Copyright © 1999 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4 th Edition Chapter One 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 a c = b c, c  0. Division Property 5.If a = b, then either may replace the other in any statement without changing the truth or falsity of the statement. Substitution Property Properties of Equality 1-1-1

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 1-1-2

Quantity-Rate-Time Formulas 1-1-3

Systems of Linear Equations 1-2-4

[a, b]a  x  b [] ab x Closed [a, b)a  x <b b [ a ) x Half-open (a, b]a <x  b ] a b x ( Half-open (a, b)a <x <b ab x () Open Interval Inequality Notation Notation Line Graph Type Interval Notation

[b,)x  b b x [ Closed ( b,  )x >b b x ( Open ( –,–, a]x  a a x ] Closed (– ,, a)x <a a x ) Open Interval Inequality Notation Notation Line Graph Type  Interval Notation

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 between 4 and 5.) 6.If a < b and c is positive, then a c < b c. 7.If a < b and c is negative, then a c > b c.   Division Property (Note difference between 6 and 7.) For a, b, and c any real numbers: Inequality Properties 1-3-6

| x –c | =d{c –d,c +d} | x –c | <d(c– d,c +d) x 0 < |–c | < dc,cd (–dc )  ( c, +) | x –c | >d d +, (  ,c– )  ( cd  ) Absolute Value Equations and Inequalities 1-4-7

Imaginary Unit: i Complex Number: a + bia and b real numbers Imaginary Number: a + bi b  0 Pure Imaginary Number:0 + 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 1-5-8

Natural numbers (N) Negative Integers Zero Integers (Z) Noninteger rational numbers Rational numbers (Q) Irrational numbers (I) Real numbers (R) Complex numbers (C) Imaginary numbers N  Z  Q  R  C Subsets of the Set of Complex Numbers 1-5-9

Quadratic Formula Discriminant and Roots 2 c DiscrimantRoots of ax + bx + = 0 2 ab ca b – 4 ac,, andreal numbers, 

Squaring Operation on Equations EquationSolution Set x= 3{3} x 2 = 9{–3, 3}

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 polynomial inequality (and draw the graph, if required.) Key Steps in Solving Polynomial Inequalities