Beginning and Intermediate Algebra - Wallace Math Courses ...

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ISBN #978-1-4583-7768-5 Copyright 2010, Some Rights Reserved CC-BY. Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. (http://creativecommons.org/licenses/by/3.0/) Based on a work at http://wallace.ccfaculty.org/book/book.html. You are free: • to Share: to copy, distribute and transmit the work • to Remix: to adapt the work Under the following conditions: • Attribution: You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). With the understanding that: • Waiver: Any of the above conditions can be waived if you get permission from the copyright holder. • Public Domain: Where the work or any of its elements is in the public domain under applicable law, that status is in no way affected by the license. • Other Rights: In no way are any of the following rights affected by the license: − Your fair dealing or fair use rights, or other applicable copyright exceptions and limitations; − The author’s moral rights; − Rights other persons may have either in the work itself or in how the work is used such as publicity or privacy rights • Notice: For any reuse or distribution, you must make clear to others the license term of this work. The best way to do this is with a link to the following web page: http://creativecommons.org/licenses/by/3.0/ This is a human readable summary of the full legal code which can be read at the following URL: http://creativecommons.org/licenses/by/3.0/legalcode 2
Special thanks to: My beautiful wife, Nicole Wallace who spent countless hours typing problems and my two wonderful kids for their patience and support during this project Another thanks goes to the faculty reviewers who reviewed this text: Donna Brown, Michelle Sherwood, Ron Wallace, and Barbara Whitney One last thanks to the student reviewers of the text: Eloisa Butler, Norma Cabanas, Irene Chavez, Anna Dahlke, Kelly Diguilio, Camden Eckhart, Brad Evers, Lisa Garza, Nickie Hampshire, Melissa Hanson, Adriana Hernandez, Tiffany Isaacson, Maria Martinez, Brandon Platt, Tim Ries, Lorissa Smith, Nadine Svopa, Cayleen Trautman, and Erin White 3
Page 1: Beginning and Intermediate Algebra
Page 5 and 6: Chapter 6: Factoring 6.1 Greatest C
Page 7 and 8: 0.1 Pre-Algebra - Integers Objectiv
Page 9 and 10: Multiplication and division of inte
Page 11 and 12: 45) (4)( − 6) Find each quotient.
Page 13 and 14: 9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
Page 15 and 16: Example 21. 13 6 − 9 6 4 6 2 3 Sa
Page 17 and 18: Find each quotient. 37) − 2 ÷ 7
Page 19 and 20: the expression that is to be evalua
Page 21 and 22: Solve. 1) − 6 · 4( − 1) 3) 3+(
Page 23 and 24: It will be more common in our study
Page 25 and 26: 0.4 Practice - Properties of Algebr
Page 27 and 28: Chapter 1 : Solving Linear Equation
Page 29 and 30: Addition Problems To solve equation
Page 31 and 32: 8x = − 24 8 8 x = − 3 Division
Page 33 and 34: 1.2 Linear Equations - Two-Step Equ
Page 35 and 36: − 1 − 1 Divide both sides by
Page 37 and 38: 1.3 Solving Linear Equations - Gene
Page 39 and 40: Example 63. − 3x +9=6x − 27 Not
Page 41 and 42: 4[ − 2]+9=−15 +8(2) Finish mult
Page 43 and 44: 1.4 Solving Linear Equations - Frac
Page 45 and 46: Example 72. � � 3 5 4 x + = 3 2
Page 47 and 48: 1.5 Solving Linear Equations - Form
Page 49 and 50: Example 79. A = πr 2 +πrs for s S
Page 51 and 52: 1.5 Practice - Formulas Solve each
Page 53 and 54:
|x| = 5 Absolute value can be posit
Page 55 and 56:
Example 90. 7+|2x − 5| =4 Notice
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1.7 Solving Linear Equations - Vari
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Once we have found the constant of
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1.7 Practice - Variation Write the
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sure. If the pressure of a certain
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Fifteen more than three times a num
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When we started with our first, sec
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1.8 Practice - Number and Geometry
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30. The perimeter of a college bask
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Our equation comes from the future
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4x + 2=102 Solve the two − step e
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years ago, the bronze plaque was on
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1.10 Solving Linear Equations - Dis
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Rate Time Distance Bob r + 2 3 Fred
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2 =t Our solution fort, she catches
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Find the distance he rode. 8. A man
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The first plan is flying 25 mph slo
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2.1 Graphing - Points and Lines Obj
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E B D F G C A Our Solution The main
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x y − 3 − 4 0 − 2 3 0 Our com
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2.2 Graphing - Slope Objective: Fin
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When mathematicians began working w
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Example 131. Find the value of x be
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9) 10) Find the slope of the line t
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y = − 2 x + 3 Our Solution 3 Iden
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2.3 Practice - Slope-Intercept Writ
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2.4 Graphing - Point-Slope Form Obj
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Find the equation of the line throu
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Write the point-slope form of the e
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m = 2 5 m = 2 5 Example 147. Slope
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2.5 Practice - Parallel and Perpend
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Chapter 3 : Inequalities 3.1 Solve
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than 4. If we have an expression su
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− 5 − 5 − 2x � 6 Divide bot
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Solve each inequality, graph each s
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As the graphs overlap, we take the
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3.2 Practice - Compound Inequalitie
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edition of a college algebra text!
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It is important to remember as we a
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Chapter 4 : Systems of Equations 4.
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ested in the point that is a soluti
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that parallel lines have the same s
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4.2 Systems of Equations - Substitu
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tion. This process is described and
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World View Note: French mathematici
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31) 2x + y = 2 3x + 7y = 14 33) x +
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6x +9y = 6 New second equation −
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Just as with graphing and substutio
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4.4 Systems of Equations - Three Va
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just the ordered pair (x, y). In th
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15x − 12y +9z = − 12 5x − 4y
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23) 3x + 3y − 2z = 13 6x + 2y −
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are given to use. This means someti
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− 0.5 − 0.5 c=17 We have c, num
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− 0.06x − 0.06y = − 240 Add e
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14) A bank contains 27 coins in dim
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4.6 Systems of Equations - Mixture
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Amount Part Total Start 40 3 120 Ad
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4(30 −n)+2.5n=105 Substitute into
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16) A certain grade of milk contain
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175
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5.1 Polynomials - Exponent Properti
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A quicker method to arrive at the s
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Example 210. Example 211. 7a 3 (2a
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5.2 Polynomials - Negative Exponent
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As we simplified our fraction we to
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5.2 Practice - Negative Exponents S
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Example 223. Convert 3.21 × 10 5 t
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5.3 Practice - Scientific Notation
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World View Note: Ada Lovelace in 18
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21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
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Just as we distribute a monomial th
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Example 243. (2x − 5)(4x 2 − 7x
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5.6 Polynomials - Multiply Special
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Be very careful when we are squarin
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5.7 Polynomials - Divide Polynomial
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3. Change the sign of the terms and
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Example 261. 2x 3 + 42 − 4x x + 3
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Chapter 6 : Factoring 6.1 Greatest
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ers using mental math, then we take
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6.1 Practice - Greatest Common Fact
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Example 273. 10ab + 15b + 4a+6 Spli
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example the binomials are (a + b) a
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6.3 Factoring - Trinomials where a
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As the past few examples illustrate
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6.3 Practice - Trinomials where a =
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Example 296. 10x 2 − 27x +5 Multi
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6.5 Factoring - Factoring Special P
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Example 306. 4x 2 + 20xy + 25y 2 Mu
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6.5 Practice - Factoring Special Pr
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Example 313. Example 314. Example 3
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6.7 Factoring - Solve by Factoring
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0=(2x +3)(2x − 3) One factor must
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6.7 Practice - Solve by Factoring S
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7.1 Rational Expressions - Reduce R
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However, if there is more than just
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41) 8m+16 20m − 12 43) 2x2 − 10
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Example 333. 25x2 24y4 · 9y8 55x7
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Simplify each expression. 1) 8x2 9
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7.3 Rational Expressions - Least Co
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As the above example illustrates, w
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7.4 Rational Expressions - Add & Su
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13 12 Our Solution The same process
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7.4 Practice - Add and Subtract Add
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2(12) 1(12) − 3 4 5(12) 1(12) + 6
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LCD =(x + 3)(x − 3) Multiply each
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23) 25) 27) 29) x − 4+ 9 2x +3 x
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(20)(9)=6x Multiply 180 = 6x Divide
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3.5 6 The man hasashadow of 3.5 fee
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33) Kali reduced the size of a pain
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LCD will be more involved. We will
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x=5 or − 2 Our Solution World Vie
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7.8 Rational Expressions - Dimensio
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� 435 1 �� 1 lbs = 16 435
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To focus on the process of conversi
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7.8 Practice - Dimensional Analysis
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Chapter 8 : Radicals 8.1 Square Roo
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√ process is being able to transl
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Simplify. √ 1) 245 √ 3) 36 √
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Example 384. 3√ 54 We are working
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8.3 Radicals - Adding Radicals Obje
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Simiplify 1) 2 5 √ + 2 5 √ + 2
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√ √ √ √ 4 50 + 6 30 − 8 3
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The previous example could have bee
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8.5 Radicals - Rationalize Denomina
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2x2 √ √ 10 − 6x 3x Our Soluti
Page 307 and 308:
Example 408. 2 5 √ − 3 7 √ 5
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37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
Page 311 and 312:
1 (3) 4 Evaluate exponent 1 81 Our
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8.6 Practice - Rational Exponents W
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Example 420. ab2 3√ a2 4√ b Rew
Page 317 and 318:
8.7 Practice - Radicals of Mixed In
Page 319 and 320:
Example 426. Example 427. i 35 Divi
Page 321 and 322:
√ Dividing with complex numbers a
Page 323 and 324:
Simplify. 1) 3 − ( − 8+4i) 3) (
Page 325 and 326:
Chapter 9 : Quadratics 9.1 Solving
Page 327 and 328:
Example 444. + 1 +1 Add1to both sid
Page 329 and 330:
2 − 2=0 Subtract 0=0 True! It wor
Page 331 and 332:
Solve. √ 1) 2x + 3 √ 3) 6x −
Page 333 and 334:
x4 4√ = ± 4√ 16 Simplify roots
Page 335 and 336:
5√ 4x +1 = ± 3 Clear radical by
Page 337 and 338:
9.3 Quadratics - Complete the Squar
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x2 + 5 25 x+ 3 36 � x + 5 �2 6
Page 341 and 342:
Example 464. − 7 3 � 3 3 + 3 3
Page 343 and 344:
9.4 Quadratics - Quadratic Formula
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x = Example 468. √ 30 ± 900 + 11
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9.4 Practice - Quadratic Formula So
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12x 2 − 17x + 6=0 Our Solution If
Page 351 and 352:
9.5 Practice - Build Quadratics fro
Page 353 and 354:
Example 479. a −2 − a −1 −
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12 = (x − 3) 2 or 13 =(x − 3) 2
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9.7 Quadratics - Rectangles Objecti
Page 359 and 360:
Example 486. 2 Our Solution The len
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4(x 2 + 10x − 24) =0 Factor trino
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that require 24 square meters for p
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1(12x) 3 + 1(12x) x 1 1 5 + = 3 x 1
Page 367 and 368:
6x +6(x − 1)=5x(x − 1) Reduce e
Page 369 and 370:
16) A sink can be filled from the f
Page 371 and 372:
These simultaneous product equation
Page 373 and 374:
9.10 Quadratics - Revenue and Dista
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96n(n − 2) 96n(n − 2) +4n(n −
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upstream, the current will pull aga
Page 379 and 380:
12) A pilot flying at a constant ra
Page 381 and 382:
The above method to graph a parabol
Page 383 and 384:
It is important to remember the gra
Page 385 and 386:
Chapter 10 : Functions 10.1 Functio
Page 387 and 388:
Example 504. Which of the following
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2x − 3 � 0 Set up an inequality
Page 391 and 392:
Solve. 10.1 Practice - Function Not
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10.2 Functions - Operations on Func
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(x − 5)(x +1) (x − 5) Divide ou
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We can also evaluate a composition
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23) f(x) = x 2 − 5x g(x) = x + 5
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10.3 Functions - Inverse Functions
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(3x +6)(3 − 4x) 3 − 4x (4x+8)(3
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10.3 Practice - Inverse Functions S
Page 407 and 408:
Example 530. 8 3x = 32 Rewrite 8 as
Page 409 and 410:
10.4 Practice - Exponential Functio
Page 411 and 412:
log3x =7 Identify base, 3, answer,
Page 413 and 414:
10.5 Practice - Logarithmic Functio
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A = 25000(1.01625) 20 Evaluate expo
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The variable e is a constant simila
Page 419 and 420:
j. All of the above compounded cont
Page 421 and 422:
Using the diagram at right, find ea
Page 423 and 424:
10.7 Practice - Trigonometric Funct
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21) 23) 25) 27) A B A 5 38 ◦ x x
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37) 39) A A 37.1 ◦ C x x C 13.1 4
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θ 17 From angle θ the given sides
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function to find the missing angle
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Find the measure of each angle indi
Page 435 and 436:
27) 29) 14 A A θ C 15 7 C 14 θ B
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437
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7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
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54) 60v − 7 55) − 3x + 8x 2 56)
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45) 12 46) All real numbers 47) No
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15) wx 3 = 1458 16) h = 1.5 j 17) a
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1) B(4, − 3) C(1, 2) D( − 1, 4)
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1) y = 2x +5 2) y = − 6x +4 3) y
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23) x =4 24) y − 4 = 7 (x − 1)
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25) 5�x
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1) ( − 2, 4) 2) (2,4) 3) No solut
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27) 56, 144 28) 1.5, 3.5 29) 30 30)
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42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
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5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
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7) (b +8)(b + 4) 8) (b − 10)(b
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11) n(n − 1) 12) (5x +3)(x − 5)
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5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
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7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
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21) 0, 5 22) − 2, 5 3 23) 4,7 24)
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42) − 18xz 4x3yz3 � 4 8.3 1) 6
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16 3 √ + 4 5 √ 43 18) 19) 2 √
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8.8 1) 11 − 4i 2) − 4i 3) − 3
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√ √ 33) 4+i 39, 4 −i 39 34)
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9) ± 2, ± 4 10) 2,3, − 1 ±i 3
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1) 2) 3) 4) 5) 6) (-2,0) (0,-8) (-1
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11) 100 12) − 74 13) 1 5 14) 27 1
Page 487 and 488:
10.5 1) 9 2 = 81 2) b −16 = a 3)
Page 489:
21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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