College Algebra
Chapter 1
Equations and Inequalities
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Solving equations using
Addition and multiplication properties of equality
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Solving equations using
Addition and multiplication properties of equality
General Approach
I.
Simplify the equation
•Clear fractions or decimals as needed/desired
•Combine any like terms
II. Solve the equation
•Use additive property of equality to write the
equation with all variable terms on one side
and constants on the other
•Simplify each side
•Use the multiplicative property of equality to
obtain solution form
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Solving equations using
Addition and multiplication properties of equality
General Approach
I.
Simplify the equation
•Clear fractions or decimals as needed/desired
•Combine any like terms
II.
Solve the equation
•Use additive property of equality to write the
equation with all variable terms on one side
and constants on the other
•Simplify each side
•Use the multiplicative property of equality to
obtain solution form
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Solving equations using
Addition and multiplication properties of equality
General Approach
I.
Simplify the equation
•Clear fractions or decimals as needed/desired
•Combine any like terms
II.
Solve the equation
•Use additive property of equality to write the
equation with all variable terms on one side
and constants on the other
•Simplify each side
•Use the multiplicative property of equality to
obtain solution form
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Linear Equation Terminology
Algebraic expression
Sum or difference of algebraic terms
Equation
Statement that two expressions are equal
Linear equations
Three tests – exponents, divisors, multiplication
Standard form
Ax+By=C, where A and B are not both zero
Conditional equation
Equation is true for one value and false for all others
Identity
Equation that is always true
Contradictions
Equations that are never true
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
No Solution
0
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
5
1
 x  3   x  9   3 x  1
9
3
33

x
25
1
1
x  7   10  x  9
3
3
0  50
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Linear Equation Terminology
Literal equation
An equation that has two or more unknowns
Formula
Equation that models a known relationship between
two or more quantities
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Literal Equations:
V  l  w  h solve for w
a 2  b2  c 2
solve for b
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Literal Equations:
d
x2  x1 2   y2  y1 
solve for x2
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Consecutive Integer Problems
Sum of 3 even integers is 66. What are they?
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Mixture Problems
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Mixture Problems
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving
Homework pg 80 1-86
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Inequalities and Solution Sets Terminology
Solution Set
The set of numbers that satisfy an inequality
Set Notation
Method of writing a solution set. Ex {x|x>k}
Number Line
Method of writing a solution set
Interval Notation
Method of writing a solution set. Ex
Boundary Point
x  k , 
Marks the location of a number on a number line also called an endpoint
Inclusion
Uses brackets [ ] to indicate
Not Included
Uses parenthesis ( ) to indicate
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Inequalities and Solution Sets Terminology
Set Notation
n | n  1
The set of all n
Such that
n is greater than or equal to 1
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Inequalities and Solution Sets Terminology
Interval Notation
n  1,  
n is an element of
This set
Smaller number to left
Larger number to right
[,] brackets if boundary points is included
(,) parenthesis if boundary point is not included
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Write the following in set notation, number line graph,
and interval notation
x is less than 10
Set notation
x | x  10
Number line graph
10
Interval notation
n   ,10
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Write the following in set notation, number line graph,
and interval notation
n is greater than or equal to 2
Set notation
n | n  2
Number line graph
2
Interval notation
n  2, 
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Write the following in set notation, number line graph,
and interval notation
X is less than 5 and greater than or equal to -1
Set notation
x | 1  x  5
Number line graph
-1
Interval notation
x  1,5
5
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Solving Inequalities
Beware of multiplying or dividing by a (-)
3
p  2  12
8
80
p
3
2
1 5
 x 
3
2 6
x
1
2
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Solving Inequalities
x  2
n 1
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Solving Compound Inequalities
2 x 0
 10  k  8
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Solving Compound Inequalities
r  8 or r  5
x  0 or x  8
College Algebra Chapter 1.2 Linear Inequalities in One Variable
Homework pg 90 1-94
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Zero factor property
Given that A and B represent real numbers or
real-valued expressions, if A  B  0 , then
either A  0 or B  0
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
2 x  20x  3x
Solve
3
2
2 x  3x  20x  0
3

Set equal to zero
2

x 2 x 2  3x  20x  0
x2 x  5x  4  0
x  0 or 2 x  5  0 or x  4  0
5
x  0 or x   or x  4
2
Remove common factors
aka undistribute
Factor trinomial
Solve each factor
equal to zero
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solve
t  4t  7  54
t  13 or 2
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
x 2  15  2 x
x5
x  3
a 2  17  8
a3
a  3
m2  8m  16
m4
b 2  5b  24
b  8
b3
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving rational equations
1. Identify and exclude values that
cause a zero denominator
2. Multiply by the LCD and simplify
3. Solve resulting equation
4. Check in original equation and
step 1
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving rational equations
1.
2
1
4

 2
m m 1 m  m
m6
2.
3.
4.
Identify and exclude values that
cause a zero denominator
Multiply by the LCD and simplify
Solve resulting equation
Check in original equation and step 1
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving rational equations
1.
14
2x
x
 1
x7
x7
x  3 or x  7
x7
2.
3.
4.
Identify and exclude values that
cause a zero denominator
Multiply by the LCD and simplify
Solve resulting equation
Check in original equation and step 1
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving rational equations
1.
2.
3.
4.
Identify and exclude values that
cause a zero denominator
Multiply by the LCD and simplify
Solve resulting equation
Check in original equation and step 1
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving rational equations
1.
2.
3.
4.
Identify and exclude values that
cause a zero denominator
Multiply by the LCD and simplify
Solve resulting equation
Check in original equation and step 1
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving Radical Equations
1. Isolate radical term
2. Square both sides
3. Simplify
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving Radical Equations
x  1 12  10
3
1.
2.
3.
Isolate radical term
Square both sides
Simplify
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving Radical Equations
1.
2.
3.
Isolate radical term
Square both sides
Simplify
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving Radical Equations
1.
2.
3.
Isolate radical term
Square both sides
Simplify
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving Radical Equations
x  2  2x  4  10
1.
2.
3.
Isolate radical term
Square both sides
Simplify
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Rational Exponents
3x  1  9  15
3
4
15
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Rational Exponents
8x

3
2
11
 17  
8
16
x
25
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solving with U-Substitution
2
3
1
3
x  3x  10  0
x  125 or x  8
College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Homework pg 101 1-100
College Algebra Chapter 1.4 Complex Numbers
Imaginary unit i represents the number whose square is -1
i 2  1
i  1
College Algebra Chapter 1.4 Complex Numbers
i  1
i  1
2
i 3  1  1
i4  1
i 5  i 4  i  1
College Algebra Chapter 1.4 Complex Numbers
Complex numbers are those that can be
written in the form a  bi , where a and
b are real numbers and i   1 . The
expression a  bi is called the standard
form of a complex numbers
a  bi
Real part
Imaginary part
College Algebra Chapter 1.4 Complex Numbers
Identify the values of a and b
a=-5
-5+3i
b=3
a=2
b=-3
2   49
a=2
b=7
 12
a=0
b= 12
a=7
b=0
2-3i
7
College Algebra Chapter 1.4 Complex Numbers
Identify the values of a and b
4  3  25
20
a=0.2
b=0.75
College Algebra Chapter 1.4 Complex Numbers
Adding and subtracting complex numbers
4  2i   3  4i 
2  3i   10  6i 
College Algebra Chapter 1.4 Complex Numbers
Adding and subtracting complex numbers
 5  4i    2 
2i

2  3i    5  2i 
College Algebra Chapter 1.4 Complex Numbers
Multiplying Complex Numbers
24  3i 
10  i  6
College Algebra Chapter 1.4 Complex Numbers
Multiplying Complex Numbers
4  3i 2  i 
6  5i  2  4i 
College Algebra Chapter 1.4 Complex Numbers
Multiplying by the Complex Conjugate
2  3i 
3  i 
College Algebra Chapter 1.4 Complex Numbers
Division of Complex Numbers
2
5i
6i
7  3i
College Algebra Chapter 1.4 Complex Numbers
Homework pg 113 1-78
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Standard Form and Coefficients
ax  bx  c  0
x
2 x 2  10  0
z  12  3z  0
2
1 2
x  6x
4
 3x 2  9 x  5  2 x 3  0
a=2
b=0
c=-10
a=-3 b=1
c=-12
a=0.25
b=-6
Not quadratic
c=0
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Square root property
If
p 2  k then p  k
aka p   k
x 9
2
x  3 or  3
x  3
or
p k
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
solve
y  28  0
2
p  36  0
2
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
solve
 3x 2  28  23
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Completing the Square
1.
2.
3.
4.
Subtract the constant c from both sides
Divide both sides by the leading coefficient a
Take  12 linear coefficien t  and add the result to both sides
Factor the left hand side as a binomial square:
simplify the right hand side
5. Solve using the SQR property of equality
2
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Completing the Square
1.
2.
3.
4.
5.
Subtract the constant c from both sides
Divide both sides by the leading
coefficient a
 1 

 2 linear coefficien t 
Take
and add the
 

result to both sides
Factor the left hand side as a binomial
square: simplify the right hand side
Solve using the SQR property of equality
2
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Quadratic Formula
If
ax  bx  c  0
2
Discriminant
 b  b  4ac
then x 
2a
2
if
b2  4ac  0
if
b2  4ac  0
1 real root
2 real roots
if
b2  4ac  0
2 complex roots
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Solve
Quadratic Formula
 9  177 9  177 
,


8
8


No Solution
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Solve
{6,-3}
Quadratic Formula
{6,1}
College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Homework pg 126 1-130
College Algebra Chapter 1 Review
Solve
-10
All Real Numbers
No Solution
2
College Algebra Chapter 1 Review
Solve and graph, state solution in set and interval notation
College Algebra Chapter 1 Review
Solve by the method of your choice
{5, -5}
{2/7, -7}
{-4, 6}
{7.483, -7.483}
{5/8, -8/5}
{8, 7}
No solution
{0.732, -2.732}
{1.309, -3.309}
{12.426, -4.426}
College Algebra Chapter 1 Review
Solve
{6}
{3}
{-1}
{9, 2}
College Algebra Chapter 1 Review
Homework pg 129 1-58