Download x 3 + 3x 4 = 2

Chapter 2: Solving Equations and Inequalities (Page 1 of 40)
2.1 Linear Equations and Problem Solving
Example 1
Equations with Fractions
x 3x
Solve
+
=2
3 4
Example 2
Solve
To simplify an
equation containing
fractions, first
multiply every term in
the equation by the
LCD of the terms.
Then solve the simpler
equation.
Watch for Extraneous Solutions
1
3
6x
=
− 2
x−2 x+2 x −4
2.1 Linear Equations and Problem Solving (Page 2 of 40)
Mathematical Modeling With Linear Equations
The process of translating phrases or sentences into algebraic
expressions or equations is called mathematical modeling. A
mathematical model is an equation describing a real-life situation.
One method of modeling involves two stages: (1) Use the verbal
statement of the problem to form a verbal model of the problem;
(2) Assign variables to quantities and form the mathematical model
(equation).
Verbal
Description
Verbal
Model
Mathematical
Model (equation)
Example 1
Using a Verbal Model
You accept a job for which the annual salary will be $32,300. This
salary includes a year-end bonus of $500. You will be paid twice a
month. What will be your gross pay (before taxes) of each
paycheck?
Example 2
Find the Percent of a Raise
You accepted a job that pays $8 an hour. You are told that after a
two-month probationary period, you wage will increase to $9 per
hour. What percent raise will you earn after the probationary
period?
2.1 Linear Equations and Problem Solving (Page 3 of 40)
Example 3
Find the Percent Monthly Expenses
Your family has an annual income of $57,000 and the following
monthly expenses: mortgage ($1100), car payment ($375), food
($295), utilities ($240) and credit cards ($220). What percent of
your total family income are your total monthly expenses?
Example 4
Dimensions of a Room
A rectangular kitchen is twice as long as it is wide, and its
perimeter is 84 feet. Find the dimensions of the kitchen.
Example 5
A Distance Problem
A plane is flying non-stop from Atlanta to
Portland, a distance of 2700 miles. After 1.5
hours in the air, the plane flies over Kansas
city (a distance of 820 miles from Atlanta).
Estimate the time it will take the plane to fly
from Atlanta to Portland.
2.1 Linear Equations and Problem Solving (Page 4 of 40)
Example 6
Similar Triangles
To determine the height of the Aon
Center Building (in Chicago), you
measure the shadow cast by the
building and find it to be 142 feet
long. Then you measure the shadow
cast by a 4-foot post and find it to be
6 inches long. Estimate the height of
the building.
Example 7
Simple Interest, I = Prt
You invest a total of $10,000 at 4.5% and 5.5% interest. During
one year the two accounts earned $508.75. How much did you
invest in each account?
2.1 Linear Equations and Problem Solving (Page 5 of 40)
Example 8
An Inventory Problem
A store has $30,000 of inventory in single-disc DVD players and
multi-disc DVD players. The profit on a single-disc player is 22%
and the profit on the multi-disc player is 40%. The profit for the
entire stock is 35%. How much was invested in each type of DVD
player?
Example 9
Using a Formula
A cylindrical can has a volume of 200 cubic centimeters
and a radius of 4 centimeters. Find the height of the can.
The volume of a cylinder is V = π r 2 h .
See “Common Formulas” on page 170.
2.2 Solve Equations Graphically (Page 6 of 40)
2.2 Solving Equations Graphically
Intercepts in a Graph
1. An x-intercept is an ordered pair in the form (x, 0). To find the
x-intercepts, set y = 0 and solve for x.
2. A y-intercept is an ordered pair in the form (0, y). To find the
y-intercepts, set x = 0 and solve for y.
Example 1
Find the Intercepts
Find the x- and y-intercepts in the graph of 2x + 5y = 10 .
The Zeros of a Function
The zeros of a function f are the x-values for which f (x) = 0 .
Example 2
Verify the Zeros of a Function
a. Verify that the numbers -2 and 3 are zeros of f (x) = x 2 − x − 6
.
b. What are the x-intercepts of the graph of f (x) = x 2 − x − 6 .
The Zeros and x-Intercepts
The following statements are equivalent:
1.
The point (a, 0) is an x-intercept in the graph of f.
2.
The number a is a zero of f.
2.2 Solve Equations Graphically (Page 7 of 40)
3.
The number a is a solution to f (x) = 0 . That is, f (a) = 0
Finding Solutions Graphically
Example 3
Find Solutions Graphically
Use a graphing utility to approximate the solutions of
2x 2 − 3x = −2
Example 4
Find Solutions Graphically
Use a graphing utility to approximate the solutions of x 2 + 3 = 5x
2.2 Solve Equations Graphically (Page 8 of 40)
Example 5
Finding Points of Intersection
a. Find the points of intersection in the graphs of y = x + 2 and
y = x 2 − 2x − 2 .
b. Solve x + 2 = x 2 − 2x − 2
Example 6
Finding Points of Intersection
Find the points of intersection in the graphs of 2x − 3y = −2 and
4x − y = 6 .
Example 7
Finding Points of Intersection
a. Find the points of intersection in the graphs of y = x 2 − 3x + 4
and y = x 3 + 3x 2 − 2x − 1.
b. Solve x 2 − 3x + 4 = x 3 + 3x 2 − 2x − 1
2.2 Solve Equations Graphically (Page 9 of 40)
2.3 Complex Numbers (Page 10 of 40)
2.3 Complex Numbers
So far we have said the equation x 2 = −1 has no real solutions.
However, by introducing complex numbers, we can solve such
equations.
The Imaginary Unit i
The imaginary unit, written i, is the number whose square is –1.
That is,
i 2 = −1
or
i = −1
Complex Number a + bi
A complex number in standard form is written a + bi, where a
and b are real numbers. If b = 0, then a + bi = a is a real number.
If b ≠ 0 , then a + bi is an imaginary number. If a = 0 and b ≠ 0 ,
then a + bi = bi is called a pure imaginary number.
Complex
Number
2 + 5i
− 23 − 43 i
5
Real Part Complex Part Real, Imaginary,
or Pure Imaginary
a
bi
−i 2
Example 1
Add & Subtract Complex Numbers
(4 + 7i) + (1− 6i)
a.
b.
(1+ 2i) − (4 + 2i)
2.3 Complex Numbers (Page 11 of 40)
Example 2
Multiply Complex Numbers
−3(5 − 6i)
(1+ 2i)(4 + 2i)
a.
b.
c.
(3 + 2i)2
d.
(3 + 4i)(3 − 4i)
Complex Conjugates and Their Product
The numbers a + bi and a – bi are complex conjugates of each
other. The product of a complex number and its conjugate is a real
number, that is,
(a + bi)(a − bi) = a 2 + b 2
Example 3
Multiply Complex Conjugate
Multiply each complex number by its conjugate.
3+ i
3 − 4i
a.
b.
Example 4
Simplify a Quotient
3 − 4i
Simplify
.
4 − 2i
Fact: A quotient is
not simplified until
the denominator is a
real number.
2.3 Complex Numbers (Page 12 of 40)
Square Root of a Negative Number If a is a positive real number, then
For example:
−a = i a
− 4 = −1 4 = i 4 = 2i .
Example 5
Simplify each complex number and write in a + bi form.
1. −18
2. − −24
3.
−3 −12
5. −6i 3 + i 2
4. (−2 + −3)2
6. (−2i)−3
2.4 Solving Equations Algebraically (Page 13 of 40)
2.4 Solving Quadratic Equations Algebraically
A quadratic equation in general form is a second-degree
polynomial equation written in the form
ax 2 + bx + c = 0 ,
where a ≠ 0 . Furthermore, ax 2 is called the quadratic term, bx is
the linear term and c is the constant term.
The methods we study to solve quadratic equations are factoring,
extracting square roots, completing the square, and using the
Quadratic Formula. The factoring method requires the
Zero-Factor Property:
If ab = 0 , then a = 0 or b = 0.
Example 1
Solve a Quadratic Equation by Factoring
2x 2 + 9x + 7 = 3
a.
Solve and check
b.
Solve and check
6x 2 − 3x = 0
2.4 Solving Equations Algebraically (Page 14 of 40)
Extracting Square Roots
The equation u 2 = d , where d > 0, has exactly two solutions:
u= d
and
u=− d
Example 2
Extracting Square Roots
2
a. Solve 4x = 12
b. Solve
9(x − 3)2 = 28
Completing The Square
To complete the square on the expression x 2 + bx , add (b / 2)2 , that is
add the square of half the linear coefficient. Then,
2
2
b⎞
⎛ b⎞
⎛
2
x + bx + ⎜ ⎟ = ⎜ x + ⎟ .
⎝ 2⎠
⎝
2⎠
Example 3
For each show the constant term of the perfect-square trinomial is
one-half of the coefficient of its linear term squared
1. (x + 5)2 = x 2 + 10x + 25
2. (x − 7)2 = x 2 − 14x + 49
Example 4
Find the value of c so the expression is a perfect square trinomial.
Then factor each perfect square trinomial.
1. x 2 + 12x + c
2. x 2 − 8x + c
3. x 2 + 4x + c
2.4 Solving Equations Algebraically (Page 15 of 40)
Steps To Solve ax 2 + bx + c = 0 by
Completing the Square
1. Write the equation in the form ax 2 + bx = − c .
That is, get all the variable terms on one side
of the equation and the constant term on the
other side.
2. The completing the square procedure requires
the leading coefficient to be one, so divide
both sides by a. Then
b
c
x2 + x = −
a
a
b
x a perfect square trinomial,
a
add half the linear coefficient squared to both
sides of the equation.
Solve
3x 2 − 18x − 3 = 0
1. 3x 2 − 18x = 3
2.
3x 2 − 18x 3
=
3
3
x 2 − 6x = 1
3. To make x 2 +
Since ⎡⎣ 12 ( −6 ) ⎤⎦ = (−3)2 = 9
3. x 2 − 6x + 9 = 1+ 9
x 2 − 6x + 9 = 10
4. Factor the perfect square trinomial:
(x + half the linear coefficient)2
4. (x − 3)2 = 10
5. Solve by taking square roots.
5.
2
(x − 3)2 = ± 10
x − 3 = ± 10
6. Check your solutions.
x = 3 ± 10
x = 3 ± 10
x ≈ −0.1623, 6.1623
You can also verify the graph of
y = 3x 2 − 18x − 3 has x-intercepts
at (− 0.162, 0) and (6.162, 0) .
6.
2.4 Solving Equations Algebraically (Page 16 of 40)
Example 3
Completing the Square
2
Solve x + 2x − 6 = 0 by completing the square.
1.
Write the equation in the form
ax 2 + bx = − c . That is, get all
the variable terms on one side of
the equation and the constant
term on the other side.
2.
The completing the square
procedure requires the leading
coefficient to be one, so divide
both sides by a. Then
b
c
x2 + x = −
a
a
3.
To make x 2 +
4.
Factor the perfect square
trinomial:
b
x a perfect
a
square trinomial, add half the
linear coefficient squared to both
sides of the equation.
(x + half the linear coefficient)2
5.
Solve by taking square roots.
6.
Check your solutions.
2.4 Solving Equations Algebraically (Page 17 of 40)
Example 4
Completing the Square
Solve 2x 2 + 8x + 3 = 0 by completing the square.
Example 5
Completing the Square
Solve 3x 2 − 4x − 5 = 0 by completing the square.
2.4 Solving Equations Algebraically (Page 18 of 40)
Derivation of the Quadratic Formula
Solve ax 2 + bx + c = 0 by completing the square.
The Quadratic Formula
The solutions the quadratic equation ax 2 + bx + c = 0 are given by the
Quadratic Formula
−b ± b 2 − 4ac
x=
2a
Example 6
Solve by the Quadratic Formula
Solve 3x 2 − 4x − 5 = 0 by using the quadratic formula.
2.4 Solving Equations Algebraically (Page 19 of 40)
Example 7
No Real Solutions, No x-Intercepts
1. Solve 2x 2 − 3x + 5 = 0 .
2. Find the zeros and x-intercepts of f (x) = 2x 2 − 3x + 5 .
Example 8
Two Real Solutions, Two x-Intercepts
1. Solve 2x 2 − 4x − 3 = 0 .
2. Find the zeros and x-intercepts of f (x) = 2x 2 − 4x − 3 .
Example 9
One Real Solutions, One x-Intercepts
2
1. Solve 2x − 4x + 2 = 0 .
2. Find the zeros and x-intercepts of f (x) = 2x 2 − 4x + 2 .
2.4 Solving Equations Algebraically (Page 20 of 40)
Example 10 Finding Room Dimensions
A rectangular bedroom is three feet longer than it is wide and has
an area of 154 square feet. Find the dimensions of the room.
The Position Equation
The height of an object above the earth’s surface is governed by the
position equation,
s = −16t 2 + v0t + s0 ,
where
s = the height of the object in feet
v0 = the initial velocity of the object
s0 = the initial height of the object
Example 11 Falling Time
A construction worker on the 24th floor of a building project
accidentally drops a wrench and yells “look out below!” Could a
person at ground level hear the warning in time to get out of the
way? The speed of sound is 110 feet per second. Assume each
floor is 10 feet high.
2.4 Solving Equations Algebraically (Page 21 of 40)
Example 12 Modeling Number of Internet Users
From 2000 to 2007, the estimated number of Internet users I (in
millions) in the United States can be modeled by the quadratic
equation
I = −1.163t 2 + 17.19t + 125.9 ,
0≤t ≤7
where t is the number of years since 2000. Use the model to
determine when the number of Internet users is predicted to be 180
million? Solve algebraically and graphically.
Exercise 110 Dimensions of a Corral
A rancher has 100 meters of fencing to enclose
two identical, adjacent rectangular corrals (as
diagramed). If the rancher wants to enclose a
total area of 350 square meters, what are the
dimensions of each corral?
2.5 Solving Other Types of Equations Algebraically (Page 22 of 40)
2.5 Solving Other Types of Equations Algebraically
Polynomial Equations
an x n + an−1 x n−1 ++ a2 x 2 + a1 x + a0 = 0
General Form:
Linear:
a1 x + a0 = 0
a2 x 2 + a1 x + a0 = 0
Quadratic:
a3 x 3 + a2 x 2 + a1 x + a0 = 0
Cubic:
Example 1
Solve Polynomial Equation by Factoring
3x 4 = 48x 2
Solve
Example 2
Solve Polynomial Equation by Factoring
2x 3 − 6x 2 + 6x − 18 = 0
Solve
Example 3
Solve Equation of the Quadratic Type
x 4 − 3x 2 + 2 = 0
Solve
2.5 Solving Other Types of Equations Algebraically (Page 23 of 40)
Example 4
Solve Equations With Radicals
a.
Solve
2x + 7 − x = 2
b.
Solve
2x − 5 − x − 3 = 1
Example 5
Solve
Solve an Equation with a Rational
Exponent
(x − 4)2/3 = 25
2.5 Solving Other Types of Equations Algebraically (Page 24 of 40)
Example 6
Solve an Equation with Fractions
2
3
Solve
=
−1
x x−2
Solving Absolute Value Equations
Let E be any expression and k be a real number, k ≥ 0 .
If E = k , then E = k or E = −k
For Example, if x = 4 , then x = 4 or x = - 4.
Example 7
Solve
Solve an Absolute Value Equation
x 2 − 3x = − 4x + 6
2.5 Solving Other Types of Equations Algebraically (Page 25 of 40)
Example 8
Reduced Rates
A ski club chartered a bus for a ski trip at a cost of $480. To lower
the cost of the bus per person, the club invited nonmembers to go
along. After 5 nonmembers joined the trip, the fare per skier
decreased by $4.80. How many club members went on the trip?
Compound Interest Formula
r⎞
⎛
A = P ⎜ 1+ ⎟
⎝ n⎠
nt
A = Amount invested after t years.
P = initial principal deposited
r = annual interest rate
n = number of compounding periods per year
Example 9
Compound Interest
When you were born, your grandparents deposited $5000 in longterm investments in which the interest was compounded quarterly.
Today, on your 25th birthday, the value of the investment is
$25,062.59. What is the annual interest rate for the investment?
1.8 Other Types of Inequalities (Page 26 of 40)
2.6 Solving Inequalities Algebraically and Graphically
A number satisfies an inequality in one variable if substituting the
number for the variable results in a true statement. Such numbers are
called solutions to the inequality. The solution set to an inequality is
the set of all solutions to the inequality. To solve an inequality means
to find all the solutions to the inequality. Two inequalities are said to be
equivalent if they have the same solutions set. Some example of linear
inequalities in one variable follow:
−3
4 − x < 3 , 3x + 2 ≤ 5 , −8(2 − 3x) > 7 ,
x≥5
7
Graphs of Solution Sets and Interval Notation
Since the solution set to a linear inequality is generally an infinite set,
we can express the solution set as an inequality, a graph, or as an
interval using what is referred to as interval notation.
In Words Inequality
Numbers less
than 4
Numbers less
than or equal to 4
Numbers greater
than 4
Numbers greater
than or equal to 4
Numbers
between 0 and 4
Numbers
between 0 and 4,
including 4
Numbers
between 0 and 4,
including 0 and 4
x<4
Graph
0
4
0
4
0
4
0
4
0<x<4
0
4
0<x≤4
0
4
0≤x≤4
0
4
x≤4
x>4
x≥4
Interval
Notation
1.8 Other Types of Inequalities (Page 27 of 40)
Properties of Inequalities
Let a, b, c and d be real numbers or expressions.
1.
Transitive Property
a < b and b < c
a<c
2.
Addition of Inequalities
a < b and c < d
a+c <b+d
3.
Addition Property
a<b
4.
Multiplication Property
For c > 0, a < b
ac < bc
For c < 0, a < b
ac > bc
a+c <b+c
There are corresponding inequalities for >, ≥ and ≤ inequalities.
Example 1
Solving Linear Inequalities
Solve each of the following. Express the solution set as an inequality, a
graph, and in interval notation.
1. 8 − 2x > 11
2. −3(4 − x) ≥ 14 + x
3. 2[x − (2x + 3)] ≤ 7 − 2(5 − x)
1.8 Other Types of Inequalities (Page 28 of 40)
Example 2
Solving Double Inequalities
Solve each of the following. Express the solution set as an inequality, a
graph, and in interval notation.
1. 17 < 5 − 3x ≤ 29
2.
−3 ≤ 6x − 1 < 3
1.8 Other Types of Inequalities (Page 29 of 40)
Absolute Value Property for Inequalities
For an expression E and positive number k.
E < k is equivalent to −k < E < k
1.
E > k is equivalent to E > k or E < −k
2.
Example 3
Solving Absolute Value Inequalities
Solve without a calculator. Describe the solution set as an inequality, in
a graph and in interval notation. Verify with a graphing calculator.
x ≤5
1.
2.
x >5
Example 4
Solving Absolute Value Inequalities
Solve 2x − 3 ≤ 9 without a calculator. Describe the solution set as an
inequality, in a graph and in interval notation. Verify with a graphing
calculator.
1.8 Other Types of Inequalities (Page 30 of 40)
Example 5
Solving Absolute Value Inequalities
Solve without a calculator. Describe the solution set as an inequality, in
a graph and in interval notation. Verify with a graphing calculator.
3t + 4 > 12
1.
2.
7− x+2 ≥ 3
1.8 Other Types of Inequalities (Page 31 of 40)
Example 6
Solve a Polynomial Inequality
x 2 − 2x < 3
Solve
Fact: A polynomial can only change signs at its zeros (the values of x
that make the polynomial equal to zero). These zeros are the
critical numbers of the inequality, which divide the number line
into test intervals for the inequality.
STEP 1
Write the inequality so that
one side is a polynomial
and the other side is zero.
STEP 2
Find all the real zeros of
the polynomial, that is, the
critical numbers of the
inequality.
STEP 3
Arrange the critical
numbers along a “working
number line” showing the
test intervals.
STEP 4
Evaluate the polynomial at
a test value in each test
interval. The polynomial
will have the same sign for
every value on each test
interval.
1.8 Other Types of Inequalities (Page 32 of 40)
Example 7
Solving a Polynomial Inequality
1. Solve x 2 − x < 6
2. Solve 2x 3 − 3x 2 − 32x > −48
1.8 Other Types of Inequalities (Page 33 of 40)
Example 8
Unusual Solution Sets
a. Solve x 2 + 2x + 4 > 0
b. Solve x 2 + 2x + 1 ≤ 0
c.
Solve x 2 + 3x + 5 < 0
d. Solve x 2 − 4x + 4 > 0
1.8 Other Types of Inequalities (Page 34 of 40)
Example 9
Solving a Rational Inequality
2x − 7
Solve
≤ 3.
x−5
Fact: A rational expression can only change signs at the zeros of the
numerator and denominator. These zeros are the critical
numbers of the inequality, which divide the number line into
test intervals for the inequality.
Example 10 Finding the Domain
Find the domain of 64 − 4x 2
Recall: The domain
of an expression is the
set of all x-values so
that the expression is a
real number.
1.8 Other Types of Inequalities (Page 35 of 40)
Example 11 Comparative Shopping
You need to choose between two different cell phone plans. Plan A costs
$49.99 per month for 500 minutes plus $0.40 for each additional minute.
Plan B costs $45.99 per month for 500 minutes and $0.45 for each
additional minute. Beyond the 500 included minutes, how many
additional minutes must you use for plan B to cost more than plan A?
Example 12 Height of a Projectile
A projectile is fired straight upward from ground level with an initial
velocity of 384 feet per second. During what time period will its height
exceed 2000 feet? Recall: s(t) = −16t 2 + v0 t + s0 .
1.8 Other Types of Inequalities (Page 36 of 40)
2.7 Linear Models and Scatter Plots
Correlation
Example 1
Scatter Plots
The data in the table shows the outstanding household credit market debt
D (in trillions of dollars) from 1998 through 2004.
a.
Construct a scatter plot of the data.
Year
1998
1999
2000
2001
2002
2003
2004
b.
Household credit
market debt, D
(trillions of dollars)
6.0
6.4
7.0
7.6
804
9.2
10.3
Find the linear regression equation that models D.
1.8 Other Types of Inequalities (Page 37 of 40)
Linear Regression Function
The linear regression line is the line that mathematically best fits the
data. The linear regression equation, or linear regression function, is
the equation of the regression line.
Finding the Linear Regression Equation on the TI-83
1. Enter the independent variable data values into list 1 (L1) and the
corresponding dependent variable values into list 2 (L2). To access
your lists press STAT followed by ENTER.
STAT / 1:Edit
2. Press STAT PLOT followed by
ENTER. Then set the Plot1
settings as shown. Press
ZOOM / 9:ZoomStat
to view the scattergram.
3. Run the linear regression program:
STAT / CALC / 4:LinReg (ax+b) Lx , Ly , Y1
On the TI-83 a is the slope and (0, b) is the “y”-intercept (i.e. the
vertical axis intercept). Write a and b to three decimal places.
3. Rewrite the equation using the variables in the application.
1.8 Other Types of Inequalities (Page 38 of 40)
Example 2
The American life span has been increasing over the last century. Let
L(t) represent the life expectancy at birth for
Birth
Life
an American born t years after 1970.
Year
Expectancy
1. Create a scatter plot of the data on your
1970
70.8
1975
72.6
calculator and determine if a linear
1980
73.7
model is appropriate. If it is, then find
1985
74.7
the linear regression model for the data.
1990
1995
2000
75.4
76.1
76.9
2. Use the model to predict the life expectancy for someone born in
2010.
3. Use the model to predict the birth year for someone whose life
expectancy at birth is 66 years.
4. Interpret the meaning of the slope in this application.
1.8 Other Types of Inequalities (Page 39 of 40)
Example 3
Let P represent the salmon population (in
millions) at t years since 1950.
1. Create a scatter plot of the data on
your calculator and determine if a
linear model is appropriate. If it is,
then find the linear regression model
for the data.
2. Predict the salmon population in year
1955.
Number of
Years
since 1950
t
10
15
20
25
30
35
40
Salmon
Population
(millions)
P
10.02
10.00
7.61
3.15
4.59
3.11
2.22
3. Predict the year when the salmon population was 6.4 million.
4. Explain the meaning of the slope in this situation.
1.8 Other Types of Inequalities (Page 40 of 40)
Example 4
The table shows the average salaries for
professors at four-year colleges and
universities. Let s represent the average salary
(in thousands of dollars) at t years since 1970.
a. Verify the linear regression equation for
s is s = 1.70t + 6.71 .
b. Predict the average salary in 2008.
c.
Year
Average Salary
(thousands of dollars)
1975
1980
1985
1990
1995
2000
16.6
22.1
31.2
41.9
49.1
57.7
Predict when the average salary will be $75,000.
d. Explain the meaning of the slope in this situation.