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College Algebra Test #01 Review: Equations and Inequalities Page 1 of 16
Section 1.1: Linear Equations, Formulas, and Problem Solving
Big Idea: You can solve any linear equation by combining like terms and using the Properties of
Equality to create simpler equivalent equations until you find the solution.
Big Skill: You should be able to solve linear equations in one variable, even if there are only
letters in the equation!
A. SOLVING LINEAR EQUATIONS USING PROPERTIES OF EQUALITY
 An equation is a statement that two mathematical expressions are equal. A linear
equation in one variable has only one variable, and the only operations performed on that
variable are multiplication by a number or addition. Example: 3  x 1  x   x  7






A common use of equations is to figure out the correct number to put into a calculation to
get a desired answer.
Finding the number that satisfies the equation (or makes the equality be true) is called
“solving the equation”.
Numbers that make the equation true are called solutions or roots of the equation.
The usual way to solve an equation is to keep re-writing the equation as a sequence of
equivalent equations, each one simpler than the one before.
Equivalent equations are two equations that have the same solutions (even if they look
different from each other).
Equations can be shown to be equivalent by using the Properties of Equality:
The Additive Property of Equality
If A, B, and C represent algebraic expressions, and
A = B, then
A + C = B + C.
The Multiplicative Property of Equality
If A, B, and C represent algebraic expressions, and
A = B, then
AC = BC, and
A B

(provided that C  0).
C C
Algorithm for Solving Linear Equations in One Variable
 Eliminate parentheses using the distributive property.
 (To simplify equations with fractions, you might want to multiply both sides of the
equation by the least common denominator of all the fractions at this point, or even as a
first step.)
 Combine like terms.
 Use the Additive Property of Equality to get all variable terms on one side, and all other
terms on the other side. Simplify both sides (by combining like terms).
 Use the Multiplicative Property of Equality to isolate the variable, as in x = 5.
 Check your answer in the original equation.
 For applications, answer the question completely and also include any units of measure.
College Algebra Test #01 Review: Equations and Inequalities Page 2 of 16
B. IDENTITIES AND CONTRADICTIONS
 A conditional equation is an equation that is only true under the condition that the
variable is equal to its solution (or solutions).
 An identity is an equation that is always true regardless of the value of the variable. You
can tell you have an identity because it will have an equivalent equation like 2 = 2, which
is true regardless of the value of the variable (since the variable does not appear in the
equation). We write the solution to an identity like: x | x   or x   ,   .

A contradiction is an equation that is always false regardless of the value of the variable.
You can tell you have a contradiction because it will have an equivalent equation like 2 =
-5, which is false regardless of the value of the variable (since the variable does not
appear in the equation). We write the solution to a contradiction using the null set or the
null symbol: x   or x  .
C. SOLVING FOR A SPECIFIED VARIABLE IN LITERAL EQUATIONS
 A literal equation is an equation with two or more variables. Example: 2A2B + CD =
B/A
 A formula is a type of literal equation in which a quantity is modeled using a
mathematical relationship or dependency on one or more other variables. Example: F =
ma
 It is common to say “solve for B in terms of A, C, and D” to indicate that you are
supposed to create a formula from a literal equation by manipulating the equation until
you isolate B on one side of the equation and the variables A, C, and D on the other side.
 Being able to re-arrange literal equations is useful because then you can derive your own
formulas based on given equations. Exciting! Cool! Powerful!
D. USING THE PROBLEM_SOLVING GUIDE
 Gather and Organize Information
o List given info
o List given formulas
o Clarify what you are asked to find
 Make the Problem Visual
o Draw and label a diagram
o Create a table of values
 Develop and Equation Model
o Assign a variable to what you are asked to find
o Write the equation model, using verbal phrases instead of mathematical
expressions, if necessary
 Use the Model and Given Information to Solve the Problem
College Algebra Test #01 Review: Equations and Inequalities Page 3 of 16
Section 1.2: Linear Inequalities in One Variable
Big Idea: An inequality is used when a range of numbers are desired as the result of a
calculation.
Big Skill: You should be able to solve linear inequalities using the Properties of Inequalities.
Examples of when you might want to use an inequality:
 To get an A in this class, you must get between 92% and 100%  0.92  score  1.00
 You can only afford a car that costs less than $4,000  price < $4,000
 To escape the earth’s gravity, a rocket’s launch speed must be more than 25,000 mph 
speed > 25,000 mph
A. INEQUALITIES AND SOLUTION SETS
 While conditional linear equations in one variable have one solution, linear inequalities
usually have an infinite number of solutions (especially when the answers can be real
numbers)
 Solution set: the set of all numbers that satisfy an inequality
 Three ways to represent a solution set:
o Set notation
o Interval notation (recall that brackets [] mean to include the endpoints and
parentheses mean ( ) mean that the endpoints are not included
o Number line graph
Inequality
x is greater than k
Comparison of the Representations of an Inequality
Set Notation Interval Notation
Number Line
x   k, 
 x | x  k
x is greater than or
equal to k
x is less than k
x is less than or equal
to k
x is greater than a and
less than b
x is greater than or
equal to a and less
than b
x is greater than or
equal to a and less
than or equal to b
x is greater than a and
less than or equal to b
College Algebra Test #01 Review: Equations and Inequalities Page 4 of 16
x is less than a or
greater than b
x is less than or equal
to a or greater than b
x is less than or equal
to a or greater than or
equal to b
x is less than a or
greater than or equal
to b
B. SOLVING LINEAR INEQUALITIES
 The usual way to solve a linear inequality is to keep re-writing the inequality as a
sequence of equivalent inequality, each one simpler than the one before.
 Inequalities can be shown to be equivalent by using the Properties of Inequality:
The Additive Property of Inequality
If A, B, and C represent algebraic expressions, and
A < B, then
A + C < B + C.
Summary: Like quantities can be can be added (or subtracted) to both sides of an inequality
without changing the inequality.
The Multiplicative Property of Equality
If A, B, and C represent algebraic expressions,
and
A < B, then
AC < BC,
if C is a positive quantity (i.e., C > 0)
If A, B, and C represent algebraic expressions,
and
A < B, then
AC > BC,
if C is a negative quantity (i.e., C < 0)
Summary: Like positive quantities can
multiply both sides of an inequality without
changing the inequality.
Summary: If like negative quantities multiply
both sides of an inequality, then the inequality
reverses.
College Algebra Test #01 Review: Equations and Inequalities Page 5 of 16
C. SOLVING COMPOUND INEQUALITIES
 Compound Inequality: when the range of desired solutions must obey more than one
inequality at once.
o Example: If you want to be in the second tax bracket, then you must make $8,350
or more but less than $33,950
 income  $8,350 and income < $33,950
 $8,350  income < $33,950
o Example: If you want to be avoid a parade route, you must be more than 5 blocks
east or west of block #22
 location > block #22 + 5 or location < block #22 - 5
 location > block #27 or location < block #17
 Since there is more than one inequality involved in a compound inequality, we will have
to figure out how to combine multiple inequalities. This can be done using the notions of
set unions and set intersection
 Intersection of two sets (i.e., A  B): A new set formed that includes only the elements
that are common to both sets. This is the first income tax example. Intersections always
got with the word “and.”
 Union of two sets (i.e., A  B): A new set formed that includes all elements that are in
either set. This is the second parade example. Unions always got with the word “or.”
D. APPLICATIONS OF INEQUALITIES
 Writing the domain of functions
o Denominators can not be zero
o The argument to a square root must be greater than or equal to zero
 Any real-world problem where a range of values is acceptable for an answer.
College Algebra Test #01 Review: Equations and Inequalities Page 6 of 16
Section 1.3: Absolute Value Equations and Inequalities
Big Idea: Absolute values can be thought of as stating the distance from a point. When
combined with inequalities, they result in compound inequalities.
Big Skill: You should be able to solve absolute value equations and inequalities using the
Property of Absolute Value Equations.
Examples of when you might want to use absolute value equations or inequalities:
 State the two x values that are 5 units away from x = 1
 if x is greater than 1, then x – 1 = 5
 if x is less than 1, then 1 – x = 5  x – 1 = -5
 x – 1 = 5 or x – 1 = -5
 |x – 1| = 5
 If you want to be avoid a parade route, you must be more than 5 blocks east or west of
block #22
 location > block #27 or location < block #17
 |location – block #22| > 5
A. SOLVING ABSOLUTE VALUE EQUATIONS
 Isolate the absolute value using the Properties of Equality, then finish the solution using
the Properties of Absolute Values:
Property of Absolute Value Equations
If X, represents an algebraic expression, and k is a positive real number,
then |X| = k
implies X = k or X = -k
Multiplicative Property of Absolute Value
If A, and B represent algebraic expressions,
then |A B| = |A||B|
B. SOLVING “LESS THAN” ABSOLUTE VALUE INEQUALITIES
 Use the below Property of Absolute Values Inequality:
Property 1 of Absolute Value Inequalities
If X, represents an algebraic expression, and k is a positive real number,
then |X| < k
implies –k < X < k
C. SOLVING “GREATER THAN” ABSOLUTE VALUE INEQUALITIES
 Use the below Property of Absolute Values Inequality:
Property 2 of Absolute Value Inequalities
If X, represents an algebraic expression, and k is a positive real number,
then |X| > k
implies X < -k or X > k
D. APPLICATIONS INVOLVING ABSOLUTE VALUE
College Algebra Test #01 Review: Equations and Inequalities Page 7 of 16
Section 1.4: Complex Numbers
Big Idea: .Calling the square root of negative one “i” allows us to state the solution of many
algebra equations that would otherwise be unsolvable with real numbers.
Big Skill: You should be able to perform arithmetic (add, subtract, multiply, and divide) with
complex numbers.
A. Identifying and Simplifying Imaginary and Complex Numbers
Imaginary Numbers and the Imaginary Unit
 Imaginary Numbers are those of the form  k , where k is a positive real number.
 The imaginary unit i represents the number whose square root is -1:
i2 = -1 and i  1
Rewriting imaginary numbers

For any positive real number k,
k  1  k  i k .
Complex Numbers
 Complex numbers are numbers that can be written in the form a + bi, where a and b are
real numbers and i  1 is the imaginary unit.
 The real number a is called the real part of the complex number.
 The real number b is called the imaginary part of the complex number.
 The standard form of a complex number is a + bi.
B. Add and Subtract Complex Numbers
 Adding, subtracting, multiplying, or dividing complex numbers results in an answer that
is also a complex number.
 To add complex numbers, add “like terms”:
 a  bi    c  di    a  c   b  d  i

To subtract complex numbers, subtract “like terms”:
 a  bi    c  di    a  c   b  d  i
C. Multiply Complex Numbers and Find powers of i
 To multiply complex numbers, use the distributive property, then combine “like terms”
(like using FOIL):
 One thing to keep in mind: the product property of radicals does not apply when the
radicand is a complex number, so evaluate the square root of the negative number first,
then multiply the radicals.
Correct:
25 4 

25i
 4i 
Incorrect:
25 4 
  5i  2i 

 10i 2
 10
 25 4 
 100
 10

College Algebra Test #01 Review: Equations and Inequalities Page 8 of 16
Product of Complex Conjuagtes
 For a complex number a + bi and its conjugate a – bi, their product (a + bi)( a – bi), is the
real number a2 + b2.
 To compute powers of i, use an exponent that is the remainder of dividing the original
exponent by 4. This works because the powers of i repeat themselves every four factors
of i, as can be seen from making a list of powers of i.
D. Divide Complex Numbers
 To divide complex numbers, multiply the numerator and denominator by the conjugate
of the denominator, then multiply and simplify. Note: the denominator will always
multiply out to a difference of squares, which will be a real number.
Example:
4  3i 4  3i 1  2i


1  2i 1  2i 1  2i
 4  3i 1  2i 

1  2i 1  2i 

4  8i  3i  6i 2
12   2i 
4  11i  6
1  4i 2
2  11i

1 4
2 11i


5
5

2
College Algebra Test #01 Review: Equations and Inequalities Page 9 of 16
Section 1.5: Solving Quadratic Equations
Big Idea: Quadratic equations can be solved under any circumstance if we allow complexvalued solutions.
Big Skill: You should be able to solve any quadratic equation by factoring and using the zero
product property, completing the square and using the square root property of equality, and by
using the quadratic formula.
A. Solve Quadratic equations Using the Zero Product Property
 A quadratic equation is an equation in the variable x with constants a, b, and c
(sometimes called parameters) that can be written in the form ax 2  bx  c  0 .
 The standard form for a quadratic equation is as above, where all non-zero terms are on
the left hand side, and those terms are written in descending powers of the variable.
Quadratic Equations
A quadratic equation can be written in the form ax 2  bx  c  0 with a, b, c  and a  0 .
Zero Product Property
If A and B represent real numbers or real-valued expressions,
and A  B  0 ,
then A  0 or B  0 .

To solve a quadratic equation using the zero product property:
o Get all terms on one side of the equation (which means zero will be on the other
side)
o Factor the non-zero side
o Set each factor equal to zero (justified by the zero product property)
o Solve each mini “factor equation.”
B. Solve Quadratic equations Using the Square Root Property of Equality
Square Root Property of Equality
If X represents an algebraic expression,
and X 2  k ,
then X  k or X   k
which is also written as X   k .

To solve a quadratic equation using the square root property of equality:
o Manipulate the equation until the left hand side is a perfect square, and there is a
single number on the right hand side.
o Apply the square root property of equality.
College Algebra Test #01 Review: Equations and Inequalities Page 10 of 16
C. Solve Quadratic Equations by Completing the Square
To Solve a Quadratic Equation by Completing the Square:
(i.e, writing a quadratic trinomial as a perfect square trinomial plus a constant)
 Get the constant term on the right hand side of the equation.
i.e., if x 2  bx  c  0 , then write the equation as x 2  bx  c

Make sure the coefficient of the square term is 1.

Identify the coefficient of the linear term; multiply it by ½ and square the result.
1 
i.e., Find the number b in x 2  bx  c and compute  b 
2 

2
Add that number to both sides of the equation.
2
1 
1 
i.e., x  bx   b   c   b 
2 
2 
2
2

Write the resulting perfect square trinomial as the square of the binomial .
2
1 

1 
i.e.,  x  b   c   b 
2 

2 

2
Use the square root property to solve the equation.
D. Solve Quadratic Equations Using the Quadratic Equation
The Quadratic Formula
The solution(s) to the quadratic formula ax 2  bx  c  0 (for a  0) are given by the quadratic
formula:
x
b  b 2  4ac
2a
Proof:
The quadratic formula is derived by completing the square on the standard from of a quadratic
equation:
 Get the constant term on the right hand side of the equation.
ax 2  bx  c  0

ax 2  bx  c
Make sure the coefficient of the square term is 1.
ax 2  bx  c
ax 2  bx
c

a
a
b
c
x2  x  
a
a
College Algebra Test #01 Review: Equations and Inequalities Page 11 of 16

Identify the coefficient of the linear term; multiply it by ½ and square the result.
2


b2
1 b




4a 2
2 a
Add that number to both sides of the equation.
b
c
x2  x  
a
a
2
b
b
c b2
x2  x  2    2
a
4a
a 4a
2
b
b
b2 c
x2  x  2  2 
a
4a
4a a
Write the resulting perfect square trinomial as the square of the binomial .
b
b2
b2 c
2
x  x 2  2 
a
4a
4a a
2
b 
b 2 4ac

x   2  2
2a  4a 4a

b  b 2  4ac

x


 
2a 
4a 2

Use the square root property to solve the equation.
2

b 
b 2  4ac

x





2a 
4a 2

2
x
b
b 2  4ac

2a
2a
x
b
b 2  4ac

2a
2a
x
b  b 2  4ac
2a
College Algebra Test #01 Review: Equations and Inequalities Page 12 of 16
E. Use the Discriminant to Identify Solutions
 b2 – 4ac is called the discriminant. The discriminant is important because it determines
the nature of the solutions (roots) of the quadratic equation.
Examples of the Nature of the Roots of a Quadratic Equation with Rational Coefficients:
1. If b2 – 4ac is positive and a perfect square, then there are two solutions that are real,
rational, and unequal. In this case, you can also solve the quadratic equation by
factoring.
Example: 2 x 2  5 x  1.125  0
2. If b2 – 4ac is positive but not a perfect square, then there are two solutions that are
real, irrational, and unequal.
Example: 1.5 x 2  7 x  3  0
3. If b2 – 4ac = 0, then there is just one solution (a repeated root) that is real and
rational (or we can say that the two solutions are equal). This case can also be solved
by factoring.
Example: 9 x 2  24 x  16  0
4. If b2 – 4ac is negative, then there are two solutions that are complex and unequal.
Example: 4 x 2  2 x  9  0
F. Solve Applications of Quadratic Equations
College Algebra Test #01 Review: Equations and Inequalities Page 13 of 16
Section 1.6: Solving Other Types of Equations
Big Idea: Being able to solve polynomial equations is the key to solving rational, radical, and
“quadratic-like” equations.
Big Skill: You should be able to solve these four types of algebraic equations.
A. Polynomial Equations of Higher Degree
 A polynomial equation of degree n is an equation of the form:
a0  a1 x  a2 x 2  a3 x 3   an x n  0
 Polynomials are the most important topic in algebra because any equation that can be
written using addition, subtraction, multiplication, division, integer powers, or roots
(which are rational powers) can be solved by converting the equation into a polynomial
equation.
x2  1
2
 For example, the equation
  x  2  can be transformed into the polynomial
x3
5
4
2
equation x  6 x  38 x  80 x  47  0 . Both of these equations have the same solutions,
one of which is x = 1, as is shown in the graph below:
Steps for solving any polynomial equation (degree 2 or higher):
 Expand the polynomial equation (if needed), and collect all terms on one side; combine
like terms.
 Factor the polynomial on the one side.
 Set each factor equal to zero (this is justified by the zero-product property rule).
 Solve each first degree equation for the variable.
 Check answers in original equation.
 NOTE: The degree of the polynomial tells how many solutions there must be.
College Algebra Test #01 Review: Equations and Inequalities Page 14 of 16
College Algebra Test #01 Review: Equations and Inequalities Page 15 of 16
B. Rational Equations
Solving Rational Equations (Book’s Method)
 Identify and exclude any values that cause a zero denominator.
 Multiply both sides by the LCD and simplify.
 Solve the resulting equation.
 Check all solutions in the original equation.
Solving Rational Equations (Another Method)
 Get all terms on one side of the equation.
 Write all terms as equivalent fractions with a common denominator.
 Add the terms.
 Factor the numerator and cancel any common factors in the denominator.
 Set the numerator to zero and solve the resulting “mini equations”.
 Check all solutions in the original equation.
C. Radical Equations and Equations with Rational Exponents
Solving a radical equation with one radical expression:
 Isolate the radical expression (i.e., get it on one side of the equation by itself with all
other terms on the other side).
 Raise both sides of the equation to the power of the index of the radical.
 Solve the resulting equation.
 Check your answer. It is common to get answers that are not truly solutions. These
answers are called extraneous solutions.
Solving a radical equation with two radical expressions:
 Isolate ONE of the radical expressions.
 Raise both sides of the equation to the power of the index of the radical to eliminate the
first radical.
 Solve the resulting radical equation (that now only has one radical).
 Check your answer. Eliminate any extraneous solutions.
The Power Property of Equality:
If n u  v ,
Then
 u
n
n
 vn
Which simplifies to u  v n
College Algebra Test #01 Review: Equations and Inequalities Page 16 of 16
D. Equations in Quadratic Form
Example:
x4  5x2  6  0
x 
2 2
 5  x2   6  0
Let u  x 2
then u 2   x 2 
2
u 2  5u  6  0
 u  2  u  3  0
u  2 or u  3
x 2  2 or x 2  3
x   2 or x   3
E. Applications