Transcript Slide 1
Polynomial and
Rational
Functions
Chapter 3
Quadratic
Functions and
Models
Section 3.1
Quadratic Functions
Quadratic function: Function of the
form
f(x) = ax2 + bx + c
(a, b and c real numbers, a ≠ 0)
Quadratic Functions
Example. Plot the graphs of f(x) =
x2, g(x) = 3x2 and
30
20
10
-10
-8
-6
-4
-2
2
-10
-20
-30
4
6
8
10
Quadratic Functions
Example. Plot the graphs of f(x) =
{x2, g(x) = {3x2 and
30
20
10
-10
-8
-6
-4
-2
2
-10
-20
-30
4
6
8
10
Parabolas
Parabola: The graph of a quadratic
function
If a > 0, the parabola opens up
If a < 0, the parabola opens down
Vertex: highest / lowest point of a
parabola
Parabolas
Axis of symmetry: Vertical line
passing through the vertex
Parabolas
Example. For the function
f(x) = {3x2 +12x { 11
(a) Problem: Graph the function
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Parabolas
Example. (cont.)
(b) Problem: Find the vertex and axis of
symmetry.
Answer:
Parabolas
Locations of vertex and axis of
symmetry:
Set
Set
Vertex is at:
Axis of symmetry runs through vertex
Parabolas
Example. For the parabola defined by
f(x) = 2x2 { 3x + 2
(a) Problem: Without graphing, locate the
vertex.
Answer:
(b) Problem: Does the parabola open up
or down?
Answer:
x-intercepts of a Parabola
For a quadratic function
f(x) = ax2 + bx + c:
Discriminant is b2 { 4ac.
Number of x-intercepts depends on the
discriminant.
Positive discriminant: Two x-intercepts
Negative discriminant: Zero x-intercepts
Zero discriminant: One x-intercept
(Vertex lies on x-axis)
x-intercepts of a Parabola
Graphing Quadratic Functions
Example. For the function
f(x) = 2x2 + 8x + 4
(a) Problem: Find the vertex
Answer:
(b) Problem: Find the intercepts.
Answer:
Graphing Quadratic Functions
Example. (cont.)
(c) Problem: Graph the function
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Graphing Quadratic Functions
Example. (cont.)
(d) Problem: Determine the domain and
range of f.
Answer:
(e) Problem: Determine where f is
increasing and decreasing.
Answer:
Graphing Quadratic Functions
Example.
Problem: Determine the quadratic
function whose vertex is (2, 3) and
whose y-intercept is 11.
Answer:
14
12
10
8
6
4
2
-14 -12 -10 -8 -6 -4 -2
-2
-4
-6
-8
-10
-12
-14
2
4
6
8
10 12 14
Graphing Quadratic Functions
Method 1 for Graphing
Complete the square in x to write the
quadratic function in the form
y = a(x { h)2 + k
Graph the function using
transformations
Graphing Quadratic Functions
Method 2 for Graphing
Determine the vertex
Determine the axis of symmetry
Determine the y-intercept f(0)
Find the discriminant b2 { 4ac.
If b2 { 4ac > 0, two x-intercepts
If b2 { 4ac = 0, one x-intercept
(at the vertex)
If b2 { 4ac < 0, no x-intercepts.
Graphing Quadratic Functions
Method 2 for Graphing
Find an additional point
Use the y-intercept and axis of symmetry.
Plot the points and draw the graph
Graphing Quadratic Functions
Example. For the quadratic function
f(x) = 3x2 { 12x + 7
(a) Problem: Determine whether f has a
maximum or minimum value, then find
it.
Answer:
Graphing Quadratic Functions
Example. (cont.)
(b) Problem: Graph f
Answer:
10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Quadratic Relations
Quadratic Relations
Example. An engineer collects the
following data showing the speed s of
a Ford Taurus and its average miles
per gallon, M.
Quadratic Relations
Speed, s
Miles per Gallon, M
30
18
35
20
40
23
40
25
45
25
50
28
55
30
60
29
65
26
65
25
70
25
Quadratic Relations
Example. (cont.)
(a) Problem: Draw a scatter diagram of
the data
Answer:
Quadratic Relations
Example. (cont.)
(b) Problem: Find the quadratic function
of best fit to these data.
Answer:
Quadratic Relations
Example. (cont.)
(c) Problem: Use the function to
determine the speed that maximizes
miles per gallon.
Answer:
Key Points
Quadratic Functions
Parabolas
x-intercepts of a Parabola
Graphing Quadratic Functions
Quadratic Relations
Polynomial
Functions and
Models
Section 3.2
Polynomial Functions
Polynomial function: Function of the form
f(x) = anxn + an {1xn {1 + + a1x + a0
an, an {1, …, a1, a0 real numbers
n is a nonnegative integer (an 0)
Domain is the set of all real numbers
Terminology
Leading coefficient: an
Degree: n (largest power)
Constant term: a0
Polynomial Functions
Degrees:
Zero function: undefined degree
Constant functions: degree 0.
(Non-constant) linear functions: degree 1.
Quadratic functions: degree 2.
Polynomial Functions
Example. Determine which of the
following are polynomial functions? For
those that are, find the degree.
(a) Problem: f(x) = 3x + 6x2
Answer:
(b) Problem: g(x) = 13x3 + 5 + 9x4
Answer:
(c) Problem: h(x) = 14
Answer:
(d) Problem:
Answer:
Polynomial Functions
Graph of a polynomial function will be
smooth and continuous.
Smooth: no sharp corners or cusps.
Continuous: no gaps or holes.
Power Functions
Power function of degree n:
Function of the form
f(x) = axn
a 0 a real number
n > 0 is an integer.
Power Functions
The graph depends on whether n is
even or odd.
Power Functions
Properties of f(x) = axn
Symmetry:
If n is even, f is even.
If n is odd, f is odd.
Domain: All real numbers.
Range:
If n is even, All nonnegative real numbers
If n is odd, All real numbers.
Power Functions
Properties of f(x) = axn
Points on graph:
If n is even: (0, 0), (1, 1) and ({1, 1)
If n is odd: (0, 0), (1, 1) and ({1, {1)
Shape: As n increases
Graph becomes more vertical if |x| > 1
More horizontal near origin
Graphing Using Transformations
Example.
Problem: Graph f(x) = (x { 1)4
Answer:
4
2
-4
-2
2
-2
-4
4
Graphing Using Transformations
Example.
Problem: Graph f(x) = x5 + 2
Answer:
4
2
-4
-2
2
-2
-4
4
Zeros of a Polynomial
Zero or root of a polynomial f:
r a real number for which f(r) = 0
r is an x-intercept of the graph of f.
(x { r) is a factor of f.
Zeros of a Polynomial
Zeros of a Polynomial
Example.
Problem: Find a polynomial of degree 3
whose zeros are {4, {2 and 3.
Answer:
40
30
20
10
-10
-5
5
-10
-20
-30
-40
10
Zeros of a Polynomial
Repeated or multiple zero or root of
f:
Same factor (x { r) appears more than
once
Zero of multiplicity m:
(x { r)m is a factor of f and (x { r)m+1 isn’t.
Zeros of a Polynomial
Example.
Problem: For the polynomial, list all zeros
and their multiplicities.
f(x) = {2(x { 2)(x + 1)3(x { 3)4
Answer:
Zeros of a Polynomial
Example. For the polynomial
f(x) = {x3(x { 3)2(x + 2)
(a) Problem: Graph the polynomial
Answer:
40
20
-4
-2
2
-20
-40
4
Zeros of a Polynomial
Example. (cont.)
(b) Problem: Find the zeros and their
multiplicities
Answer:
Multiplicity
Role of multiplicity:
r a zero of even multiplicity:
f(x) does not change sign at r
Graph touches the x-axis at r, but does not
cross
40
20
-4
-2
2
-20
-40
4
Multiplicity
Role of multiplicity:
r a zero of odd multiplicity:
f(x) changes sign at r
Graph crosses x-axis at r
40
20
-4
-2
2
-20
-40
4
Turning Points
Turning points:
Points where graph changes from
increasing to decreasing function or vice
versa
Turning points correspond to local
extrema.
Theorem.
If f is a polynomial function of degree
n, then f has at most n { 1 turning
points.
End Behavior
Theorem. [End Behavior]
For large values of x, either positive
or negative, that is, for large |x|, the
graph of the polynomial
f(x) = anxn + an{1xn{1 + + a1x + a0
resembles the graph of the power
function
y = a n xn
End Behavior
End behavior of:
f(x) = anxn + an{1xn{1 + + a1x + a0
Analyzing Polynomial Graphs
Example. For the polynomial:
f(x) =12x3 { 2x4 { 2x5
(a) Problem: Find the degree.
Answer:
(b) Problem: Determine the end behavior.
(Find the power function that the graph
of f resembles for large values of |x|.)
Answer:
Analyzing Polynomial Graphs
Example. (cont.)
(c) Problem: Find the x-intercept(s), if
any
Answer:
(d) Problem: Find the y-intercept.
Answer:
(e) Problem: Does the graph cross or
touch the x-axis at each x-intercept:
Answer:
Analyzing Polynomial Graphs
Example. (cont.)
(f) Problem: Graph f using a graphing
utility
80
Answer:
60
40
20
-4
-2
2
-20
-40
-60
-80
4
Analyzing Polynomial Graphs
Example. (cont.)
(g) Problem: Determine the number of
turning points on the graph of f.
Approximate the turning points to 2
decimal places.
Answer:
(h) Problem: Find the domain
Answer:
Analyzing Polynomial Graphs
Example. (cont.)
(i) Problem: Find the range
Answer:
(j) Problem: Find where f is increasing
Answer:
(k) Problem: Find where f is decreasing
Answer:
Cubic Relations
Cubic Relations
Example. The following data
represent the average number of miles
driven (in thousands) annually by
vans, pickups, and sports utility
vehicles for the years 1993-2001,
where x = 1 represents 1993, x = 2
represents 1994, and so on.
Cubic Relations
Year, x
Average Miles Driven, M
1993, 1
12.4
1994, 2
12.2
1995, 3
12.0
1996, 4
11.8
1997, 5
12.1
1998, 6
12.2
1999, 7
12.0
2000, 8
11.7
2001, 9
11.1
Cubic Relations
Example. (cont.)
(a) Problem: Draw a scatter diagram of
the data using x as the independent
variable and M as the dependent
variable.
Answer:
Cubic Relations
Example. (cont.)
(b) Problem: Find the cubic function of
best fit and graph it
Answer:
Key Points
Polynomial Functions
Power Functions
Graphing Using Transformations
Zeros of a Polynomial
Multiplicity
Turning Points
End Behavior
Analyzing Polynomial Graphs
Cubic Relations
The Real Zeros of
a Polynomial
Function
Section 3.6
Division Algorithm
Theorem. [Division Algorithm]
If f(x) and g(x) denote polynomial
functions and if g(x) is a polynomial whose
degree is greater than zero, then there are
unique polynomial functions q(x) and r(x)
such that
where r(x) is either the zero polynomial or
a polynomial of degree less than that of
g(x).
Division Algorithm
Division algorithm
f(x) is the dividend
q(x) is the quotient
g(x) is the divisor
r(x) is the remainder
Remainder Theorem
First-degree divisor
Has form g(x) = x { c
Remainder r(x)
Either the zero polynomial or a polynomial
of degree 0,
Either way a number R.
Becomes f(x) = (x { c)q(x) + R
Substitute x = c
Becomes f(c) = R
Remainder Theorem
Theorem. [Remainder Theorem]
Let f be a polynomial function. If
f(x) is divided by x { c, the
remainder is f(c).
Remainder Theorem
Example. Find the remainder if
f(x) = x3 + 3x2 + 2x { 6
is divided by:
(a) Problem: x + 2
Answer:
(b) Problem: x { 1
Answer:
Factor Theorem
Theorem. [Factor Theorem]
Let f be a polynomial function. Then
x { c is a factor of f(x) if and only if
f(c) = 0.
If f(c) = 0, then x { c is a factor of
f(x).
If x { c is a factor of f(x), then
f(c) = 0.
Factor Theorem
Example. Determine whether the
function
f(x) = {2x3 { x2 + 4x + 3
has the given factor:
(a) Problem: x + 1
Answer:
(b) Problem: x { 1
Answer:
Number of Real Zeros
Theorem. [Number of Real Zeros]
A polynomial function of degree n,
n ¸ 1, has at most n real zeros.
Rational Zeros Theorem
Theorem. [Rational Zeros Theorem]
Let f be a polynomial function of
degree 1 or higher of the form
f(x) = anxn + an{1xn{1 + + a1x + a0
an 0, a0 0, where each coefficient
is an integer. If p/q, in lowest terms,
is a rational zero of f, then p must be
a factor of a0 and q must be a factor
of an.
Rational Zeros Theorem
Example.
Problem: List the potential rational zeros
of
f(x) = 3x3 + 8x2 { 7x { 12
Answer:
Finding Zeros of a Polynomial
Determine the maximum number of
zeros.
If the polynomial has integer
coefficients:
Degree of the polynomial
Use the Rational Zeros Theorem to find
potential rational zeros
Using a graphing utility, graph the
function.
Finding Zeros of a Polynomial
Test values
Test a potential rational zero
Each time a zero is found, repeat on the
depressed equation.
Finding Zeros of a Polynomial
Example.
Problem: Find the rational zeros of the
polynomial in the last example.
f(x) = 3x3 + 8x2 { 7x { 12
Answer:
Finding Zeros of a Polynomial
Example.
Problem: Find the real zeros of
f(x) = 2x4 + 13x3 + 29x2 + 27x + 9
and write f in factored form
Answer:
Factoring Polynomials
Irreducible quadratic: Cannot be factored
over the real numbers
Theorem.
Every polynomial function (with real
coefficients) can be uniquely factored into a
product of linear factors and irreducible
quadratic factors
Corollary.
A polynomial function (with real
coefficients) of odd degree has at least one
real zero
Factoring Polynomials
Example.
Problem: Factor
f(x)=2x5 { 9x4 + 20x3 { 40x2 + 48x {16
Answer:
Bounds on Zeros
Bound on the zeros of a polynomial
Positive number M
Every zero lies between {M and M.
Bounds on Zeros
Theorem. [Bounds on Zeros]
Let f denote a polynomial whose
leading coefficient is 1.
f(x) = xn + an{1xn{1 + + a1x + a0
A bound M on the zeros of f is the
smaller of the two numbers
Max{1, ja0j + ja1j + + jan-1j},
1 + Max{ja0j ,ja1j , … , jan-1j}
Bounds on Zeros
Example. Find a bound to the zeros
of each polynomial.
(a) Problem:
f(x) = x5 + 6x3 { 7x2 + 8x { 10
Answer:
(b) Problem:
g(x) = 3x5 { 4x4 + 2x3 + x2 +5
Answer:
Intermediate Value Theorem
Theorem. [Intermediate Value Theorem]
Let f denote a continuous function. If
a < b and if f(a) and f(b) are of
opposite sign, then f has at least one
zero between a and b.
Intermediate Value Theorem
Example.
Problem: Show that
f(x) = x5 { x4 + 7x3 { 7x2 { 18x + 18
has a zero between 1.4 and 1.5.
Approximate it to two decimal places.
Answer:
Key Points
Division Algorithm
Remainder Theorem
Factor Theorem
Number of Real Zeros
Rational Zeros Theorem
Finding Zeros of a Polynomial
Factoring Polynomials
Bounds on Zeros
Intermediate Value Theorem
Complex Zeros;
Fundamental
Theorem of Algebra
Section 3.7
Complex Polynomial
Functions
Complex polynomial function: Function
of the form
f(x) = anxn + an {1xn {1 + + a1x + a0
an, an {1, …, a1, a0 are all complex numbers,
an 0,
n is a nonnegative integer
x is a complex variable.
Leading coefficient of f: an
Complex zero: A complex number r with
f(r) = 0.
Complex Arithmetic
See Appendix A.6.
Imaginary unit: Number i with
i2 = {1.
Complex number: Number of the form z =
a + bi
a and b real numbers.
a is the real part of z
b is the imaginary part of z
Can add, subtract, multiply
Can also divide (we won’t)
Complex Arithmetic
Conjugate of the complex number
a + bi
Number a { bi
Written
Properties:
Complex Arithmetic
Example. Suppose z = 5 + 2i and
w = 2 { 3i.
(a) Problem: Find z + w
Answer:
(b) Problem: Find z { w
Answer:
(c) Problem: Find zw
Answer:
(d) Problem: Find
Answer:
Fundamental Theorem of
Algebra
Theorem. [Fundamental Theorem of
Algebra]
Every complex polynomial function
f(x) of degree n ¸ 1 has at least one
complex zero.
Fundamental Theorem of
Algebra
Theorem.
Every complex polynomial function f(x) of
degree n ¸ 1 can be factored into n linear
factors (not necessarily distinct) of the
form
f(x) = an(x { r1)(x { r2) (x { rn)
where an, r1, r2, …, rn are complex
numbers. That is, every complex
polynomial function f(x) of degree n ¸ 1
has exactly n (not necessarily distinct)
zeros.
Conjugate Pairs Theorem
Theorem. [Conjugate Pairs Theorem]
Let f(x) be a polynomial whose
coefficients are real numbers. If a + bi
is a zero of f, then the complex
conjugate a { bi is also a zero of f.
Conjugate Pairs Theorem
Example. A polynomial of degree 5
whose coefficients are real numbers
has the zeros {2, {3i and 2 + 4i.
Problem: Find the remaining two zeros.
Answer:
Conjugate Pairs Theorem
Example.
Problem: Find a polynomial f of degree 4
whose coefficients are real numbers and
that has the zeros {2, 1 and 4 + i.
Answer:
Conjugate Pairs Theorem
Example.
Problem: Find the complex zeros of the
polynomial function
f(x) = x4 + 2x3 + x2 { 8x { 20
Answer:
Key Points
Complex Polynomial Functions
Complex Arithmetic
Fundamental Theorem of Algebra
Conjugate Pairs Theorem
Properties of
Rational
Functions
Section 3.3
Rational Functions
Rational function: Function of the
form
p and q are polynomials,
q is not the zero polynomial.
Domain: Set of all real numbers
except where q(x) = 0
Rational Functions
is in lowest terms:
The polynomials p and q have no
common factors
x-intercepts of R:
Zeros of the numerator p when R is in
lowest terms
Rational Functions
Example. For the rational function
(a) Problem: Find the domain
Answer:
(b) Problem: Find the x-intercepts
Answer:
(c) Problem: Find the y-intercepts
Answer:
Graphing Rational
Functions
Graph of
10
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Graphing Rational Functions
As x approaches 0,
is unbounded in the positive
direction.
Write f(x) ! 1
Read “f(x) approaches infinity”
Also:
May write f(x) ! 1 as x ! 0
May read: “f(x) approaches infinity as x
approaches 0”
Graphing Rational Functions
Example. For
Problem: Use transformations to graph f.
Answer:
4
2
-6
-4
-2
2
-2
-4
4
Asymptotes
Horizontal asymptotes:
Let R denote a function.
Let x ! {1 or as x ! 1,
If the values of R(x) approach some
fixed number L, then the line y = L is a
horizontal asymptote of the graph of R.
Asymptotes
Vertical asymptotes:
Let x ! c
If the values jR(x)j ! 1, then the line
x = c is a vertical asymptote of the
graph of R.
Asymptotes
Asymptotes:
Oblique asymptote: Neither horizontal
nor vertical
Graphs and asymptotes:
Graph of R never intersects a vertical
asymptote.
Graph of R can intersect a horizontal or
oblique asymptote (but doesn’t have to)
Asymptotes
A rational function can have:
Any number of vertical asymptotes.
1 horizontal and 0 oblique asymptote
0 horizontal and 1 oblique asymptotes
0 horizontal and 0 oblique asymptotes
There are no other possibilities
Vertical Asymptotes
Theorem. [Locating Vertical
Asymptotes]
A rational function
in lowest terms, will have a vertical
asymptote x = r if r is a real zero of
the denominator q.
Vertical Asymptotes
Example. Find the vertical
asymptotes, if any, of the graph of
each rational function.
(a) Problem:
Answer:
(b) Problem:
Answer:
Vertical Asymptotes
Example. (cont.)
(c) Problem:
Answer:
(d) Problem:
Answer:
Horizontal and Oblique
Asymptotes
Describe the end behavior of a
rational function.
Proper rational function:
Degree of the numerator is less than the
degree of the denominator.
Theorem.
If a rational function R(x) is proper,
then y = 0 is a horizontal asymptote
of its graph.
Horizontal and Oblique
Asymptotes
Improper rational function R(x): one
that is not proper.
May be written
where
is proper. (Long division!)
Horizontal and Oblique
Asymptotes
If f(x) = b, (a constant)
If f(x) = ax + b, a 0,
Line y = b is a horizontal asymptote
Line y = ax + b is an oblique asymptote
In all other cases, the graph of R
approaches the graph of f, and there
are no horizontal or oblique
asymptotes.
This is all higher-degree polynomials
Horizontal and Oblique
Asymptotes
Example. Find the hoizontal or
oblique asymptotes, if any, of the
graph of each rational function.
(a) Problem:
Answer:
(b) Problem:
Answer:
Horizontal and Oblique
Asymptotes
Example. (cont.)
(c) Problem:
Answer:
(d) Problem:
Answer:
Key Points
Rational Functions
Graphing Rational Functions
Vertical Asymptotes
Horizontal and Oblique Asymptotes
The Graph of a
Rational Function;
Inverse and Joint
Variation
Section 3.4
Analyzing Rational Functions
Find the domain of the rational
function.
Write R in lowest terms.
Locate the intercepts of the graph.
x-intercepts: Zeros of numerator of
function in lowest terms.
y-intercept: R(0), if 0 is in the domain.
Test for symmetry – Even, odd or
neither.
Analyzing Rational Functions
Locate the vertical asymptotes:
Zeros of denominator of function in
lowest terms.
Locate horizontal or oblique
asymptotes
Graph R using a graphing utility.
Use the results obtained to graph by
hand
Analyzing Rational Functions
Example.
Problem: Analyze the graph of the
rational function
Answer:
Domain:
R in lowest terms:
x-intercepts:
y-intercept:
Symmetry:
Analyzing Rational Functions
Example. (cont.)
Answer: (cont.)
Vertical asymptotes:
Horizontal asymptote:
Oblique asymptote:
Analyzing Rational Functions
Example. (cont.)
Answer: (cont.)
4
2
-4
-2
2
-2
-4
4
Analyzing Rational Functions
Example.
Problem: Analyze the graph of the
rational function
Answer:
Domain:
R in lowest terms:
x-intercepts:
y-intercept:
Symmetry:
Analyzing Rational Functions
Example. (cont.)
Answer: (cont.)
Vertical asymptotes:
Horizontal asymptote:
Oblique asymptote:
Analyzing Rational Functions
Example. (cont.)
Answer: (cont.)
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Variation
Inverse variation:
Let x and y denote 2 quantities.
y varies inversely with x
If there is a nonzero constant such that
Also say: y is inversely proportional to
x
Variation
Joint or Combined Variation:
Variable quantity Q proportional to the
product of two or more other variables
Say Q varies jointly with these
quantities.
Combinations of direct and/or inverse
variation are combined variation.
Variation
Example. Boyle’s law states that for a
fixed amount of gas kept at a fixed
temperature, the pressure P and
volume V are inversely proportional
(while one increases, the other
decreases).
Variation
Example. According to Newton, the
gravitational force between two
objects varies jointly with the
masses m1 and m2 of each object
and inversely with the square of the
distance r between the objects,
hence
Key Points
Analyzing Rational Functions
Variation
Polynomial and
Rational
Inequalities
Section 3.5
Solving Inequalities
Algebraically
Rewrite the inequality
Left side: Polynomial or rational
expression f. (Write rational expression
as a single quotient)
Right side: Zero
Should have one of following forms
f(x)
f(x)
f(x)
f(x)
>
¸
<
·
0
0
0
0
Solving Inequalities
Algebraically
Determine where left side is 0 or
undefined.
Separate the real line into intervals
based on answers to previous step.
Solving Inequalities
Algebraically
Test Points:
Select a number in each interval
Evaluate f at that number.
If the value of f is positive, then
f(x) > 0 for all numbers x in the
interval.
If the value of f is negative, then
f(x) < 0 for all numbers x in the
interval.
Solving Inequalities
Algebraically
Test Points (cont.)
If the inequality is strict (< or >)
Don’t include values where x = 0
Don’t include values where x is undefined.
If the inequality is not strict (· or ¸)
Include values where x = 0
Don’t include values where x is undefined.
Solving Inequalities
Algebraically
Example.
Problem: Solve the inequality x5 ¸ 16x
Answer:
Key Points
Solving Inequalities Algebraically