Beginning and Intermediate Algebra - Wallace Math Courses ...

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41) x2 − 1 2x −4 · x2 − 4 x2 − x − 2 ÷ x2 + x − 2 3x − 6 43) x2 + 3x + 9 x 2 + x − 12 · x2 +2x − 8 x 3 − 27 ÷ x2 − 4 x 2 − 6x+9 252 42) a 3 + b 3 a 2 + 3ab+2b 2 · 3a − 6b 3a2 − 3ab+ 3b2 ÷ a2 − 4b2 a + 2b 44) x2 + 3x − 10 x2 +6x+5 · 2x2 − x − 3 2x2 8x+20 ÷ + x − 6 6x+15
7.3 Rational Expressions - Least Common Denominators Objective: Idenfity the least common denominator and build up denominators to match this common denominator. As with fractions, the least common denominator or LCD is very important to working with rational expressions. The process we use to find the LCD is based on the process used to find the LCD of intergers. Example 338. Find the LCD of8and 6 Consider multiples of the larger number 8, 16, 24 . 24 is the first multiple of8that is also divisible by 6 24 Our Solution When finding the LCD of several monomials we first find the LCD of the coefficients, then use all variables and attach the highest exponent on each variable. Example 339. Find the LCD of 4x 2 y 5 and 6x 4 y 3 z 6 First find the LCD of coefficients 4 and 6 12 12 is the LCD of4and 6 x 4 y 5 z 6 Use all variables with highest exponents on each variable 12x 4 y 5 z 6 Our Solution The same pattern can be used on polynomials that have more than one term. However, we must first factor each polynomial so we can identify all the factors to be used (attaching highest exponent if necessary). Example 340. Find the LCD of x 2 +2x − 3 and x 2 −x − 12 Factor each polynomial (x − 1)(x +3) and(x − 4)(x +3) LCD uses all unique factors (x − 1)(x+3)(x − 4) Our Solution Notice we only used (x + 3) once in our LCD. This is because it only appears as a factor once in either polynomial. The only time we need to repeat a factor or use an exponent on a factor is if there are exponents when one of the polynomials is factored 253
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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
Page 201 and 202: 5.6 Polynomials - Multiply Special
Page 203 and 204: Be very careful when we are squarin
Page 205 and 206: 5.7 Polynomials - Divide Polynomial
Page 207 and 208: 3. Change the sign of the terms and
Page 209 and 210: Example 261. 2x 3 + 42 − 4x x + 3
Page 211 and 212: Chapter 6 : Factoring 6.1 Greatest
Page 213 and 214: ers using mental math, then we take
Page 215 and 216: 6.1 Practice - Greatest Common Fact
Page 217 and 218: Example 273. 10ab + 15b + 4a+6 Spli
Page 219 and 220: example the binomials are (a + b) a
Page 221 and 222: 6.3 Factoring - Trinomials where a
Page 223 and 224: As the past few examples illustrate
Page 225 and 226: 6.3 Practice - Trinomials where a =
Page 227 and 228: Example 296. 10x 2 − 27x +5 Multi
Page 229 and 230: 6.5 Factoring - Factoring Special P
Page 231 and 232: Example 306. 4x 2 + 20xy + 25y 2 Mu
Page 233 and 234: 6.5 Practice - Factoring Special Pr
Page 235 and 236: Example 313. Example 314. Example 3
Page 237 and 238: 6.7 Factoring - Solve by Factoring
Page 239 and 240: 0=(2x +3)(2x − 3) One factor must
Page 241 and 242: 6.7 Practice - Solve by Factoring S
Page 243 and 244: 7.1 Rational Expressions - Reduce R
Page 245 and 246: However, if there is more than just
Page 247 and 248: 41) 8m+16 20m − 12 43) 2x2 − 10
Page 249 and 250: Example 333. 25x2 24y4 · 9y8 55x7
Page 251: Simplify each expression. 1) 8x2 9
Page 255 and 256: As the above example illustrates, w
Page 257 and 258: 7.4 Rational Expressions - Add & Su
Page 259 and 260: 13 12 Our Solution The same process
Page 261 and 262: 7.4 Practice - Add and Subtract Add
Page 263 and 264: 2(12) 1(12) − 3 4 5(12) 1(12) + 6
Page 265 and 266: LCD =(x + 3)(x − 3) Multiply each
Page 267 and 268: 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
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Page 301 and 302: 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
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Example 408. 2 5 √ − 3 7 √ 5
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37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
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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
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8.7 Practice - Radicals of Mixed In
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Example 426. Example 427. i 35 Divi
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√ Dividing with complex numbers a
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Simplify. 1) 3 − ( − 8+4i) 3) (
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Chapter 9 : Quadratics 9.1 Solving
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Example 444. + 1 +1 Add1to both sid
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2 − 2=0 Subtract 0=0 True! It wor
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Solve. √ 1) 2x + 3 √ 3) 6x −
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x4 4√ = ± 4√ 16 Simplify roots
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5√ 4x +1 = ± 3 Clear radical by
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9.3 Quadratics - Complete the Squar
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x2 + 5 25 x+ 3 36 � x + 5 �2 6
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Example 464. − 7 3 � 3 3 + 3 3
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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
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9.5 Practice - Build Quadratics fro
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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
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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
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6x +6(x − 1)=5x(x − 1) Reduce e
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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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