Beginning and Intermediate Algebra - Wallace Math Courses ...

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are the other two sides (legs), then we can use the following formula, a 2 + b 2 = c 2 to find a missing side. Often when solving triangles we use trigonometry to find one part, then use the angle sum and/or the Pythagorean Theorem to find the other two parts. Example 557. Solve the triangle 5 35 ◦ We have one angle and one side. We can use these to find either other side. We will find the other leg, the adjacent side to the 35 ◦ angle. The 5 is the opposite side, so we will use the tangent to find the leg. tan35◦ = 5 Tangent is opposite over adjacent x 0.700 5 = Evaluate tangent, put it over one so we have a proportion 1 x 0.700x = 5 Find cross product 0.700 0.700 Divide both sides by 0.700 x = 7.1 The missing leg. a 2 +b 2 = c 2 We can now use pythagorean thorem to find hypotenuse, c 5 2 + 7.1 2 = c 2 Evaluate exponents 25 + 50.41 = c 2 Add 5 75.41 = c 2 Square root both sides 8.7 =c The hypotenuse 90 ◦ − 35 ◦ To find the missing angle we subtract from 90 ◦ 55 ◦ 55 ◦ The missing angle 7.1 8.7 35 ◦ Our Solution In the previous example, once we found the leg to be 7.1 we could have used the sine function on the 35 ◦ angle to get the hypotenuse and then any inverse trig 430
function to find the missing angle and we would have found the same answers. The angle sum and pythagorean theorem are just nice shortcuts to solve the problem quicker. Example 558. Solve the triangle 9 tan−1 � � 9 3 Evaluate fraction tan−1 (3) Evaluate tangent 3 71.6 ◦ The angle on the right side In this triangle we have two sides. We will first find the angle on the right side, adjacent to 3 and opposite from the 9. Tangent uses opposite and adjacent To find an angle we use the inverse tangent. 90 ◦ − 71.6 ◦ Subtract angle from 90 ◦ to get other angle 18.4 ◦ The angle on the left side a 2 +b 2 = c 2 Pythagorean theorem to find hypotenuse 9 2 + 3 2 = c 2 Evaluate exponents 81 +9=c 2 Add 90 = c2 Square root both sides √ 3 10 or 9.5 = c The hypotenuse 9.5 71.6 9 3 ◦ 18.4◦ Our Solution World View Note: Ancient Babylonian astronomers kept detailed records of the starts, planets, and eclipses using trigonometric ratios as early as 1900 BC! 431
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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
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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
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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
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
Page 389 and 390: 2x − 3 � 0 Set up an inequality
Page 391 and 392: Solve. 10.1 Practice - Function Not
Page 393 and 394: 10.2 Functions - Operations on Func
Page 395 and 396: (x − 5)(x +1) (x − 5) Divide ou
Page 397 and 398: We can also evaluate a composition
Page 399 and 400: 23) f(x) = x 2 − 5x g(x) = x + 5
Page 401 and 402: 10.3 Functions - Inverse Functions
Page 403 and 404: (3x +6)(3 − 4x) 3 − 4x (4x+8)(3
Page 405 and 406: 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
Page 415 and 416: A = 25000(1.01625) 20 Evaluate expo
Page 417 and 418: 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
Page 425 and 426: 21) 23) 25) 27) A B A 5 38 ◦ x x
Page 427 and 428: 37) 39) A A 37.1 ◦ C x x C 13.1 4
Page 429: θ 17 From angle θ the given sides
Page 433 and 434: Find the measure of each angle indi
Page 435 and 436: 27) 29) 14 A A θ C 15 7 C 14 θ B
Page 437 and 438: 437
Page 439 and 440: 7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
Page 441 and 442: 54) 60v − 7 55) − 3x + 8x 2 56)
Page 443 and 444: 45) 12 46) All real numbers 47) No
Page 445 and 446: 15) wx 3 = 1458 16) h = 1.5 j 17) a
Page 447 and 448: 1) B(4, − 3) C(1, 2) D( − 1, 4)
Page 449 and 450: 1) y = 2x +5 2) y = − 6x +4 3) y
Page 451 and 452: 23) x =4 24) y − 4 = 7 (x − 1)
Page 453 and 454: 25) 5�x
Page 455 and 456: 1) ( − 2, 4) 2) (2,4) 3) No solut
Page 457 and 458: 27) 56, 144 28) 1.5, 3.5 29) 30 30)
Page 459 and 460: 42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
Page 461 and 462: 5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
Page 463 and 464: 7) (b +8)(b + 4) 8) (b − 10)(b
Page 465 and 466: 11) n(n − 1) 12) (5x +3)(x − 5)
Page 467 and 468: 5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
Page 469 and 470: 7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
Page 471 and 472: 21) 0, 5 22) − 2, 5 3 23) 4,7 24)
Page 473 and 474: 42) − 18xz 4x3yz3 � 4 8.3 1) 6
Page 475 and 476: 16 3 √ + 4 5 √ 43 18) 19) 2 √
Page 477 and 478: 8.8 1) 11 − 4i 2) − 4i 3) − 3
Page 479 and 480: √ √ 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)
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21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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