Beginning and Intermediate Algebra - Wallace Math Courses ...

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Example 431. Example 432. Example 433. Example 434. (3i)(7i) Multilpy coefficients and i ′ s 21i 2 Simplify i 2 = − 1 21( − 1) Multiply − 21 Our Solution 5i(3i − 7) Distribute 15i 2 − 35i Simplify i 2 = − 1 15( − 1) − 35i Multiply − 15 − 35i Our Solution (2 − 4i)(3 +5i) FOIL 6+10i − 12i − 20i 2 Simplify i 2 = − 1 6+10i − 12i − 20( − 1) Multiply 6+10i − 12i+20 Combine like terms 6+20 and 10i − 12i 26 − 2i Our Solution (3i)(6i)(2 − 3i) Multiply first two monomials 18i 2 (2 − 3i) Distribute 36i 2 − 54i 3 Simplify i 2 = − 1 andi 3 = −i 36( − 1) − 54( −i) Multiply − 36 + 54i Our Solution Remember when squaring a binomial we either have to FOIL or use our shortcut to square the first, twice the product and square the last. The next example uses the shortcut Example 435. (4 − 5i) 2 Use perfect square shortcut 4 2 = 16 Square the first 2(4)( − 5i)=−40i Twice the product (5i) 2 = 25i 2 = 25( − 1)=−25 Square the last, simplify i 2 = − 1 16 − 40i − 25 Combine like terms − 9 − 40i Our Solution 320
√ Dividing with complex numbers also has one thing we need to be careful of. If i is − 1, and it is in the denominator of a fraction, then we have a radical in the denominator! This means we will want to rationalize our denominator so there are no i’s. This is done the same way we rationalized denominators with square roots. Example 436. 7+3i − 5i 7 +3i − 5i � � i i 7i + 3i 2 − 5i 2 7i+3( − 1) − 5( − 1) 7i − 3 5 Just a monomial in denominator, multiply by i Distributeiin numerator Simplify i 2 = − 1 Multiply Our Solution The solution for these problems can be written several different ways, for example − 3+7i or 5 −3 7 + i, The author has elected to use the solution as written, but it is 5 5 important to express your answer in the form your instructor prefers. Example 437. 2 − 6i 4+8i � 2 − 6i 4 − 8i 4 +8i 4 − 8i � 8 − 16i − 24i + 48i 2 16 − 64i 2 8 − 16i − 24i + 48( − 1) 16 − 64( − 1) 8 − 16i − 24i − 48 16 + 64 − 40 − 40i 80 − 1 −i 2 Binomial in denominator, multiply by conjugate,4−8i FOIL in numerator, denominator isadifference of squares Simplify i 2 = − 1 Multiply Combine like terms 8 − 48 and − 16i − 24i and 16 + 64 Reduce, divide each term by 40 Our Solution 321
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Beginning and Intermediate Algebra
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Special thanks to: My beautiful wif
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Chapter 6: Factoring 6.1 Greatest C
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0.1 Pre-Algebra - Integers Objectiv
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Multiplication and division of inte
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45) (4)( − 6) Find each quotient.
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9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
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Example 21. 13 6 − 9 6 4 6 2 3 Sa
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Find each quotient. 37) − 2 ÷ 7
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the expression that is to be evalua
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Solve. 1) − 6 · 4( − 1) 3) 3+(
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It will be more common in our study
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0.4 Practice - Properties of Algebr
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Chapter 1 : Solving Linear Equation
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Addition Problems To solve equation
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8x = − 24 8 8 x = − 3 Division
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1.2 Linear Equations - Two-Step Equ
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− 1 − 1 Divide both sides by
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1.3 Solving Linear Equations - Gene
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Example 63. − 3x +9=6x − 27 Not
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4[ − 2]+9=−15 +8(2) Finish mult
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1.4 Solving Linear Equations - Frac
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Example 72. � � 3 5 4 x + = 3 2
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1.5 Solving Linear Equations - Form
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Example 79. A = πr 2 +πrs for s S
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1.5 Practice - Formulas Solve each
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|x| = 5 Absolute value can be posit
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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
Page 269 and 270: (20)(9)=6x Multiply 180 = 6x Divide
Page 271 and 272: 3.5 6 The man hasashadow of 3.5 fee
Page 273 and 274: 33) Kali reduced the size of a pain
Page 275 and 276: LCD will be more involved. We will
Page 277 and 278: x=5 or − 2 Our Solution World Vie
Page 279 and 280: 7.8 Rational Expressions - Dimensio
Page 281 and 282: � 435 1 �� 1 lbs = 16 435
Page 283 and 284: To focus on the process of conversi
Page 285 and 286: 7.8 Practice - Dimensional Analysis
Page 287 and 288: Chapter 8 : Radicals 8.1 Square Roo
Page 289 and 290: √ process is being able to transl
Page 291 and 292: Simplify. √ 1) 245 √ 3) 36 √
Page 293 and 294: Example 384. 3√ 54 We are working
Page 295 and 296: 8.3 Radicals - Adding Radicals Obje
Page 297 and 298: Simiplify 1) 2 5 √ + 2 5 √ + 2
Page 299 and 300: √ √ √ √ 4 50 + 6 30 − 8 3
Page 301 and 302: The previous example could have bee
Page 303 and 304: 8.5 Radicals - Rationalize Denomina
Page 305 and 306: 2x2 √ √ 10 − 6x 3x Our Soluti
Page 307 and 308: Example 408. 2 5 √ − 3 7 √ 5
Page 309 and 310: 37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
Page 311 and 312: 1 (3) 4 Evaluate exponent 1 81 Our
Page 313 and 314: 8.6 Practice - Rational Exponents W
Page 315 and 316: Example 420. ab2 3√ a2 4√ b Rew
Page 317 and 318: 8.7 Practice - Radicals of Mixed In
Page 319: Example 426. Example 427. i 35 Divi
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
Page 339 and 340: 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
Page 345 and 346: x = Example 468. √ 30 ± 900 + 11
Page 347 and 348: 9.4 Practice - Quadratic Formula So
Page 349 and 350: 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 −
Page 355 and 356: 12 = (x − 3) 2 or 13 =(x − 3) 2
Page 357 and 358: 9.7 Quadratics - Rectangles Objecti
Page 359 and 360: Example 486. 2 Our Solution The len
Page 361 and 362: 4(x 2 + 10x − 24) =0 Factor trino
Page 363 and 364: that require 24 square meters for p
Page 365 and 366: 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
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These simultaneous product equation
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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
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12) A pilot flying at a constant ra
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The above method to graph a parabol
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It is important to remember the gra
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Chapter 10 : Functions 10.1 Functio
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Example 504. Which of the following
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2x − 3 � 0 Set up an inequality
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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
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Example 530. 8 3x = 32 Rewrite 8 as
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10.4 Practice - Exponential Functio
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log3x =7 Identify base, 3, answer,
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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
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j. All of the above compounded cont
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Using the diagram at right, find ea
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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
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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
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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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