Copyright © 2001 by the McGraw-Hill Companies, Inc.

Barnett/Ziegler/ByleenCollege Algebra with Trigonometry, 7th Edition

Chapter Two

Equations & Inequalities

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

4. If a = b, then ac =

bc , c 0. Division Property

5. If a = b, then either may replace the other inany statement without changing the truthor falsity of the statement.

Substitution Property

Properties of Equality

2-1-11

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 with 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.

Strategy for Solving Word Problems

2-1-12

R = QT Rate =

QuantityTime

Q = RT Quantity = (Rate)(Time)

T = QR Time =

QuantityRate

If Q is distance D, then

R = DT D = RT T =

DR

[Note: R is an average or uniform rate.]

Quantity-Rate-Time Formulas

2-1-13

ax + by = h System of two linearcx + dy = k equations in two variables

Note that x and y are variables and a, b, c, d, h, and k are real constants.

A pair of numbers x = x0 and y = y0 is a solution of this system if each

equation is satisfied by the pair. The set of all such pairs of numbers iscalled the solution set for the system. To solve a system is to find itssolution set.

To solve a system by substitution, choose one of the two equations in asystem and solve for one variable in terms of the other.

Then substitute the result in the other equation and solve the resultinglinear equation in one variable.

Finally, substitute this result back into the expression obtained in thefirst step to find the second variable.

Systems of Linear Equations

2-2-14

[a, b] a x b [ ]a b

x Closed

[a, b) a x < bb

[a

) x Half-open

(a, b] a < x b ]a b

x( Half-open

(a, b) a < x < ba b

x( ) Open

Interval InequalityNotation Notation Line Graph Type

Interval Notation

2-3-15-1

[b , ) x bb

x[Closed

( b, ) x > b bx(

Open

( –, a] x a ax]

Closed

( –, a) x < a ax)

Open

Interval InequalityNotation Notation Line Graph Type

Interval Notation

2-3-15-2

1. If a < b and b < c, then a < c. Transitive Property

2. If a < b, then a + c < b + c. Addition Property

3. If a < b, then a – c < b – c. Subtraction Property

4. If a < b and c is positive, then ca < cb .

5. If a < b and c is negative, then ca > cb .

Multiplication Property(Note difference between4 and 5.)

6. If a < b and c is positive, then ac <

bc .

7. If a < b and c is negative, then ac >

bc .

Division Property(Note difference between6 and 7.)

For a, b, and c any real numbers:

Inequality Properties

2-3-16

| x – c | = d {c – d, c + d}

d d

c – d c c + dx

| x – c | < d (c – d, c + d) c – d c c + dx

x 0 < | – c | < d c , c d( – d c) (c, + )

xc – d c c + d

| x – c | > d d + , (

, c – ) (c d )

xc – d c c + d

Absolute Value Equations and Inequalities

2-4-17

Imaginary Unit: i

Complex Number: a + bi a and b real numbers

Imaginary Number: a + bi b 0

Pure Imaginary Number: 0 + bi = bi b 0

Real Number: a + 0 i = a

Zero: 0 + 0 i = 0

Conjugate of a + bi : a – bi

Particular Kinds of Complex Numbers

2-5-18

Naturalnumbers (N)

NegativeIntegers

Zero Integers (Z)

Nonintegerrationalnumbers

Rationalnumbers (Q)

Irrationalnumbers (I)

Realnumbers (R)

Complexnumbers (C)

Imaginarynumbers

N Z Q R C

Subsets of the Set of Complex Numbers

2-5-19

If ax2 + bx + c = 0, a ? 0, then

x = –b ± b2 – 4ac

2a

Quadratic Formula

Discriminant and Roots

2 cDiscrimant Roots of ax + bx + = 02 a b c ab – 4ac , , and real numbers, 0

Positive Two distinct real roots

0 One real root (a double root)

Negative Two imaginary roots, one the conjugate of the other

2-6-20

If both sides of an equation are squared, then the solutionset of the original equation is a subset of the solution set ofthe new equation.

Every solution of the new equation must be checked in theoriginal equation to eliminate extraneous solutions.

Squaring Operation on Equations

Equation Solution Set

x= 3 {3}x2= 9 {–3, 3}

2-7-21

Step 1. Write the polynomial inequality in standard form (a form where the right-hand side is 0.)

Step 2. Find all real zeros of the polynomial (the left side of the standard form.)

Step 3. Plot the real zeros on a number line, dividing the number line into intervals.

Step 4. Choose a test number (that is easy to compute with) in each interval, and evaluate the polynomial for each number (a small table is useful.)

Step 5. Use the results of step 4 to construct a sign chart, showing the sign of the polynomial in each interval.

Step 6. From the sign chart, write down the solution of the original polynomialinequality (and draw the graph, if required.)

Key Steps in Solving Polynomial Inequalities

2-8-22