Math 017 Materials With Exercises

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$Math 171 Review for the Final Exam$

$Math 171 Test 2$

$MATH 171 SYLLABUS$

A solution, the solution set, and what it means to solve an equationEquations are often used to state the equality of two expressions containing one or more variables(often called unknowns).The solution setA value of the variable for which the equation is true (that is, avalue for which the left hand side of the equation is equal to theright hand side) is called a solution. The solution set of theequation is the set of all solutions of the equation.For example, x 7 is a solution of the equation x 2 9 (when we replace x with 7, the right-handside is equal to the left-hand side of the equation 7 2 9 ). Later on, we will learn that x 7 is theonly solution of this equation, and thus, since there are no other solutions, x 7 is also the solutionset of x 2 9 . Consider the equation x 2 36. One can check that x 6 is a solution of x2 36(substitute 6 for x to get 6 2 36 ). It is not the solution set, because it is not the only solution of thisequation. For example, x 6is another solution, since ( 6)2 36 . The solution set of an equationconsists of all solutions. Therefore, in the previous example, the solution set would need to includeboth 6 and 6 .To Solve an EquationTo solve an equation is to find the set of all solutions of theequation, or prove that it does not have a solution.3Example 8.2 Determine which of the following numbers are solutions of the equation x 6 x.a) x 2b) x 2Solution:a) Evaluate the left-hand side of the equation when x 2 .x 3 2 3 8 .Evaluate the right-hand side of the equation when x 2 .6 x 6 2 8.Since both the left-hand and right-hand side of the equations are equal, 2 is a solution.b) Evaluate the left-hand side of the equation when x 2.3 3x ( 2) 8Evaluate the right-hand side of the equation when x 2.6 x 6 ( 2) 6 2 4 .Since the left-hand side is not equal to the right-hand side ( 8 4 ), 2 is not asolution.Why do we solve equations? Here is a very basic example. Suppose that you bought some candies andyou did not know how many you bought but you know that you spent 80 cents and that each candycost 2 cents. To find out how many candies you actually bought, you set up an equation 2x 80,where x represents number of candies you bought. If you solve this equation, you would find thatx 40 and this means that you bought 40 candies. Now, let us learn how to solve certain types ofequations.78
Operations that can be performed on any equation without changing the set of solutionsThe following operations can be done to any equation without changing its solution.Any quantity can be added to, or subtracted from both sides of an equation.For example, If x y then x 2 y 2If x y then x 2 y 2Both sides of an equation can be multiplied, or divided by any nonzero quantity.For example, If x y then 2x 2yx yIf x y then 2 2It is important to remember that, any time we perform any operation on one side of an equation,exactly the same operation must be performed on the other side. We always perform the sameoperation on both sides.Solving linear equations in one variableThere are many types of equations, and depending on the type of equation, different solving techniquesare involved. We will learn how to solve linear equations in one variable (the formal definition of alinear equation will be provided later).To solve a linear equation we perform the operations of adding, subtracting, multiplyingor dividing both sides of an equation by suitable quantities with the goal of isolating thevariable (often x) on one side, and „bringing all numbers to the other side‟ of the equation.The numerical value obtained on the other side is the solution.We will learn how to implement the above idea by solving several equations.■ Solve the following equation x 2 7 .We need to isolate x , and to this end, we must somehow „get rid of 2‟ on the left-hand side of theequation. 2 is added to x , so if we subtract 2 (opposite operation to addition), we will get zero, and 2will no longer be on the left-hand side. But if we subtract 2 from the left-hand side of the equation, thesame operation must be performed on the right-side. Thus we need to subtract 2 from both sides. Wecould say that we are „bringing 2 to the other side‟x 2 7Subtract 2 from both sides.x 2 2 7 2Simplify.x 5The solution of x 2 7 is x 5.■79
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Math 017MaterialsWithExercises
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TABLE OF CONTENTSLesson 1Variables
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Lesson 1___________________________
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e) „minus x ‟ or „the opposit
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For example,Evaluate 5 x when x 10
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3xd) x( y)e) 2f) 2x yz w(t)Ex.4 T
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Ex.16 Let x 3. Rewrite the express
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e) x5f) x2Ex.28 Substitute“undefi
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Lesson 2___________________________
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Solution:Please, notice the use of
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g)i)4( a 2)h)8y (x)4(ab)j) ( a d)
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1 2Ex.19 If possible, evaluate when
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Lesson 3___________________________
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Commutative Propertyof Additionx y
Page 30 and 31: 2 x 2 xFor example, and thus als
Page 32 and 33: Example 3.10 Determine which of the
Page 34 and 35: Exercises with Answers (For answers
Page 36 and 37: Ex.16 Write the following expressio
Page 38 and 39: s) ( 0.2xy)( 0.3z)t)4x1Ex.24 Is 1
Page 40 and 41: Lesson 4___________________________
Page 42 and 43: Numerical coefficientsNumerical Coe
Page 44 and 45: For your convenience, we will displ
Page 46 and 47: 9 818081a) 9 98097 6 76134 4 4 4
Page 48 and 49: Ex.8Simplify by raising to the indi
Page 50 and 51: 73a 2xxi) 2aj)2( 42a)x(3xk)4x2 ) 3
Page 52 and 53: Lesson 5___________________________
Page 54 and 55: FactorizationThe Distributive Law l
Page 56 and 57: Simplification of algebraic fractio
Page 58 and 59: Mistake 5.1When factoring 3 from 3x
Page 60 and 61: Ex.13 Factor2a) 5 a from the follow
Page 62 and 63: 24x ( 5x)g)xa bi)7(a b)bc ( b e)
Page 64 and 65: Adding and subtracting like termsIf
Page 66 and 67: ) This should be written as 4x 2
Page 68 and 69: Ex.4 Circle all terms that are like
Page 70 and 71: 682 122k) 6 ( d a) a dl) 3a1(4a
Page 72 and 73: Lesson 7___________________________
Page 74 and 75: For instance, the area of a rectang
Page 76 and 77: a) a b 2bc) a bd) ae)b)aba 22a a
Page 78 and 79: Ex.24 Express the expressiona) m s
Page 82 and 83: ■ Solve the following equation 3x
Page 84 and 85: sides (subtracting 4 x from both si
Page 86 and 87: Mistake 8.2While solving an equatio
Page 88 and 89: Ex.12 Determine whether the followi
Page 90: Lesson 9___________________________
Page 94 and 95: xy x 1 yFactor x .x( y 1)1yDivide
Page 96 and 97: a) b b) a c) tEx.15. SolveEx.16. So
Page 98 and 99: For each number you should check if
Page 100 and 101: ■ Solve the inequality 4x 9 1.
Page 102 and 103: ■ Solve the following inequality
Page 104 and 105: Ex.10 Graph the following number se
Page 106 and 107: d) z 3e) z 13 Ex.22 Knowing x 1
Page 108 and 109: Example 11.2 The expression( 22 2x
Page 110 and 111: Solution:a) To match the form y mx
Page 112 and 113: a x and b 3 , so now we can rewri
Page 114 and 115: Ex.9 For each of the following expr
Page 116 and 117: 114xc) y 5d) y 3x 2 0e) 3y 6x1f
Page 118 and 119: APPENDIX A:Lesson 1ANSWERS TO EXERC
Page 120 and 121: 49C 3255 2(L+W)62mc75a) a b expon
Page 122 and 123: 3(m n)a(x1)g)h)2tx19s s s sThe opp
Page 124 and 125: 52233x1 2 x5 3 1513 a) m ( 2m) 2mb
Page 126 and 127: c) Numerator: one term, 3ab (with t
Page 128 and 129: 18 a) 9 b) 3 c) 3 d) 3e) 6 f) 279
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11 a) x 1 b)12 a) x abcx a b42 b
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c) x 2d) a 7e) x 10f) a 225 a)
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x or 4x ( 7) 0; a 4; b 7g) x (
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3. Let m = ─3. Evaluate the follo
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1. Write the following statements a
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6. Solve the following equations:8.
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8. Factor the following expressions
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___________________________________
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3 ;b) x 3 y 6 xy 2 A xy 28. a) x
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Math 017 Materials With Exercises

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