Chapter 2 Review
163
(A) What is the average rate that the ball falls during the first second (from x 0 to x 1 second)? During
the second second? During the third second?
By definition, average rate involves the distance an object travels over an interval of time, as in part A. How
can we determine the rate of an object at a given instant of time? For example, how fast is the ball falling at
exactly 2 seconds after it was released? We will approach this problem from two directions, numerically and
algebraically.
(B) Complete the following table of average rates. What number do these average rates appear to approach?
Time interval
[1.9, 2]
[1.99, 2]
[1.999, 2]
[1.9999, 2]
Distance fallen
Average rate
64 16t2
(C) Show that the average rate over the time interval [t, 2] is
. Simplify this algebraic expression and
2t
discuss its values for t very close to 2.
(D) Based on the results of parts B and C, how fast do you think the ball is falling at 2 seconds?
Chapter 2 Review
2-1
LINEAR EQUATIONS AND APPLICATIONS
A solution or root of an equation is a number in the domain or
replacement set of the variable that when substituted for the
variable makes the equation a true statement. An equation is an
identity if it is true for all values from the domain of the variable and a conditional equation if it is true for some domain
values and false for others. Two equations are equivalent if they
have the same solution set. The properties of equality are used
to solve equations:
1. If a b, then a c b c.
Addition Property
2. If a b, then a c b c.
Subtraction Property
3. If a b, then ca cb, c 0.
Multiplication Property
a b
4. If a b, then , c 0.
c c
Division Property
5. If a b, then either may replace Substitution Property
the other in any statement without
changing the truth or falsity of
statement.
An equation that can be written in the standard form ax b 0, a 0, is a linear or first-degree equation.
Strategy for Solving Word Problems
1. Read the problem carefully—several times if necessary—that is, until you understand the problem,
know what is to be found, and know what is given.
2. Let one of the unknown quantities be represented by
a variable, say x, and try to represent all other
unknown quantities in terms of x. This is an important step and must be done carefully.
3. If appropriate, draw figures or diagrams and label
known and unknown parts.
4. Look for formulas connecting the known quantities
to the unknown quantities.
5. Form an equation relating the unknown quantities to
the known quantities.
6. Solve the equation and write answers to all questions
asked in the problem.
7. Check and interpret all solutions in terms of the original problem—not just the equation found in step
5—since a mistake may have been made in setting
up the equation in step 5.
2 Equations and Inequalities
If Q is quantity produced or distance traveled at an average or
–rate–
–time
uniform rate R in T units of time, then the quantity–
formulas are
R
2-2
Q
T
T
Q RT
Q
R
a b
7. If a b and c 0, then .
c c
SYSTEMS OF LINEAR EQUATIONS AND
APPLICATIONS
ax by h
(1)
cx dy k
where x and y are the two variables. The ordered pair of numbers (x0, y0) is a solution to system (1) if each equation is satisfied by the pair. The set of all such ordered pairs of numbers is
called the solution set for the system. To solve a system is to
find its solution set. To solve a system by substitution, solve either equation for either variable, substitute in the other equation,
solve the resulting linear equation in one variable, and then substitute this value into the expression obtained in the first step to
find the other variable.
If one equation in a system is a demand equation and
the other is a supply equation, then the solution produces the
equilibrium price and the equilibrium quantity. If one equation in a system is a cost equation (often formed by using fixed
costs and variable costs) and the other is a revenue equation,
then the solution produces the number of units that must be
manufactured to break even.
The inequality symbols , , , are used to express inequality relations. Line graphs, interval notation, and the set
operations of union and intersection are used to describe inequality relations. A solution of a linear inequality in one variable is a value of the variable that makes the inequality a true
statement. Two inequalities are equivalent if they have the
same solution set. Inequality properties are used to solve
inequalities:
Transitive Property
2. If a b, then a c b c.
Addition Property
3. If a b, then a c b c.
Subtraction Property
aFbFc
1. If a b and b c, then a c.
5. If a b and c 0, then ca cb.
Division Property
2-4
ABSOLUTE VALUE IN EQUATIONS AND
INEQUALITIES
The absolute value of a number x is the distance on a real number line from the origin to the point with coordinate x and is
given by
x x
x
if x 0
if x 0
The distance between points A and B with coordinates a and
b, respectively, is d(A, B) b a, which has the following
geometric interpretations:
x c d
Distance between x and c is equal to d.
x c d
Distance between x and c is less than d.
0 x c d
Distance between x and c is less than d,
but x c.
x c d
Distance between x and c is greater
than d.
Equations and inequalities involving absolute values are solved
using the following relationships for p 0:
1. x p is equivalent to x p or x p.
LINEAR INEQUALITIES
4. If a b and c 0, then ca cb.
a b
.
c c
The order of an inequality reverses if we multiply or divide both sides of an inequality statement
by a negative number.
A system of two linear equations with two variables is a system
of the form:
2-3
6. If a b and c 0, then
aGdddEbdEddGc
164
Multiplication Property
2. x p is equivalent to p x p.
3. x p is equivalent to x p or x p.
These relationships also hold if x is replaced with ax b. For x
any real number, x2 x.
2-5
COMPLEX NUMBERS
A complex number in standard form is a number in the form
a bi, where a and b are real numbers and i is the imaginary
unit. If b 0, then a bi is also called an imaginary number.
If a 0, then 0 bi bi is also called a pure imaginary number. If b 0, then a 0i a is a real number. The complex
zero is 0 0i 0. The conjugate of a bi is a bi. Equality, addition, and multiplication are defined as follows:
Chapter 2 Review Exercise
1. a bi c di
if and only if
a c and b d
4. Using the quadratic formula:
2. (a bi) (c di) (a c) (b d )i
3. (a bi)(c di) (ac bd ) (ad bc)i
Since complex numbers obey the same commutative, associative, and distributive properties as real numbers, most operations with complex numbers are performed by using these properties and the fact that i2 1. The property of conjugates,
(a bi)(a bi) a2 b2
can be used to find reciprocals and quotients. If a 0, then the
principal square root of the negative real number a is
a ia.
2-6
QUADRATIC EQUATIONS AND APPLICATIONS
A quadratic equation in standard form is an equation that can
be written in the form
ax2 bx c 0
a0
where x is a variable and a, b, and c are constants. Methods of
solution include:
165
x
b b2 4ac
2a
If the discriminant b2 4ac is positive, the equation has two
distinct real roots; if the discriminant is 0, the equation has one
real double root; and if the discriminant is negative, the equation has two imaginary roots, each the conjugate of the other.
2-7
EQUATIONS REDUCIBLE TO QUADRATIC FORM
A square root radical can be eliminated from an equation by
isolating the radical on one side of the equation and squaring
both sides of the equation. The new equation formed by squaring both sides may have extraneous solutions. Consequently,
every solution of the new equation must be checked in the
original equation to eliminate extraneous solutions. If an
equation contains more than one radical, then the process of isolating a radical and squaring both sides can be repeated until all
radicals are eliminated. If a substitution transforms an equation
into the form au2 bu c 0, where u is an expression in
some other variable, then the equation is an equation of quadratic type that can be solved by quadratic methods.
2-8
POLYNOMIAL AND RATIONAL INEQUALITIES
1. Factoring and using the zero property:
mn0
m 0 or n 0 (or both)
if and only if
2. Using the square root property:
If A2 C, then A C
3. Completing the square:
x2 bx 2 x 2b b
2
2
An inequality is in standard form if the right side is 0. If the left
side is a polynomial, then the real zeros of this polynomial divide the real number line into intervals with the property that the
polynomial has constant sign over each interval. Selecting a test
number in each interval and constructing a sign chart produces
the solution to the inequality. If the left side of an inequality is a
rational expression of the form P/Q, where P and Q are polynomials, then the real zeros of both polynomials are used to divide the real number line into intervals over which P/Q has constant sign. Since division by zero is never allowed, the real
zeros of Q must always be excluded from the solution set.
Chapter 2 Review Exercise
Work through all the problems in this chapter review and
check answers in the back of the book. Answers to all review
problems are there, and following each answer is a number
in italics indicating the section in which that type of problem
is discussed. Where weaknesses show up, review appropriate
sections in the text.
2.
5x 4 x x 2
1
3
2
4
3. y 4x 9
y x 6
Solve and graph Problems 4–8.
A
4. 3(2 x) 2 2x 1
5. y 9 5
Solve Problems 1–3.
6. 3 2x 5
7. x2 x 20
1. 0.05x 0.25(30 x) 3.3
8. x2 4x 21