Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18, 21, 24, 30,...} Then A B= and A B= Sets of Numbers Name(s) for the set 1, 2, 3, 4, Natural Numbers Positive Integers Symbol(s) for the set, -3, -2, -1 Negative integers 0, 1, 2, 3, 4, Non-negative Integers Whole Numbers, -3, -2, -1, 0, 1, 2, 3, Integers Note: is the German word for number.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 2 4 5, 17 13, 1 29 Rational Numbers Note: This is the first letter of 1. Circle One: a) 5 is a rational number!!! TRUE!! FALSE b) π 4 is a rational number!!! TRUE!! FALSE c) 16 5 is a rational number!! TRUE!! FALSE d) 17 5 is a rational number!! TRUE!! FALSE e) 0 is a rational number!!! TRUE!! FALSE Definition of a rational number: If a, b! and b 0, then a b!. Translation: Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either or Examples! 1 4 = ( This is a terminating decimal.) 4 9 = (This is a repeating decimal.)
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 3 You know how to express a terminating decimal as a fraction. For example.25 = 25, but how 100 do you express a repeating decimal as a fraction? 2. Express these repeating decimals as the quotient of two integers: a).373737 or.37 Let x =.37373737... Then 100x = b).123 Let x =.123123123... Then There are a lot of rational numbers, but there are even more irrational numbers. Irrational numbers cannot be expressed as the quotient of two integers. Their decimal expansions never terminate nor do they repeat. Examples of irrational numbers are The union of the rational numbers and the irrational numbers is called the or simply the. It is denoted by.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 4 4. What properties are being used? Name of Property a) ( 40) + 8(x + 5) = 8(x + 5) + ( 40) a + b = b + a a,b! b) 8(x + 5) + ( 40) = 8x + 40 + ( 40) a(b + c) = ab + ac a,b,c! c) 8x + 40 + ( 40) = 8x + (40 + ( 40)) (a + b) + c = a + (b + c) a,b,c! d) 0 + 5 = 5 + 0 = 5 0 + a = a + 0 = a a! e) 7 + ( 7) = ( 7) + 7 = 0 For each a!, another element of! denoted by a such that a + ( a) = ( a) + a = 0 f) 6 1 = 1 6 = 6 1 a = a 1 = a a! For each a!, a 0 another g) 4 1 4 = 1 4 4 = 1 element of! denoted by 1 a a 1 a = 1 a a = 1 such that
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 5 Properties of Real Numbers Notice that the commutative property considers the of the numbers. When demonstrating the associative property, the order of the numbers stays the same but the changes. When do we use the commutative property of addition? Consider 7 + 28 + 43.! The rules for order of operation tell us to add left to right, but the commutative property of addition allows us to change the order to make our life easier: 7 + 29 + 43 = When do we use the associative property of multiplication? Consider 16 25. 16 25 = Is subtraction commutative? Example: Is division commutative? Example: Consider ( 12 3) 4 = while 12 ( 3 4) = So is subtraction associative? Consider ( 12 3) 4 = while 12 ( 3 4) = So is division associative?
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 6 Notice that there are 2 identities: The additive identity is.! The multiplicative identity is. Notice there are 2 kinds of inverses. The additive inverse of 4 is. Note that The additive inverse of -9 is. Note that What is true about the sum of a number and its additive inverse? Every real number has an additive inverse. In rigorous settings, subtraction is defined as addition with the additive inverse. In other words 5 2 = 5 + ( 2) or more generally a b = a + ( b) The multiplicative inverse of 4 is. Note that The multiplicative inverse of -9 is Note that What is true about the product of a number and its multiplicative inverse? Almost all of the real numbers have a multiplicative inverse. Which real number does NOT have a multiplicative inverse?
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 7 Absolute Value 4 = 9 = Formal definition for absolute value!!!! x = It will help if you learn to think of absolute value as (Notice that -4 is 4 units away from the origin.) Distance between two points: 3.) Find the distance between a) 4 and 1 b) -1 and 5 c) 4 and -2 d) -3 and -4 e) 1 and x f) x and y
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 8 Notice that! 4 1 = 1 4! and! 3 2 = 2 ( 3). In general x y = Notice that! 5 + 4 =! whereas 5 + 4 = So, in general x + y Understanding Zero True or False :!! True or False :!! True or False :!! 0 6 = 0! 6 0 = 0!! 0 0 = 0!! For a, b!, a = 0 and b Another way to say this: a fraction is zero when its is zero and its is not. This concept will be very important when we solve rational equations like 1 x +1 = x + 2 3 x Notation inconsistency: When we write 4x it means the product of 4 and x. When we write 4 1 3 it means the sum of 4 and 1 3 4 1 3 is usually called a mixed number whereas the corresponding is termed an improper fraction. There is nothing improper about an improper fraction. Extra Problems: Text: 1-9, 22, 23
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 9 Chapter 1B - Exponents and Radicals Multiplication is shorthand notation for repeated addition 4 + 4 + 4 + 4 + 4 = 5 4 Exponents are shorthand notation for repeated multiplication. So 2 2 2 2 2 = xixixix = You probably know that 3 1 = and x 5 = this is because by definition x 1 =. In general if m Ν, then x m = Properties of exponents Example In general x 4 x 3 x 8 x 5 x m x n x m x n x 3 x 3 x 0 for x 0 Note: 0 0 0 and 0 0 1 0 0 is an More Properties of Exponents Example In general ( x 2 ) 3 (xy) 3 (xy) m x 2 y x m y
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 10 1. Simplify a) ( 8) 2 b) 8 2 c) 3 0 d) 1 2 6 e) 1 x 2! e) x 3 4 f) 3 x 1 g) 25x 5 y 2 15x 2 y 1 3 h) 12x 3 y 2 4x 4 y 1 2
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 11 Roots or Radicals TRUE or FALSE: 4 = ±2. Important distinction: x 2 = 25 has solutions: x = and x = 25 represents exactly number. 25 =. Recall that 2 5 = 32. Now suppose you have x 5 = 32 with the instructions: Solve for x. How do you express x in terms of 32? The vocabulary word is In this case x equals the. The notation is x = or x = This second notation is called a fractional exponent. ( ) 1 4 4 Examples:! 81 = 81 = because 3 =. ( ) 1 3!!! 125 = because 5 =. Properties of Roots (just like properties of exponents!) Example In general 3 27i8 m xy 3 27 3 8
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 12 Example In general 36 4 36 4 m m x y 2 3 Now consider 8. Notice that ( 8 2 ) 1 3 = Also 8 1 3 2 = 2 3 So it would seem that 8 = In fact, more generally, it is true that x m n = 3 Similarly, consider 64 = 64 1 2 1 3 = 3 Whereas, 64 = 64 1 3 1 2 = 3 So it would seem that 64 In general m n x
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 13 Simplify n x n for all n!. On Worksheet 1 you graphed y = 3 x 3. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that 3 x 3 = You also graphed y = 2 x 2. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that 2 x 2 = Most people believe that 5 2 5 = and agree easily that n 2 n = n But ( 2) 4 ( ) 5 4 5 = while 2 = So hopefully you see believe n ( 2) n = 2 whenever n is and n ( 2) n = 2 whenever n is So to simplify n x n for any x! and any n! we have n x n = When simplifying, we don t want to use absolute value signs if they are unnecessary. Obviously we would write 4 = 4. But also x 4 = Also, for the time being, if n is even, n x is defined only if x > 0. In other words, for the time 4 being 4 and 17 are not defined. So x 3 is only defined if x > 0. That means that x 3 =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 14 Simplifying Radicals A radical expression is simplified when the following conditions hold: 1. All possible factors ( perfect roots ) have been removed from the radical. 2. The index of the radical is as small as possible. 3. No radicals appear in the denominator. 2. Simplify 5 a) 32 b) ( 216) 1 3 1 2 c) 9 d) 64 6 e) ( 13) 6 f) 4 16x 4 g) 3 x 5 8 h) 16 i) 125 64 2 3
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 15 Rationalizing the Denominator Rationalizing the denominator is the term given to the techniques used for eliminating radicals from the denominator of an expression without changing the value of the expression. It involves multiplying the expression by a 1 in a helpful form. 3. Simplify a) 1 2!!!!! b) 1 3 2 c) 3 1 4x!!!!!! d) 4 16x 2 27y Notice: ( x + 3) ( x 3) =. To simplify 1 2 5 we multiply it by 1 in the form of
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 16 So 1 2 5 = and 9 x + 3 5 =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 17 Adding and Subtracting Radical Expressions Terms must be alike to combine them with addition or subtraction. Radical terms are alike if they have the same index and the same radicand. (The radicand is the expression under the radical sign.) 4. Simplify ( ) a) 5 + 2 7 3 5 7 b) 3 128x 4 3 3x 18x x 2x Please notice 9 + 16 =! Whereas 25 = In other words In general True or False: 2x + 3x = 5x True or False: 3x + 3x = 6x 3x + 3x =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1B! Page 18 Interesting cases of n x n when n is even For the time being if n is even, n x is undefined if x < 0 in other words 6 2 is undefined. Moreover, we want to remember that when x > 0, x = x ; we always simplify 4 = To simplify the following, start by considering for what values of x the expression is defined 4 x 4 Here the index of the radical is even, but the expression under the radical is always positive, so the expression is defined for all values of x 4 x 4 = 4 x 5 Here the index of the radical is even, but x 5 could be negative if x was negative. We have to recognize that 4 x 5 is only defined if x > 0 so 4 x 5 = 4 x 8 Here the index of the radical is even, and the expression under the radical is always positive, so the expression is defined for all values of x 4 x 8 = 4 x 12 Here the index of the radical is even, and the expression under the radical is always positive, so the expression is defined for all values of x 4 x 12 = Life is so much easier when the index of the radical is odd, because when the index of the radical is odd, the expression is always defined 3 x 3 5 5 =!! 1 =!! 32 Extra Problems: Text: 1-32 =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 19 Chapter 1C - Polynomials Which of these are examples of polynomials? Polynomial Not a polynomial Polynomials do not have x 2 + 2x +1 x 7 + π ( 2) x 2 + 2x +1 4 x 1 1 x + 3x2 ( 5x 3 + 3x 2 2) 1 4 ( 5x 3 + 3x 2 2) 2 x 2 + sin x 5 Polynomials are expressions of the form a n x n + a n 1 x n 1 +!+ a 1 x + a 0 where a i! and n! n is called a n is called a n x n is called a 0 is called If n = 2, the polynomial is called a Example: If n = 3, the polynomial is called a Example: A polynomial with 2 terms is called a Example: A polynomial with 3 terms is called a Example:
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 20 Factoring Polynomials Factoring polynomials will help us in later chapters when we need to find the roots of polynomials. Notice that we are working with expressions, not equations. When the instructions say, Factor, you will be given an expression and asked to change its form so that it is written as a product rather than a sum. Common Factors: 12x 4 + 4x 2 + 2x Factor by Grouping: (Factoring by grouping is your best bet if you have a polynomial with four terms!) 3x 3 + x 2 12x 4 Factor quadratics using special patterns: Look what happens when you multiply and simplify: (x 3)(x + 3) (4 y)(4 + y) Difference of Squares a 2 b 2 = Look what happens when you expand (x + 5) 2 (x 2) 2 Square of a Binomial a 2 + 2ab + b 2 = a 2 2ab + b 2 = 1. Factor a) x 2 144 b) 16x 2 81 c) x 2 +10x + 25 d) 16x 2 + 8x +1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 21 Factoring quadratics, lead coefficient 1: To factor x 2 +10x +16, determine if there are 2 numbers whose product is and whose sum is. Those two numbers are and so x 2 +10x +16 = To factor x 2 2x 48, determine if there are 2 numbers whose product is and whose sum is. Those two numbers are and so x 2 2x 48 = Factoring quadratics with lead coefficient is not 1 Use the Blankety-Blank Method! To factor: 10x 2 +11x + 3 determine if there are 2 numbers whose product is and whose sum is Those two numbers are and Now rewrite 10x 2 +11x + 3 = To factor: 3x 2 14x 5 determine if there are 2 numbers whose product is and whose sum is Those two numbers are and Now rewrite 3x 2 14x 5 = Important Fact: a 2 + b 2 does not factor over the real numbers. So x 2 + 4 cannot be factored using real numbers.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 22 Factoring polynomials in quadratic form 2. Factor completely (Sometimes factors need to be factored.) a) 9x 4 36 b) x 6 5x 3 6 c) x 10 + 2x 5 +1 4x 12 9x 6 + 2 Factoring special cubics Look what happens when you multiply (a + b)(a 2 ab + b 2 ) Similarly look what happens when you multiply (a b)(a 2 + ab + b 2 )
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 23 Sum of Cubes a 3 + b 3 Difference of Cubes a 3 b 3 3. Factor completely (Sometimes factors need to be factored.) a) 27 x 3 b) 64x 3 +1 c) x6 + 8 d) x 5 4x 3 x 2 + 4 e) 3x3 + 6x 2 12x 24 f) 16x 4 25x 2
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 24 Dividing Polynomials (This is a lot like long division of integers!) 4. 4x 3 x 2 5x + 2 x 1 What is the dividend? What is the divisor? What is the quotient?
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1C! Page 25 5. 3x 3 2x 2 + 5 x 2 1 =! What is the dividend? What is the divisor? What is the quotient? Extra Problems: Text: 1, 2, 16, 17, 18, 21-38
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1D! Page 26 Chapter 1D - Rational Expressions A rational expression is the quotient of two In this section we ll add, subtract, multiply, and divide rational expressions. Our goal will be to simplify the result so that it is expressed as the quotient of two factored polynomials. We will also need to be aware of possible restrictions on the values of our independent variable(s), usually x. Rational expressions are undefined when So the rational expression x = as long as x Sometimes we will simplify a rational expression by canceling common factors. This is similar to reducing fractions 15 18 = 3 5 3 6 = 5 6 Notice that 15 18 = 3+12, but we would not cancel the 12 s! 6 +12 When simplifying rational expressions we look for common in the numerator and denominator. Simplifying Rational Expressions: 1. Simplify the rational expressions, if possible, and list any restrictions that exist for x. a) x 1 x!!!!!! Restrictions on x :
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1D! Page 27 b) x + 2!!!!!! Restrictions on x : x 2 + 4 c) x7 x 4!!!!!! Restrictions on x : x 5 3 x Addition and Subtraction of Rational Expressions 2. Simplify a) 3 x + 2 + 2!!!!! Restrictions on x : x 3 b) x 4 x +1 x2 8x +16!!!! Restrictions on x : x 2 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1D! Page 28 Multiplication and Division of Rational Expressions 3. Simplify a) 8 x + 2 i x 2 x 2 x 2 1!!!! Restrictions on x : b) x 2 4x 21 (x 2 + 7x +12)!!! Restrictions on x : x + 4 c) x 3 1 x +1 x 1 =!!!!! Restrictions on x : x 2 + 2x +1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1D! Page 29 Compound Fractions A compound (or complex) fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division. 4. Simplify a) 3 x + 1 2x 7!!!!! Restrictions on x : 5 x +1 b) 3( x + h) 2 3x 2 h = Restrictions on x : Restrictions on h : Extra Problems: Text: 1-25
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E! Page 30 Chapter 1E - Complex Numbers 16 exists! So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers. And x!, x 2 0 But there exists a number (that is not an element of! ) named i and i 2 = Since i 2 =, 1 =, so 16 = i is not used in ordinary life, and humankind existed for 1000 s of years without considering i. i is, however, a legitimate number. i, products of i, and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was termed imaginary! i and numbers like 4i and 3i are called Numbers like 2 + i and 17 3 5i are called More formally! is the set of all numbers When a complex number has been simplified into this form, it is called a is called the part. b is called the imaginary part. So for the complex number 4 3i, is the real part while is the imaginary part. 1. Put the following complex numbers into standard form: a) ( 16 + 9) + ( 13 100) = b) ( 1+ 2i) (3 4i) = c) ( 5 + 4i)(3 2i) =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E! Page 31 d) i 3 = e) i 4 = f) i 5 = Graphing Complex Numbers When we graph elements of!, we use When we graph elements of!, we use the complex plane which will seem a lot like the Cartesian plane. In the Cartesian plane, a point represents In the complex plane, a point represents The horizontal axis is the The vertical axis is the 2. Graph and label the following points on the complex plane: A 3+ 4i!!! B 2 + i C 3 2 3i!!! D 3i E 4!!! F 1 2i
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E! Page 32 Absolute Value of a Complex Number: If z! then z is defined as its Calculate 3+ 4i (How far is 3+ 4i from the origin?) 3+ 4i = Consider a + bi, an arbitrary element of!.!! What is a + bi? (How far is a + bi from the origin?) In general z = Notice that z is a real number. It is z s distance from the origin. We did not use i to calculate z. 3. Calculate 5 + i. (How far is 5 + i from the origin?) 5 + i =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E! Page 33 Distance Between Two Complex Numbers Plot and label two points in the complex plane z 1 = 3+ 5i and z 2 = 1+ 2i The distance between z 1 and z 2 is Just for fun, calculate z 1 z 2 = And z 1 z 2 = What we have seen is that for this particular z 1 and z 2, the distance between these points is equal to. But this is actually true for all complex numbers. The distance between two general points z 1 and z 2 is Complex Conjugate If z 1 = 3+ 4i, then z 1 =. If z 2 = 1 2i, then z 2 =. Graphically the complex conjugate is the of the number through the More generally if z = a + bi, then z =
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1E! Page 34 4. Put the following complex numbers into standard form. a) 2 3i =!!! b) 4i = c) 5 =!!! d) 3i 2 = Finally notice that ( 3 + 4i) ( 3 4i) = More generally ( a + bi) ( a bi) = in other words ziz = Dividing by complex numbers 5. 25 + 50i 3+ 4i Extra Problems: Text: 1-12
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 35 Chapter 2A Solving Equations The solution of an equation is a value (or a set of values) that yields a true statement when substituted for the variable in an equation. The word solve means determine Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2x +1 = 4 and 4x + 2 = x 3 The algebraic techniques to find the solutions to these equations are simple, but I want you to keep in mind that there is a geometric interpretation associated with the equation. We are looking for the intersection of two linear functions. Here are the graphs of!!!!!! Here are the graphs of and!!!!!! and!!!!!!! See how the lines intersect at!!!!! See how the lines intersect at x =!!!!!!! x = See how substituting x =!!!! See how substituting x = into 2x +1 = 4!!!!!!!! into 4x + 2 = x 3 makes the statement true.!!!!!! makes the statement true.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 36 Solving Quadratic Equations Quadratic equations are equations involving only polynomials of degree two. Examples include x 2 x 6 = 2x 2! and! 5x 2 5x = 2x 2 3x +1 Geometrically, such equations could represent the intersection of a parabola and a line or the intersection of two parabolas. x 2 x 6 = 2x 2 5x 2 5x = 2x 2 3x +1 1 2 x 2 = x 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 37 Solve Quadratic Equations by Factoring Another technique for solving quadratic equations is by factoring. This technique is based on the Zero-Product Principle. Think about it this way: If a, b! and if ab = 1, must either a = 1 or b = 1? If ab = 0, must either a = 0 or b = 0? 1. Solve a) 1 2x 2 = x Put in standard form ax 2 + bx + c = 0 Factor Set each factor equal to zero b) 6x 2 = x +15 Put in standard form ax 2 + bx + c = 0 Factor Set each factor equal to zero
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 38 Solve by completing the square. This is an important technique that we will use in other types of problems throughout the semester. There will be a problem on exam 1 in which you will be asked to solve a quadratic equation by completing the square. Students who solve the equation correctly with methods other than completing the square will not receive credit for their solution. Before we do this, let s make some observations and practice some skills. Expand the following expressions, looking for patterns: (x 3) 2 = x 2 x (x + 4) 2 = x 2 x (x 5) 2 = x 2 x Now work backward... (x ) 2 = x 2 +16x (x ) 2 = x 2 + 22x (x ) 2 = x 2 18x
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 39 2. Solve the following quadratic equations by completing the square: a) x 2 = 36 b) ( x 2) 2 = 25 c) x 2 + 2x +1 = 16 d) x 2 + 6x = 1 e) x 2 + 9x +1 = 0 f) 2x 2 + x 8 = 0
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 40 g) ax 2 + bx + c = 0 If ax 2 + bx + c = 0, then x = This is called 3. Solve using the quadratic formula: 2x 2 = 6x 3
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 41 Equations in quadratic form 4. Solve a) x 4 5x 2 6 = 0 b) x 10 + 2x 5 +1 = 0 c) 4x 12 9x 6 + 2 = 0
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 42 d) x 4 3 2 5x 3 + 4 = 0
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 43 Solving Rational Equations Rational equations are equations involving rational functions. Solving rational equations is equivalent to finding the intersection(s) between 2 rational functions. Algebraically what we do is to set one side of the equation equal to zero and remember that a fraction is zero when The mistake: Consider solving 2 x = 3 x. A lot of people want to start by cross multiplying or multiplying both sides of the equation by x. This would yield. But neither 2 nor 3 are defined. This equation does not have a solution. The risk occurs when multiplying both sides of an equation by something that could be zero. 5. Solve a) 1 2x = 2 + 1 5x
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 44 b) 4 x 4 3 x 1 = 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 45 Radical Equations To Solve Radical equations: I. Isolate one radical. II. Raise both sides of the equation to the appropriate power to remove the radical.! (Usually this means square both sides of the equation.) III. Repeat the process until all radicals have been removed IV. Check for extraneous solutions! Why is it important to check for extraneous solutions to equations involving radicals? How many solutions exist for the equation x = 3? Square both sides of the equation: What are the solutions to this equation? See how is not a solution to the original equation?! REMEMBER: (a + b) 2 a 2 + b 2 (a + b) 2 = 6. Solve a) 3 = x + 2x 3
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 46 b) 8x +17 2x + 8 = 3 c) x + x 5 = 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 47 Equations with Absolute Values!! Recall the definition of x = 7. Solve a) x = 4 b) x 3 = 2 c) 8 x + 2 = 7
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 48 d) x +1 + 3 = 2 e) x 2 x 12 = 8!!
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 49 Equations with several variables Equations in science and engineering often include many variables, and it is useful to be able to solve for one of the variables in terms of the others.!! Warm up: If 1 x = 1 2 + 1, then what is the value of x? 3 8. The following equation comes from the physics of circuits 1 R eq = 1 R 1 + 1 R 2 + 1 R 3 Solve for R eq.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2A! Page 50 9. T s = 2π m k Solve for k. 10. Solve for y x = 3y 2y 5 Extra Problems: Text: 7-56
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 51 Chapter 2B - Solving Inequalities True or False: 4 < 4!!! True or False: 4 4 Also though 3 < 4, -4 < -3 which could also be written -3 > -4. Written more generally, if a < b, then This leads to the rule that when you multiply both sides of an inequality by a negative number, you change the direction of the inequality. A solution set is the set of values that make a (mathematical) statement true. Absolute Value Inequalities Determine the solution sets for the following inequalities Number Line Inequality Interval Notation x 3 x > 3 x > 3 x < 3 This kind of bracket indicates that the endpoint of the interval is included. This kind of parenthesis means that the endpoint of the interval is not included. The interval notation for this set 3 x < 3 is!!
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 52 1. Solve a) x < 4 b) x 2 c) x + 2 < 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 53 d) 2x 4 3 e) 5 2x > 4
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 54 f) 4 x 3 > 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 55 2. Think about solving the following inequalities graphically. a) x + 3 > 0 b) ( x + 3) ( 2 x) > 0 Where does x + 3 = 0? At x = Where does ( x + 3) ( 2 x) = 0? At x = and x = These zeros divide the x-axis into 3 regions. Graphing the point (3, 0) divides the x-axis into 2 regions. Notice that when x < -3, the graph is below the x-axis. Also notice how one factor is negative, but the other factor is positive: When x < -3 the graph is the x-axis so x+3 0 x+3 < 0, but 2-x > 0. That means their product will be negative when x < -3. When -3 < x < 2, the graph is above the x- axis. Look at the factors. Both factors are When x > -3 the graph is the x-axis so x+3 0 positive: x+3 > 0 and 2-x > 0. That means their product will be positive when -3 < x < 2 Finally when x > 2, the graph is below the x- x+3 changes sign at x=-3. x+3 goes from negative to positive. axis. This corresponds to one factor being positive while the other is negative x+3 > 0 but 2-x < 0. So product of these two factors is negative.
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 56 Nonlinear Inequalities To solve nonlinear inequalities: I. Move every term to one side of the inequality by adding or subtracting. II. Factor the nonzero side. III. Determine the critical values. (Critical values are those that make the expression zero or undefined. More specifically the values that make the numerator and/or denominator zero.) IV. Let the critical numbers divide the number line (x-axis) into intervals. V. Determine the sign of each factor in each interval. VI. Use the sign of each factor to determine the sign of the entire product or quotient. 3. Solve a) (x +1)(x 2) > 0
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 57 b) x 2 x 6 TRUE or FALSE: 2 < 3 TRUE or FALSE: 2x < 3x because when x < 0, So don t multiply both sides of an inequality by a variable that could be either positive or negative. c) 2 x < 1
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 58 d) 4 x +1 > 1 2 x
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 59 Linear Inequalities 4. Solve a) 1 x > x + 3 b) 1 2x < 17 4x 8 x
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 2B! Page 60 c) 2x + 4 < x + 2 < x 2 + 3 d) 4x +1 4 2x 3 x Extra Problems: Text: 1-34