Mathematics Placement Review

Transcription

Mathematics
Placement
Review
John A. Logan College
TABLE OF CONTENTS
Contents
Letter from the Mathematics Department Chair .......................................................................................................3
Mathematics Sequence Chart ....................................................................................................................................4
Required Math Entry Testing......................................................................................................................................5
Frequently Asked Questions.......................................................................................................................................5
1.
What types of questions are on the test? ......................................................................................................5
2.
Who must take the ASSET or COMPASS Assessment before registration? ...................................................6
3.
Who is exempt? ..............................................................................................................................................6
4.
Are there arrangements for students who have a disability or learning problem? .......................................6
5.
Is there a fee charged for the test? ................................................................................................................6
6.
When and where can I take the test? ............................................................................................................6
7.
What does a student need to bring on the day of testing? ...........................................................................6
8.
Where and when can I get my test results and register?...............................................................................7
How to Use this Guide ................................................................................................................................................7
References to Other Sources ......................................................................................................................................7
STUDENT SUCCESS CENTER ...............................................................................................................................7
TEXTBOOKS ........................................................................................................................................................7
VIDEOS ...............................................................................................................................................................7
MAT 051 – Pre-Algebra ..............................................................................................................................................8
Pre-Algebra Review Questions ...............................................................................................................................9
A.
Whole Numbers and Whole Number Concepts.........................................................................................9
B.
Fractions and Mixed Numbers ...................................................................................................................9
C.
Decimals .................................................................................................................................................. 10
D.
Ratios and Proportions............................................................................................................................ 10
1
E.
Percent .................................................................................................................................................... 11
F.
Simplifying Algebraic Expressions and Solving Linear Equations ............................................................ 11
G.
Signed Numbers ...................................................................................................................................... 11
H.
Polynomials ............................................................................................................................................. 12
MAT 051 Review Answers ................................................................................................................................... 12
MAT 052 – Beginning Algebra ................................................................................................................................. 14
Beginning Algebra Review Questions ................................................................................................................. 15
A.
Exponents, Square Roots, Order of Operations, Properties of Real Numbers, Algebraic Expressions.. 15
B. Addition and Multiplication Principles, Solving Linear Equations, Word Problems, Literal Equations,
Linear Inequalities........................................................................................................................................... 15
C.
Exponents, Scientific Notation, Polynomials, Multiplication of Polynomials, Products of Binomials .... 16
D.
Factor, Greatest Common Factors, Factoring Trinomials ....................................................................... 17
E. Rectangular Coordinate System, Linear Equations in Two Variables, Graphing Linear Equations, Slope
of a Line, Equation of a Line, Graphing Linear Inequalities ............................................................................ 17
F. Solving Linear Systems of Two Equations by Graphing, Solving Linear Systems by Substitution,
Applications of Linear Systems ....................................................................................................................... 19
MAT 062 – Intermediate Algebra ............................................................................................................................ 22
Intermediate Algebra Review Questions ............................................................................................................ 23
A.
Function Notation, Graphs of Common Algebraic Functions ................................................................. 23
B.
Review Polynomial Operations, Factoring Polynomials, and Solving Equations by Factoring ............... 23
C.
Rational Exponents, Radical Expressions, Complex Numbers, and Quadratic Equations....................... 23
D.
Rational Expressions and Rational Equations ......................................................................................... 24
E.
Logarithmic and Exponential Expressions / Equations ........................................................................... 25
2
Letter from the Mathematics Department Chair
3
Mathematics Sequence Chart
4
Required Math Entry Testing
Purpose: John A. Logan College is committed to helping students achieve success in their course work. In
this effort, the college has designed a mathematics placement program to aid students in selecting the
most appropriate mathematics course, while taking into account their widely varied mathematics
backgrounds.
Frequently Asked Questions
What is COMPASS or ASSET? COMPASS and ASSET are assessment tools of academic skills. They are used as a
means to gather information regarding the background, educational needs and goals of students. A low
COMPASS or ASSET score will not keep anyone from attending college. By participating in a COMPASS or ASSET
Test, students may build a solid plan for success in the education options they choose to pursue at John A. Logan
College. Special preparations are not necessary as the assessments are designed to measure skills acquired
through high school, work experience or previous college work.
1. What types of questions are on the test?
•
For practice, visit the ACT website at www.act.org/compass for sample COMPASS test items and
www.act.org/asset for ASSET sample test items. Look for sample questions in Pre-algebra and
Algebra.
COMPASS TEST
•
COMPASS stands for Computerized Adaptive Placement Assessment & Support System. The
COMPASS assessments are a series of un-timed adaptive tests in the areas of Reading, Writing
and Mathematics. They are designed to select the range of questions that is appropriate for the
test takers ability level. This process of computer adaptive tests is typically shorter than paper
and pencil tests and test results are received immediately after testing.
•
The COMPASS Mathematics Assessment: You will be given the level to begin the Mathematics
Test based upon the level of math that you completed in high school. You will be given scratch
paper, a calculator and pencils to work out multiple choice problems for this test.
•
COMPASS math tests are adaptive. This means that the test will base each new question on how
well the student did on the preceding question. If the student keeps answering questions
correctly, they will end up with trigonometry questions. On the other hand, if the student misses
a couple of questions, then they will be given easier questions. Once the computer finds a level
that the student can reach but not go beyond, it assigns a placement score. This is not a test to
pass or fail—it is an instrument to help select the best level for the student to have a reasonable
chance of success at the time they take the test—not too low and not too high.
5
ASSET TEST
•
ASSET stands for Assessment of Skills for Successful Entry and Transfer in written (not computer)
form. The ASSET Test consists of one Reading, one Writing, and four Mathematics Tests ranging
from Numerical Skills through College Algebra. Students only take one of the four Math Tests.
Each of these tests requires 25 minutes. Normally, these tests are only administered in the high
schools and extension centers.
•
The ASSET Mathematics Assessment: You will be able to select the level to begin the
Mathematics Test based upon the level of Math that you completed in high school. You will be
given scratch paper, a calculator and pencils to work out multiple choice problems for this test.
2. Who must take the ASSET or COMPASS Assessment before registration?
•
All first time college students.
•
Transfer students without a college transcript.
•
Transfer students with a college transcript that does not indicate a successful completion of a
college level Math course.
3. Who is exempt?
•
Students who have Math scores from ASSET or COMPASS tests administered at a high school.
•
Students with passing grades in college-level Math courses.
•
Students who have undergone ASSET/COMPASS assessment at another college or university and
have transferred their scores to John A. Logan.
•
Students enrolling only in ALH 101, ALH 102, ALH 103, TRT 152, PED 100 or 103, EMT 111.
4. Are there arrangements for students who have a disability or learning
problem?
Students needing any reasonable accommodations because of a disability or learning problem should
contact Angela Calcaterra, Special Needs Coordinator, or Christy McBride, Director of Student Academic
Success Services.
5. Is there a fee charged for the test?
There is no fee for first time testing. Retesting will be $10.00 each time you retest.
6. When and where can I take the test?
Assessment tests can be taken in the Assessment Office, Room C205, or at a John A. Logan Extension
Center at a scheduled time; or on a walk-in basis during late registration.
7. What does a student need to bring on the day of testing?
A photo I.D., Social Security Number and a pencil are required. Open book or persons not testing are not
allowed in the testing room. All personal belongings (purses, cell phones, pagers, notebooks, etc.) are
not allowed in the testing room.
6
8. Where and when can I get my test results and register?
The proctor will give you your results immediately after the test if you take the computerized COMPASS
Test at the Assessment Office. If you take the written ASSET Test at an Extension Center, your results will
be mailed to you a few days after the test. After receiving your results, you may contact the Admissions
Office to register with an advisor.
How to Use this Guide
The student must decide what course they think is the highest level that they have already mastered,
and then review this material. For example, if this course is Beginning Algebra MAT 052, then review
the learning outcomes covered in MAT 052. If the student feels comfortable with all these topics, then
they should try the sample questions for this course. If they are not comfortable, then they may want to
review some before trying the sample questions. The student may use the answer key to determine
how well they did on the sample questions.
One way to review this material is to work through the chapter reviews or mastery tests at the end of
each chapter of the appropriate level textbooks. Other references are given below.
References to Other Sources
STUDENT SUCCESS CENTER
The Student Success Center's tutoring program is certified through the College Reading and Learning
Association (CRLA). The center offers tutoring to John A. Logan College students free of charge. Tutoring
is available for both transfer and vocational courses. The center offers two types of tutoring: one-onone tutoring and help-room tutoring. One-on-one tutoring is by appointment only and offers students
individualized attention. Our help rooms are less formal settings where students may drop in for
assistance any time during the hours of operation. The math help room is located in room C-218.
Requests for tutoring are made in room C-219.
TEXTBOOKS
Textbooks for all developmental courses are available on reserve in the Library.
1. MAT 051: Prealgebra, Fifth Edition, K. Elayn Martin-Gay; Prentice Hall, Upper Saddle River, New
Jersey, 2008.
2. MAT 052: Introductory Algebra, 10th Edition, Marvin L. Bittinger; Addison-Wesley Publishing
Company, Boston, Massachusetts, 2007.
3. MAT 062: Intermediate Algebra, 10th Edition, Marvin L. Bittinger; Addison-Wesley Publishing
Company, Reading, Massachusetts, 2007.
VIDEOS
MAT 051, MAT 052, MAT 062: Videotapes and DVD’s to accompany textbooks are available in the
Learning Lab.
7
MAT 051 – Pre-Algebra
Course Description
MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra.
The course is not designed for college transfer. The student must earn a grade of “C” or higher in order
to enroll in MAT 052. In addition, the student will need to enroll in MAT 052, MAT 061, and MAT 062
before progressing to transfer-level mathematics courses. This course will cover the integers, fractions
and decimals; ratio, proportion, and percent; prime numbers, factoring; exponents; and solving
equations.
Prerequisites: None
Course Objectives
At the completion of this course, the student should be able to:
1. Perform arithmetic operations with integers, rational numbers (fractional, decimal, and mixed
number forms), real numbers, algebraic expressions and polynomials.
2. Solve linear equations and inequalities in one variable.
3. Solve and graph linear equations in two variables.
4. Learn and apply the rules (laws) of exponents.
5. Perform geometric concepts of perimeter and area.
6. Understand the basic concepts of square roots.
7. Find the least common multiple (LCM) using the prime factorization method applying concepts
of prime numbers and divisibility tests.
8. Convert numbers to their fractional, decimal, and percent forms and apply these concepts to solve
basic percent problems.
9. Apply the order of operations to numerical and algebraic expressions.
10. Use the Pythagorean Theorem to solve problems involving right triangles.
11. Use algebra as a strategy to solve word problems.
12. Solve real-life applications of percent problems.
The purpose of the following questions is to help the student gauge their readiness on
the topics taught in MAT 051. (The purpose is not to teach this material.)
8
Pre-Algebra Review Questions
A.
Whole Numbers and Whole Number Concepts
Simplify problems 1 – 5 without a calculator.
7 + 2 (3 − 9)
1.
−10
2.
3.
18 ÷ ( 4 − 7 ) ⋅ 2
4.
5.
18 + 33 − 30 ÷ 6
6.
Find the perimeter of a triangle with each side measuring 7 cm.
7.
Find the volume of a box with length 15 m, width 12 m, and height 3 m.
8.
Find the area of a triangle whose base is 34 inches and height is 12 inches.
9.
Find the fencing costs of a square lot that is 85 meters on each side. The fencing costs
$14 per meter.
10.
Find the perimeter of Figure 1.
11.
Find the area of Figure 1.
Figure 1.
−2(5) + 4(−3)
−8 + 10
4 ft
2 ft
16 ft
3 ft
20 ft
B.
Fractions and Mixed Numbers
1.
1
5
1 +1
4
8
4.
3
5 22
⋅
11 21 25
7.
1 2 1
⋅ + 
4 3 2
8.
Change
⋅
79
6
2.
5  1
−− 
6  4
3.
1
2
4 −1
5
3
5.
7
1
1 ÷6
8
4
6.
6 3 3 
÷ − 
5  5 10 
2
to a mixed number.
9. Change 16.15 to a mixed number.
9
C.
Decimals
0.006 ( 0.09 )
2.
( 0.3)
3.
34.8 ÷ 4
4.
1.1 56.375
5.
.09 + 0.5 ( 6 + 0.02 )
6.
3 4 − 2 36
7.
Round 5,525.2178 to the nearest:
b.
thousandth
Change
a.
D.
2
1.
hundred
3
to a decimal.
8
8.
Change 0.275 to a fraction.
9.
10.
11 yd = ? in
11.
12.
If one cup of regular coffee contains 96 milligrams of caffeine, how much caffeine is
contained in 2.4 cups of coffee?
13.
If Richard makes $6.50 per hour at the video store and works 22 hours in one week, how
much money will he make (before deductions)?
14.
Taryn goes shopping for new clothes and buys two sweaters for $19.98 each, 1 pair of
jeans for $35.25, and three pairs of socks for $1.45 each pair. If Taryn gives the cashier
two $50.00 bills, find the amount of change she will receive.
15.
A consumer watchdog group priced a can of tomato soup at five different grocery
stores. They found the following prices: $.89, $1.39, $1.65, and $1.89. What is the
average selling price of a can of soup? Round to the nearest cent.
66 mi
1 hr
=
? ft
1 sec
Ratios and Proportions
1.
Write a ratio to compare 42 miles to 15 miles.
2.
If Kristen is paid $202.32 for 40 hours of work, how much will she be paid for 12 hours of
work?
3.
Solve the proportion:
4.
6.2 12.4
=
1.7
x
The ratio of songs to commercials on a local radio station is 3 to 8. How many
commercials would you expect if you heard 9 songs played?
10
E.
F.
Percent
1.
A 35 mm camera sells for $329. During a sale, a photo store discounts the price $79.
What is the percent of the discount?
2.
One hundred five percent of what number is 50.715?
3.
The Gill family saves 15.5% of their monthly income. If their monthly income is
$4892.50, how much do they save each month?
4.
Write 35.5 as a percent.
5.
What number is 9.2% of 1803?
6.
Write 0.035% as a decimal.
Simplifying Algebraic Expressions and Solving Linear Equations
1.
Write the following in symbols: the quotient of 10 and x is y.
2.
Find the value of the expression 7 – 2x when x = 6.
3.
Use the equation 4x + 3y = 12 to find y when x =
3
2
.
Solve problems 4 – 10.
G.
4.
x + 3 = –5
5.
7.
15x + 1 = –4x + 20
8.
10.
2.5x – 3.7 = 18.8
2
3
x = 12
3(x – 2) + 1 = 4
6.
9.
–3x + 7 = –5
0.6 x = −0.12
Signed Numbers
Simplify problems 1 – 10 without the use of a calculator.
1.
7 + (−10)
2.
−3 + (−12)
3.
6− 4 −8+5
4.
−5 − (−11)
5.
−3(8)
6.
−5(−3)
7.
(7 − 4)(3 − 5)
8.
9.
7−4
2−5
10.
−4 + 2( − 3)
5−3
11.
3(−5) + 5(3 − 7)
Find the average of −16, − 28, 14, − 10, and 30 .
11
H.
Polynomials
Simplify problems 1 – 10.
2.
6 x ⋅ 5x
(x )
4.
(2x2y3)3
5.
(6x2 + 2x – 5) + (2x2 – x – 1)
6.
(2x2 + 9x – 7) – (2x2 + x – 5)
7.
5x + 6x – 3 + 7
8.
9.
4(x + 3) + 3(2x – 9)
10.
1.
4(2x – 3)
3.
6 3
3(2x – 1)
10 y 2
3x
÷
5y
6x
MAT 051 Review Answers
A:
3. 2
1. 10
2. –5
4.
6. –6
8
15
7.a. 5500
2
35
7.b. 5525.218
3. –12
4. –11
3
5.
10
5. 40
6. 4
6. 21 cm
5
7.
12
8. 13
204 in 2
9. $4,760
9. 16
10. 50 ft
11. 68 ft
11
40
9. .375
10. 396
7. 540 m 3
8.
8.
11. 96.8 ft/sec
1
6
12. 230.4 mg caffeine
3
20
14. $20.44 change
13. $143
15. $1.46
2
C:
1. .00054
B:
D:
2. .09
1. 2
2.
7
8
13
12
3. 8.7
4. 51.25
1.
14
5
2. $60.70 or $60.72
5. 3.1
12
3. x = 3.4
5. x = 18
9. –1
4. x = 24 commercials
6. x = 4
10. –5
7. x = 1
11. –2
E:
1. x = 24%
8. x = 3
2. x = 48.3
9. x = −0.2
1. 8 x − 12
3. x = $758.34
10. x = 9
2. 30 x 2
4. 3550%
3. x18
G:
5. x = 165.876
1. –3
6. .00035
2. –15
F:
3. –1
1.
10
=y
x
2. –5
3. y = 2
4. x = −8
H:
4. 8 x 6 y 9
5. 8 x 2 + x − 6
6. 8 x − 2
4. 6
7. 11x + 4
5. –24
8. 6 x − 3
6. 15
9. 10 x − 15
7. –6
10. 4 y
8. –35
13
MAT 052 – Beginning Algebra
Course Description
MAT 052 is designed for students with less than one year of high school algebra. It is not designed for
college transfer. The student must earn a “C” or higher in order to enroll in MAT 062. In addition, the
student will need to successfully complete MAT 061 (or equivalent) and MAT 062 before progressing to
transfer-level mathematics courses. This course covers the properties of real numbers; solving
equations and inequalities in one variable; operations with polynomials in one variable as well as an
introduction to polynomials in several variables; factoring polynomials leading to solving quadratic
equations; graphing linear equations in two variables, slope, and writing equations of lines; solving
systems of linear equations; and radical notation, including solving radical equations.
Course Objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Perform arithmetic operations with real numbers, algebraic expressions, polynomials, rational
expressions, and radical expressions.
Solve linear, rational, and radical equations.
Solve linear inequalities in one or two variables.
Be able to factor polynomials, specifically binomials and trinomials, as well as be able to
determine those polynomials that cannot be factored.
Use factoring to solve quadratic equations.
Use graphs to depict solutions to linear equations and inequalities in one and two variables.
Use slope, slope-intercept form and/or point-slope form to find equations of lines and use slope
and y-intercept to determine if lines are parallel or perpendicular.
Solve systems of two linear equations in two variables by graphing, by the substitution method,
and by the elimination method.
Apply problem solving strategies and algebraic solutions to problems involving linear
expressions, equations and inequalities, rational equations, radical equations, and systems of two
equations in two variables.
Translate from a verbal statement to an algebraic statement.
Determine which law of exponents to apply, when to apply it, and how to apply it when
simplifying exponential expressions.
Determine if a problem involves direct or inverse variation and solve accordingly.
Be able to apply problem solving strategies to real life problems and problem situations.
Be able to interpret graphs and charges as they apply to real life situations.
14
Beginning Algebra Review Questions
A. Exponents, Square Roots, Order of Operations, Properties of Real Numbers, Algebraic
Expressions
 
 4
5
7
1. Simplify: − + −  .
 
8
2. Simplify: −7 − −9 .
3. Simplify: −5 (−3)(2)(−4) .
4. Evaluate: −52 + 2 − 3 ⋅ 4 .
5. Evaluate: 4 − 2 3 − ( 2 −1) .
16 + 22
=
6.
23 − 3
7. Write “four less than the square root of the product of twenty-five and four” as a numerical expression,
then give the expression’s value.
8. Evaluate:
9. Evaluate:
x 2 − 4 x + 4 for x = −5 .
x+ y
for x = 6 and y = −2 .
x− y
10. Write as an equation, “ If a number is decreased by five, the result is twice the sum of the number and
ten.” Let x represent the unknown number.
11. Simplify the given expression: 4 y − 2  y − 5 (6 − y ) .
B. Addition and Multiplication Principles, Solving Linear Equations, Word Problems, Literal
Equations, Linear Inequalities
1. Solve for x:
x 1 x +1
− =
.
3 6
9
2. Solve: 2 ( x + 1) − ( x − 7) = 3( x −1) − 2 .
3. Solve:
3a + 5
11
+5 = .
3
3
4. Solve the following formula for C: F =
9
C + 32 .
5
15
5. Solve and express the answer in set-builder notation:
1
2 5
x + < x +1 .
2
3 6
C. Exponents, Scientific Notation, Polynomials, Multiplication of Polynomials, Products of
Binomials
Write all answers using positive exponents.
2
1. Simplify:
(−3a 2b5 ) .
2. Simplify:
30c18 d 12
.
12c 6 d 4
3.
Simplify:
4. Simplify:
9 y13
3
(3 y 4 )
.
−3
( 2 a −2 b 4 )
.
5. Evaluate: −5−2 .
6. Find the product to the nearest tenth:
(5.87 ⋅10−6 )(6.73⋅107 ) .
7. Express as a decimal number: 2.37 ×104 .
−1
1
8. Simplify: 6 −   .
 4 
−1
−3
 2 x 2 y 4 
 .
9. Simplify: 
 3 xy−2 
0
10. Simplify: −30 + 5 (−3) .
11. Write in scientific notation: 0.0030417
12. A rectangle has length y and width x. A square with sides x is drawn inside the rectangle. Write a
polynomial that describes the area inside the rectangle and outside the square.
13. Subtract −2 x 2 + 3 x − 7 from 5 x 2 − x + 1 .
14. Multiply:
(2 x − 3)(4 x + 5) .
15. Multiply:
(3x − 7)
2
.
16
16. Multiply: (8 x 2 − 5 y 2 ) (3 x 2 − 2 xy + y 2 ) .
D. Factor, Greatest Common Factors, Factoring Trinomials
1. Factor completely: 6 x3 − 33 x 2 + 42 x .
2. Factor completely: 2 x 3 + 12 − 3 x − 8 x 2 .
3. Factor completely: x 2 − 8 xy + 16 y 2 .
4. Factor completely: x 2 − 3 x −18 .
5. Factor completely: −x 2 + 3 x + 28 .
6. Solve algebraically: (2 x − 5) (3 x + 7) = 0 .
7. Solve algebraically:
2
( x + 1) = 37 − 3x .
E. Rectangular Coordinate System, Linear Equations in Two Variables, Graphing Linear Equations,
Slope of a Line, Equation of a Line, Graphing Linear Inequalities
1. Which quadrant or axis contains the point (a, b) if a < 0 and b < 0 ?
2. Sketch the graph y = −2 x + 4 .
y
x
3. Given the line 2 x − 3 y = 6 , state the slope, x-intercept, and y-intercept.
17
4. From the graph of this line, determine the slope.
y
x
5. The slope of the line L1 is −2 . Determine the slope of L2 so that L2 is perpendicular to L1 and L2 is
parallel to L1.
6. According to Hooke’s Law, the force F in pounds required to stretch a spring x inches is directly
proportional to x. If 20 pounds of force stretches a spring three inches then what is the force required
to stretch it five inches?
7. Write the equation of the line through the points (−2,1) and (2,3) . Write the answer in slopeintercept form.
8. Graph the inequality y > −3 x + 5 .
y
x
18
9. Determine which of the following ordered pairs are a solution to 5 x − 3 y ≤ 30 .
b. (5, −3)
a. (5,3)
c. (3, −5)
10. Write an inequality to represent the shaded portion of the graph below:
8
y
6
4
2
x
0
-8
-6
-4
-2
0
2
4
6
8
-2
-4
-6
-8
F.
Solving Linear Systems of Two Equations by Graphing, Solving Linear Systems by Substitution,
Applications of Linear Systems
1. Solve: 6 x − 4 y = 42
10 x + 8 y = 26
2. The total receipts for a concert were $650. Adult tickets cost $3, and children’s tickets cost $2. Seventyfive more adult tickets were sold than children’s tickets. How many of each type were sold?
3. One number is 3 less than another number. If one-fifth of the larger is added to one-half the smaller
number, the result is 9. What is the smaller number?
4. Two cars departed from the same intersection, one headed east, the other west. The first car traveled at
78 mph and the second traveled at 64 mph. How many hours will pass before they are 497 miles apart?
5. After a 10% raise, your new monthly salary is $1,000. What was your old monthly salary?
6. How many gallons of a 60% acidic solution must be added to 15 gallons of a 30% solution to obtain a
51% solution?
7. You and a friend started jogging at the same location but headed in opposite directions. After 45
minutes, you were 12 miles apart. If your rate was 6 mph faster than that of your friend, what was your
rate?
19
A:
C:
1. −
D:
1. 9a 4b10
19
8
5c12 d 8
2
2. –2
2.
3. –120
y
3.
3
4. –35
4.
6. 4
25 ⋅ 4 − 4, 6
7.
8. 7
10.
2.
( 2x
3.
( x − 4y)
4.
( x − 6 )( x + 3)
2
− 3) ( x − 4 )
2
6
5. 0
9.
1. 3 x ( 2 x − 7 )( x − 2 )
a
8b12
5. −
5. − ( x − 7 )( x + 4 )
1
25
6.
6. 395.1
1
2
7. 23,700
( x − 5 ) = 2 ( x + 10 )
8. −
23
6
5
7
,−
2
3
7. −9 , 4
E:
1. III
2.
11. −8 y + 60
9.
27
8 x3 y18
y
10. 4
B:
11. 3.0417 ⋅10−3
5
1. x =
4
12. xy − x 2
2. x = 7
13. 7 x 2 − 4 x + 8
3. a = −3
14. 8 x 2 − 2 x − 15
4. C =
5.
5
( F − 32 )
9
{ x | x > −1}
x
3. slope =
2
, x-intercept =
3
15. 9 x 2 − 42 x + 49
(3,0), y-intercept = ( 0, −2 )
16.
24 x 4 − 16 x3 y − 7 x 2 y 2 + 10 xy 3 − 5 y 4
20
4. −
5.a.
F:
5
2
1. ( 5, −3)
1
2
2. 85 children’s tickets ,
160 adults’ tickets
5.b. –2
6.
100
pounds
3
7. y =
1
x+2
2
3. 12
4. 3.5 hrs
5. $909.09
6. 35 gallons
7. 11 miles per hour
8.
9. a, c
10. y ≤ x − 2
21
MAT 062 – Intermediate Algebra
Course Description
MAT 062 is designed for students with less than two years of high school algebra. It is not accepted for
college transfer. Students must earn a grade of “C” or higher in order to progress to transfer-level
mathematics courses. This course will cover linear equations and inequalities; graphs of equations – both
linear and nonlinear equations; slope and equations of lines; systems of equations; exponents; operations
with and factoring polynomials; operations with rational expressions and solving rational equations;
operations with radical expressions and solving radical equations; complex numbers; functions and graphs;
quadratic equations and graphs; exponential and logarithmic functions. The Texas Instrument TI-83 or T-84
graphing calculator or a graphing calculator approved by the instructor is recommended for this course.
Course Objectives
1. Perform arithmetic operations with real numbers, complex numbers, algebraic expressions,
polynomials, rational expressions, and radical expressions.
2. Solve linear, rational, radical, absolute value, logarithmic and exponential equations in one and two
variables.
3. Solve linear inequalities and compound inequalities in one or two variables.
4. Factor polynomials, specifically binomials and trinomials, as well as be able to determine those
polynomials that cannot be factored.
5. Use various methods to solve quadratic equations, including the quadratic formula.
6. Use graphs to depict solutions to linear equations and inequalities in one and two variables, as well as
systems of equations and inequalities in two unknowns.
7. Use slope, slope-intercept form and/or point-slope form to find equations of lines and use slope and yintercept to determine if lines are parallel or perpendicular.
8. Solve systems of equations in two and three variables by substitution or elimination.
9. Use laws of logarithms to simplify logarithmic and exponential expressions.
10. Graph conic sections, exponential and logarithmic equations.
11. Apply problem solving strategies and algebraic solutions to problems involving linear expressions,
equations and inequalities, rational equations, radical equations, and systems of equations.
12. Determine the best method for solving systems of equations and then apply that method correctly.
13. Determine which law of exponents to apply, when to apply it, and how to apply it when simplifying
exponential expressions.
14. Determine if a problem involves direct, inverse and joint variation and solve accordingly.
15. Apply problem solving strategies to real life problems and problem situations.
16. Interpret graphs and charts as they apply to real life situations.
17. Graph simple functions, and given a graph, determine whether it is a graph of a function.
22
Intermediate Algebra Review Questions
A. Function Notation, Graphs of Common Algebraic Functions
Determine the domain of each function.
x+2
2 x − 12
1.
f ( x) =
2.
f ( x) = 3 − 4x
Answer the following.
3. Let f ( x ) = 2 x 2 + 5 x − 12 . Determine the value of f ( 2 ) .
4. Find the slope of the line 2 x + 3 y = 7 .
5. Write the equation of the line connecting the points ( −4,3) and (5, −7) .
6. Find the x- and y-intercepts of the line 4 x − 5 y = 40 .
7. Factor out the Greatest Common Factor of 3x ( x − 5 y ) − 2 y ( x − 5 y ) .
8. Factor out the Greatest Common Factor of 24 x 5 y 7 − 8 x 8 y 3 + 12 x 2 y 9 .
B. Review Polynomial Operations, Factoring Polynomials, and Solving Equations by Factoring
Completely factor each polynomial.
1. 144 x 4 − 25
2.
3. 5 x 2 − 20 y 2
2 x 2 + 7 x − 30
4.
2 xm − 3 ym + 2 xn − 3 yn
Solve.
5.
( x − 12 )( x + 1) = −40
C. Rational Exponents, Radical Expressions, Complex Numbers, and Quadratic Equations
Simplify the following.
−4
1.
( −8 )
2.
(
81x 5 y
3.
(
2 2 −5 3 2 2 +5 3
4
3
4.
4
3
)
3
4
)(
5.
)
3
7 −2
( 3 − 2i )( 5 + i )
6. i 99
23
7.
7
3 + 2i
8.
( −3a k )( 6a k )
1
4
2
3
1
2
4
3
Solve for x.
9.
x 2 − 4 x + 9 − x = −1
10.
Write the quadratic formula.
11.
Find the solutions to the nearest tenth: 5 x 2 − 7 x − 2 = 0 .
12.
Find the exact solutions: 3 x 2 + 2 x + 1 = 0
13.
Calculate the distance between the points ( −4,3) and ( 2, −3) .
14.
Solve the inequality.
− x 2 < 15 − 8 x
Solve the following application.
15.
Two airplanes depart simultaneously from an airport. One flies due south; the other flies due east at a
rate 10 miles per hour faster than that of the first airplane. After 2 hours radar indicates that the
airplanes are 580 miles apart. What is the ground speed of each airplane?
D. Rational Expressions and Rational Equations
Simplify the following.
1.
3 − x2
x2 − 3
3.
x + 2 x3 − 8 x 2 + 2 x + 4
÷
⋅
x + 3 x2 − 9
x−3
2.
15 x 2 − 15 y 2
5x3 y 2 − 5x 2 y 3
÷
11x 3 + 11 y 3 4 x 3 − 4 x 2 y + 4 xy 2
4.
12 x −2 − x −1 − 1
8 x −2 + 6 x −1 + 1
6.
15
x+4 x+3
+
=
x + 5x x + 5
x
Solve for x.
5.
4
3
8
−
+
=0
x + 7 x + 10 x + 2 x + 5
2
2
24
Solve the following applications.
7. One pipe can fill a cooling tank in 3 hours less that a second pipe can. If the two pipes together can fill
seven-ninths of the tank in 4 hours, how many hours would it take each pipe alone to fill the tank?
8. A boat completes a trip from Riverton to Clear Creek and back each weekday. The distance one way is
60 miles, and the speed of the current in the river is 4 miles per hour. If the round trip takes 8 hours,
determine the speed of the boat in still water.
9. The sum of the reciprocals of two consecutive even integers is
13
. Find these integers.
84
10. A window washer can wash the windows on one side of a building in 112 working hours. With an
assistant he can do the job in 63 hours. How many hours would the assistant need to do the job alone?
11. If y is directly proportional to the square of x, and y is 37.5 when x is 5, find y when x is 12.
12. If y varies inversely as x, and y is 36 when x is 22, find y when x is 33.
E. Logarithmic and Exponential Expressions / Equations
1. Convert log 5 125 = x to an exponential expression.
2. Convert 251/ 2 = 5 to a logarithmic equation.
3. Find log 2 32 .
x2
in terms of logarithms of x, y, and z .
4. Express log
y2 z
3
5. Solve 5 x − 2 = 25 .
6. Solve 2 x = 3 .
7. Solve log 3 81 = x .
8. Solve log 4 x + log 4 ( x − 6 ) = 2
25
A:
4.
7+2
1.
( −∞, 6) ∪ ( 6, ∞ )
5. 17 − 7i
3

 −∞, 
4

6.
−i
2.
7.
21 14
− i
13 13
3. 6
2
4. −
3
5.
−10
13
y=
x−
9
9
3
4
8.
−18a k 2
9.
x=4
−b ± b 2 − 4ac
10. x =
2a
6. x-intercept: (10, 0 )
y-intercept: ( 0, − 8)
11. x = 1.6, x = −0.2
7.
( x − 5 y )( 3x − 2 y )
12. x =
8.
4 x2 y 3 (6 x3 y 4 − 2 x6 + 3 y 6 )
13. 6 2
B:
−1 ± i 2
3
14. ( −∞, 3) ∪ ( 5, ∞ )
1.
(12 x
2.
( 2 x − 5)( x + 6 )
2
− 5)(12 x 2 + 5)
3. 5 ( x + 2 y )( x − 2 y )
4.
( m + n )( 2 x − 3 y )
5.
x = 7, x = 4
15. The southbound plane has a groundspeed
of 200 mi/h, and the eastbound plane has a
groundspeed of 210 mi/h.
D:
1.
−1
2.
12
11xy 2
3.
x+2
x−2
4.
3− x
x+2
5.
x = −1
C:
1.
1
16
3
2.
27x 5 y
3.
− 67
6. No solution.
26
7. The first pipe can fill the tank in 9 hours and
the second pipe can fill the tank in 12 hours.
8. The speed of the boat in still water is 16
mi/h.
9. The two even integers are 12 and 14.
10. The assistant would need 144 hours to do
the job alone.
11. y = 216
12. y = 24
E.
1. 5 x = 125
2.
log 25 5 =
1
2
3. 5
4.
2
2
1
log x − log y − log z
3
3
3
5.
x=4
6.
x=
7.
x=4
8.
x =8
ln 3
ln 2
27

Mathematics Placement Review

Transcription

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