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May 19th

May 19th

# Algebra Placement Exam - Study Guide

## 1 Purpose of this document

There are currently four ways to become eligible to enroll in any of th 100 level math classes
at The University of Montana Western

1. Pass MATH 007 with a C- or better.

2. Earn a score of 520 or higher on the math section of the SAT or 22 or higher on the
math section of the ACT.

3. Earn a C- or better on a transferrable course from another university. This course
must be equivalent to MATH 007 or any of the 100 level math classes offered at the
University of Montana Western.

4. Pass the math placement exam with a score of 70% or higher.

The primary purpose of this document is to provide you with an inventory of the skills you
should become proficient in in order to pass the placement exam. However, it also should
serve as a reasonable study guide for MATH 007. With this in mind, pages from Prealgebra
and Algebra, by Daniel D. Benice (the MATH 007 textbook) will be cited in this document.

## 2 Algebra skills you will need for the 100 level math classes

A broad description of the skills you should posess before taking a 100 level math class
follows.

1. You should be able to state what an algebraic expression is, know the mathematical
operations you may apply to it without changing its value, and demonstrate skills at
simplifying or manipulating algebraic expressions that involve monomials, polynomials,
fractions, exponents, and roots and various combinations of these.

2. You should be able to solve single linear equations for an unknown variable (or root).
This includes equations that are obviously linear and equations that must be transformed
or simplified first.

3. you should be able to solve a system of two linear equations in two independent variables
by employing either the method of elimination or the method of substitution . In
addition, you should be able to demonstrate the connection between the solution of a
system of linear equations and the graphs of the lines described by the two equations.

4. You should be able to take an equation for a line, interpret its slope and x and y
intercepts
, and graph the line. Similarly, you should be able to look at the graph of
a line and be able to come up with its equation. Finally, you should be able to write
down the equation of a line if you are given two points on the line or one point on the
line and the slope of the line.

5. You should be able to solve quadratic equations by factoring them or by employing
either “completing the square” or the quadratic formula.

## 3 Course of study for preparing for the placement exam

If you are looking for a more general course of study that should cover the topics you will
need to master in order to pass the placement exam, or, if you simply want to have an
overview of what you will most likely be studying in Math 007, then read on.

1. Expressions are introduced in section 11.1 of Prealgebra and Algebra, pages 134 and 135.

2. Elementary simplification problems first appear in sections 11.1–11.4 of Prealgebra
and Algebra, pages 135–153. These problems involve, combining like terms, elementary
manipulation of expressions involving exponents, use of the FOIL method, and division
of polynomials and monomials .

3. Linear equations are first introduced and defined in section 12.1 of Prealgebra and
Algebra, on page 154.

4. Techniques for solving straighforward, linear equations are presented in sections 12.2-
12.5 of Prealgebra and Algebra, pages 155-165.

5. A strategy for solving slightly more general linear equations appears in sectin 12.6,
pages 165–169. These are mostly just linear equations that ought to be simplified
before you solve them, but they are good practice.

6. You can learn what it means to graph a straight line in chapter 14. The basic idea of
how to graph anything in a two-dimensional, Cartesian coordinate system is intoduced
in 14.1, pages 197–201.

7. Techniques for graphing straight lines appear in sections 14.2–14.4, pages 201–221.
You need to be familiar with how to graph lines by plotting points, and using slope
intercept method. In addition, you need to be able to write down the equation for a
line if you are given (1) two points on the line, or (2) a point on the line and the slope
of the line.

8. Be sure that you can explain the relationship between the x-intercept of a line and the
solution to the equation of the line.

9. Systems of linear equations are first introduced and defined in section 15.1 on pages
222 and 223.

10. You need to demonstrate an ability to solve a system of two linear equations (in
two independent variables) using both the method of elimination and the method of
substitution. These appear in sections 15.2 and 15.3, respectively (pages 223–230.)

11. You should be able to explain the relationship between the graphs of the two equations
in the system and the solution to the system.

12. The skill of factoring simple mathematical expressions is first developed in section 17.1,
pages 248–251.

13. Your factoring skills are specifically applied to some simple quadratic expressions (with
rational roots) in sections 17.2–17.3, pages 251–260.

14. Once you can factor simple quadratic expressions that have rational roots, use this
skill to solve simple quadratic equations. This skill is developed in section 17.4, pages
251–268.

15. You need to be able to manipulate and simplify expressions involving fractions. Fractional
expressions are introduced in section 18.1, pages 269–271. However, there are
some very specific skills you need to develop. In particular,

(a) you can learn how to apply your factoring and elementary simplification skills
in order to simplify fractions that involve monomials and polynomials in the
numerator and denomenator in section 18.2, pages 271–276;
(b) you can learn how to simplify expressions that involve products and quotients of
fractional expressions in sections 18.3 and 18.4, respectively (pages 276–281);
(c) you can learn how to simplify expression involving sums and differences of fractions
in section 18.5, pages 281–285. This requires you to develop an ability to
find a common denomenator between two fractions;
(d) you can learn how to mix and match some of these skills and apply them to some
slightly more complex fractional expressions in section 18.6, pages 286–289.

16. Once you have mastered the skills for manipulating and simplifying fractional expressions,
you can apply these to solving equations that involve fractional expressions.
Generally this means simplifying the equations until you have reduced them to either
linear or quadratic equations. Once you’ve done that, you can solve them as before.
Section 18.7, pages 289-294, addresses this.

17. Many expressions involve exponents. The laws of exponents are reviewed in section
20.1, pages 313–317. You should be able to use these laws to simplify and manipulate
expressions that involve exponents when appropriate.

18. Exponents need not be positive numbers (or even integers!). You can find out an
interpretation of what negative and fractional exponents mean in sections 20.2 and
20.5. (Section 20.4 tells you what it means to raise a quantity to the power of 0). Be
sure that you understand the relationship between fractional exponents and radicals!

19. Since radicals are intimitely related to exponents, your ability to manipulate and simplify
expressions that involve radicals will depend on your ability to work with exponents.
Some basic skills are developed in sections 21.1–21.2, pages 336–341.

20. You can learn how to combine two or more like radicals in section 21.3, pages 341–342.

21. You can learn how to take a fractional expression that involves radicals in both the
numerator and denomenator and simplify it in a way that leaves the radicals only in
one position (but not both). This is called rationalization, and it can be found in
section 21.4, pages 342–346.

22. Once you have mastered the skills for manipulating and simplifying expressions that
involve radicals, you can apply these to solving equations that involve radicals. Generally
this means simplifying the equations until you have reduced them to either linear
or quadratic equations. Once you’ve done that, you can solve them as before. Section
21.5, pages 346–351, provides instruction on this subject.

23. Finally, it is important that you recognize that not all quadratic equations have integer
or rational roots. Many have irrational roots. It is not easy to factor these equations
“by inspection,” so a more general technique is needed. One is called completing the
square and the other is called the quadratic equation. Both methods are related and
knowing either one will do. You can learn about them in sections 22.1–22.3, pages
353–362.

## 4 Practice exam

Have you studied the concepts in the previous section hard? Do you feel like you are ready
for the placement exam? If so, read on. There is a practice exam (with a key) immediately
fol lowing this document . You should take it under conditions similar to the ones you will
find on the the test day. In particular take it in during a quiet, fifty minute period, use
only a pencil or something else to write with, and put away all books, notes, calculators,
and other mathematical aids. You can miss no more than 6 out of the 20 problems on the
practice exam in order to earn a pass. Please note that this exam should be representative
of the actual placement exams, but it is not the same exam you will see. Actual placement
exams may seem harder (or easier), but they will address similar topics.

IMPORTANT: The test consists of 20 problems. You will have 50 minutes to
complete the problems. You are not allowed to use calculators, books or any
other aids during the test. Calculations may be done on provided scratch paper,
but you must turn this scratch paper in with your test. All answers must be
complete, legible and simplified to lowest terms. Record only final answers in
the blanks after the problems.

1. Simplify:

3(2x + 1) − 5

2. Find the equation of the line through point (1, 2) and (3, 8). (The equation should be
in a slope-intercept form y = mx + b.)

3. Solve the equation for x. Express your answer as a common fraction.

4. Solve an equation for x. Express your answer as a common fraction.

5. Solve the equation:

2x + 1 = 3x + 7

6. Solve the following system of equations for x and y:

7. Solve the following system of equations for x and y:

8. Simplify:

9. Simplify:

10. Simplify:

11. Solve the equation:

12. Solve the equation:

13. Solve the equation:

14. Solve the equation for x. Express your answer as a common fraction.

2(3x + 1) = 3x

15. Simplify:

16. Perform the indicated operation and simplify:

17. Simplify:

18. Perform the indicated operation and simplify:

19. Solve the quadratic equation for x: