Solution:
y
812
15
7.5
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MATHEMATICS for COLLEGE READINESS
c.
d.
Variation
Find an equation of variation where y varies inversely as and when
a.
b. ,x 10y 3.5x
35y x3yx
6.5y x35yx
Solution: d
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MATHEMATICS for COLLEGE READINESS
Variation
Y varies directly as the cube root of x. Y is 30 when x is 1. Find Y when x is 27.
a. 10b. 90c. 15d. 30
Solution:
b. 90
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MATHEMATICS for COLLEGE READINESS
The number of hours (H) it takes to do the job in inversely proportional to the number of people (P) working on it. It takes 4 people 12 hours to do the job. How long would it take 6 people to do the job?
a. 18 hoursb. 10 hoursc. 8.5 hoursd. 8 hours
Variation
Solution: d
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MATHEMATICS for COLLEGE READINESS
Equations and Modeling Slides 1 – 15Function Theory Slides 16 – 21Expressions with Exponents Slides 22 – 30Exponential and Logarithmic Functions Slides 31 – 32Simplifying Expressions Slides 33 – 36Algebraic Expressions Slides 37 – 45Quadratic Equations Slides 45 – 50Rational Expressions Slides 51 – 63Miscellaneous Slides 64 – 69VariationSlides 70 – 72
Content Guide