MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §1.1 Expressions & Real No.s

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 2 Bruce Mayer, PE Chabot College Mathematics Basic Terminology  A LETTER that can be any one of various numbers is called a VARIABLE.  If a LETTER always represents a particular number that NEVER CHANGES, it is called a CONSTANT A & B are CONSTANTS

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 3 Bruce Mayer, PE Chabot College Mathematics Algebraic Expressions  An ALGEBRAIC EXPRESSION consists of variables, numbers, and Math-Operation signs. Some Examples  When an equal sign is placed between two expressions, an EQUATION is formed →

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 4 Bruce Mayer, PE Chabot College Mathematics Translate: English → Algebra  “Word Problems” must be stated in ALGEBRAIC form using Key Words per of less than more than ratio twicedecreased byincreased by quotient of times minus plus divided byproduct ofdifference of sum of divide multiply subtract add DivisionMultiplicationSubtractionAddition

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Translation  Translate this Expression: Eight more than twice the product of 5 and an Unknown number  SOLUTION LET n ≡ the UNknown Number Eight more than twice the product of 5 and a number.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 6 Bruce Mayer, PE Chabot College Mathematics Evaluate Algebraic Expressions  When we REPLACE A VARIABLE with a number, we are SUBSTITUTING for the variable.  The calculation that follows is called EVALUATING the expression Note: the normal result for a “Evaluation” is usually a SINGLE NUMBER with NO LETTERS

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Evaluating  Evaluate  SOLUTION 8xz − y = 8·2·3 − 7 = 41 = 48 − 7 Substituting Multiplying Subtracting

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 8 Bruce Mayer, PE Chabot College Mathematics Exponential Notation The expression a n, in which n is a counting number (1, 2, 3, etc.) means n factors In a n, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a 1 = a.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 9 Bruce Mayer, PE Chabot College Mathematics Order of Operations (PEMDAS)  Perform operations in this order: 1.Grouping symbols: parentheses ( ), brackets [ ], braces { }, absolute value | |, and radicals √ 2.Exponents from left to right, in order as they occur. 3.Multiplication/Division from left to right, in order as they occur. 4.Addition/Subtraction from left to right, in order as they occur.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Order of Ops  Evaluate  SOLUTION 2(x + 3) 2 – 12 x 2 Substituting Simplifying 5 2 and 2 2 Multiplying and Dividing Subtracting = 2(2 + 3) 2 – Working within parentheses

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 11 Bruce Mayer, PE Chabot College Mathematics Formulas  A FORMULA is an equation that uses letters to represent a RELATIONSHIP between two or more quantities.  Example  The area, A, of a circle of radius r is given by the formula: r

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Temp Conversion  The formula for converting the temperature in degrees Celsius (C) to degrees Fahrenheit (F)  Use the formula to convert 86F to the Celsius form Substitute 86 for F Solve for C

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 13 Bruce Mayer, PE Chabot College Mathematics Temperature Conversion cont. Solve for C.  Thus 30 °C is the Equivalent of 86 °F

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 14 Bruce Mayer, PE Chabot College Mathematics Mathematical Modeling  A mathematical model can be a formula, or set of formulas, developed to represent a real-world situation.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Math Model  Mei-Li is 5ft 7in tall with a Body Mass Index (BMI) of approximately What is her weight?  SOLUTION 1.Familiarize. The body mass index I depends on a person’s height and weight. The BMI formula: –where W is the weight in pounds and H is the height in inches.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 16 Bruce Mayer, PE Chabot College Mathematics BMI Example cont 2.Translate. Solve the formula for W: 3.Carry Out 5 ft 7 in. = 67 in.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 17 Bruce Mayer, PE Chabot College Mathematics BMI Example cont 4.Check 5.State Answer  Mei-Li weighs about 150 Pounds 

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 18 Bruce Mayer, PE Chabot College Mathematics Set of Real Numbers  SET ≡ a Collection of Objects  Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 can be written {1, 2, 3, 4}. The numbers 1, 2, 3, and 4 are called the members or elements of this set.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 19 Bruce Mayer, PE Chabot College Mathematics Set Notation Roster notation: {2, 4, 6, 8} Set-builder notation: {x | x is an even number between 1 and 9} “The set of all xsuch thatx is an even number between 1 and 9”

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 20 Bruce Mayer, PE Chabot College Mathematics Sets of Real Numbers  Natural Numbers (Counting Numbers) Numbers used for counting: {1, 2, 3,…}  Whole Numbers The set of natural numbers with 0 included: {0, 1, 2, 3,…}  Integers The set of all whole numbers AND their opposites: {…,−3, −2, −1, 0, 1, 2, 3,…}

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 21 Bruce Mayer, PE Chabot College Mathematics Sets of Real Numbers, x  Rational Numbers (Integer Fractions) Maybe expressed as a FRACTION of two INTEGERS –Terminating Decimals (e.g.; 7/16) –Repeating NonTerminating Decimals (e.g., 2/7)  Irrational Numbers Can NOT be expressed as a Fraction of two Integers –NONterminating, NONreapeating decimals (e.g., π = ………………………)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 22 Bruce Mayer, PE Chabot College Mathematics Irrational numbers: Integers: Zero: 0 Real Numbers: Rational Numbers: Rational numbers that are not integers : Negative Integers: Whole numbers : 0, 1, 2, 3, 29 Positive integers or natural numbers: 1, 2, 3, 29 Real No. Family Tree

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 23 Bruce Mayer, PE Chabot College Mathematics Real Number Nest

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 24 Bruce Mayer, PE Chabot College Mathematics Real Number Line  An Infinite line whose points have been assigned number-coordinates is called the real number line

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 25 Bruce Mayer, PE Chabot College Mathematics 3 Sectors of the Number Line 1.The negative real numbers are the CoOrds to the left of the origin O 2.The real number zero is the CoOrd of the origin O 3.The positive real numbers are the CoOrds to the right of the origin O

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 26 Bruce Mayer, PE Chabot College Mathematics InEqualities  < means “is less than”  ≤ means “is less than or equal to”  > means “is greater than”  ≥ means “is greater than or equal to”  For any two numbers on a number line, the one to the left is said to be less than the one to the right. – –1 < 3 since –1 is to the left of 3

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 27 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §1.2 Operations with Real No.s

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 28 Bruce Mayer, PE Chabot College Mathematics Absolute Value  The ABSOLUTE VALUE of a number is its distance from zero on a number line.  The symbol |x| to represents the absolute value of a number x.  Example  |5| = |−5| = 5  5 units from 0  “Distance” is ALWAYS Positive

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 29 Bruce Mayer, PE Chabot College Mathematics Find Absolute Value  Finding Absolute Value If a number is negative, make it positive If a number is positive or zero, leave it alone  Example  Find the absolute value of each number. a) |−4.5|b) |0|c) |−3|  Solution a) |−4.5| The dist of −4.5 from 0 is 4.5, so |−4.5| = 4.5 b) |0| The distance of 0 from 0 is 0, so |0| = 0. c) |−3| The distance of –3 from 0 is 3, so |−3| = 3

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 30 Bruce Mayer, PE Chabot College Mathematics Real No. Addition Rules 1.Positive numbers: Add the numbers. The result is positive. 2.Negative numbers: Add absolute values. Make the answer negative. 3.A positive and a negative number: Subtract the smaller absolute value from the larger. Then: a)If the positive number has the greater absolute value, make the answer positive. b)If the negative number has the greater absolute value, make the answer negative. c)If the numbers have the same absolute value, the answer is 0. 4.One number is zero: The sum is the other number

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 31 Bruce Mayer, PE Chabot College Mathematics Inverse Property of Addition  For any real number a, the opposite, or additive inverse, of a, (which is −a) is such that a + (−a) = −a + a = 0  Example  Find the opposite, or additive inverse: a) 8b) −13  Solution  a) 8 −(8) = −8 b) −13 −(−13) = 13

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 32 Bruce Mayer, PE Chabot College Mathematics Subtraction  Subtraction ≡ The difference a − b is the unique number c for which a = b + c. That is, a − b = c if c is a number such that a = b + c  Subtracting by Adding the Opposite For any real numbers a and b a − b = a + (−b) –We can subtract by adding the opposite (additive inverse) of the number being subtracted (the Subtrahend)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 33 Bruce Mayer, PE Chabot College Mathematics Real No. Multiplication  The Product of a Positive and a Negative Number To multiply a positive number and a negative number, multiply their absolute values. The answer is negative: 3(−2) = −6  The Product of Two Negative Numbers To multiply two negative numbers, multiply their absolute values. The answer is positive: (−13)(−11) = +143

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 34 Bruce Mayer, PE Chabot College Mathematics Real No. Division  The quotient a  b or a/b where b ≠ 0, is that unique real number c for which a = b c.  Example  Divide  SOLUTION Because − 5 · ( − 9 ) = 45 Because − 8 · (8) = − 64

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 35 Bruce Mayer, PE Chabot College Mathematics Multiply & Divide Rules  To multiply or divide two real numbers: 1.Multiply or divide the absolute values. 2.If the signs are the same, then the answer is positive. 3.If the signs are different, then the answer is negative.  DIVISION by ZERO NEVER divide by ZERO. If asked to divide a nonzero number by zero, we say that the answer is UNDEFINED. If asked to divide 0 by 0, we say that the answer is INDETERMINATE.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 36 Bruce Mayer, PE Chabot College Mathematics Real Number Properties  COMMUTATIVE property of addition and multiplication a + b = b + a and ab = ba  ASSOCIATIVE property a + ( b + c ) = ( a + b ) + c and a ( bc ) = ( ab ) c  DISTRIBUTIVE property a(b + c) = ab + ac

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 37 Bruce Mayer, PE Chabot College Mathematics Real Number Properties  ADDITIVE IDENTITY property a + 0 = 0 + a = a  ADDITIVE INVERSE property −a + a = a +(−a) = 0  MULTIPLICATIVE IDENTITY property a 1 = 1 a = a  MULTIPLICATIVE INVERSE property a (1/a) = (1/a) a = 1(a ≠ 0)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 38 Bruce Mayer, PE Chabot College Mathematics Simplify Expressions  A TERM is a number, a variable, a product of numbers and/or variables, or a product or quotient of two numbers and/or variables.  Terms are SEPARATED by ADDITION signs. If there are SUBTRACTION signs, we can find an equivalent expression that uses addition signs.  COLLECTING LIKE TERMS is based on the DISTRIBUTIVE law

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 39 Bruce Mayer, PE Chabot College Mathematics Like (or Similar) Terms exactly  Terms in which the variable factors are exactly the same, such as 9x and −5x, are called like, or similar terms.  Like TermsUNlike Terms  7x and 8x8y and 9y 2  3xy and 9xy5ab and 4ab 2

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  Combine Terms  a) 7x + 3xb) 4a + 5b a − 6 − 5b  SOLUTION a) 7x + 3x = (7 + 3)x = 10x b) 4a + 5b a − 6 − 5b = 4a + 5b a + (−6) + (−5b) = 4a + a + 5b + (−5b) (−6) = [4a + a] + [5b + (−5b)] + [2 + (−6)] = 5a (−4) = 5a − 4

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Simplify  Remove parentheses and simplify [6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)]  SOLUTION [6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)] = [6m + 18 – 5m] – [4n + 20 – 8n + 32] Distribute = [m + 18] – [–4n + 52] Collect like terms within brackets = m n – 52 Removing brackets = m + 4n – 34 Collecting like terms

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 42 Bruce Mayer, PE Chabot College Mathematics TERMS ≠ factors  Factors are the “pieces” of a Multiplication Chain; e.g., if Then y has four factors: 7, u, v, w  TERMS are the pieces of an ADDITION Chain  Then z has Three TERMS: 7a, 3b, −5c

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 43 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §1.3 Graphing Equations

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 44 Bruce Mayer, PE Chabot College Mathematics Points and Ordered-Pairs  To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions  Consider the pair (2, 3). The numbers in such a pair are called the coordinates. The first coordinate, x, in this case is 2 and the second, y, coordinate is 3.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 45 Bruce Mayer, PE Chabot College Mathematics Plot-Pt using Ordered Pair  To plot the point (2, 3) we start at the origin, move horizontally to the 2, move up vertically 3 units, and then make a “dot” x = 2 y = 3 (2, 3)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  Plot the point (−4,3)  Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up) x = −4; y = 3 4 units left 3 units up

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 47 Bruce Mayer, PE Chabot College Mathematics Example  Read XY-Plot  Find the coordinates of pts A, B, C, D, E, F, G A B C D E F G Solution: Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: –B: (–2, 4) –C: (–3, –4) –D: (3, –2) –E: (2, 3) –F: (–3, 0) –G: (0, 2)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 48 Bruce Mayer, PE Chabot College Mathematics XY Quadrants  The horizontal and vertical axes divide the plotting plane into four regions, or quadrants

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 49 Bruce Mayer, PE Chabot College Mathematics Graphing Equations  Definitions An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true. An ordered pair that satisfies an equation is called a solution of the equation Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 50 Bruce Mayer, PE Chabot College Mathematics Eqn Graph  Bottom Line  ANY and ALL points (ordered pairs) on a Math Graph are SOLUTIONS to the Equation that generated the Graph

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 51 Bruce Mayer, PE Chabot College Mathematics Graph of an Equation  The graph of an equation in two variables, such as x and y, is the set of all ordered pairs (a, b) in the coordinate plane that satisfy the equation  y = 2x + 6 − 2x + y = 6 2x − y + 6 = 0

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 52 Bruce Mayer, PE Chabot College Mathematics Graphing by Plotting Points  Graph y = x 2 – 3  SOLUTION → Make “T-table” (–3, 6)y = (–3) 2 – 3 = 9 – 3 = 6–3 (x, y)y = x 2 – 3x X col Pick x Calc y Ordered Pair (–3, 6) is a solution to y = x 2 – 3 [ 6 = 3 2 – 3  ]

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 53 Bruce Mayer, PE Chabot College Mathematics Graph by Pt-Plot y = x 2 – 3  Pick “enough” x’s for T-table (3, 6)y = 3 2 – 3 = 9 – 3 = 63 (2, 1)y = 2 2 – 3 = 4 – 3 = 12 (1, –2)y = 1 2 – 3 = 1 – 3 = –21 (0, –3)y = 0 2 – 3 = 0 – 3 = –30 (–1, –2)y = (–1) 2 – 3 = 1 – 3 = –2–1 (–2, –1)y = (–2) 2 – 3 = 4 – 3 = 1–2 (–3, 6)y = (–3) 2 – 3 = 9 – 3 = 6–3 (x, y)y = x 2 – 3x

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 54 Bruce Mayer, PE Chabot College Mathematics Graph by Pt-Plot y = x 2 – 3  Plot (x,y) Points listed in T-table and connect the dots to complete the plot Note that most Graphs are “curves” so connect dots with curved lines

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 55 Bruce Mayer, PE Chabot College Mathematics Graph by Pt-Plot: y = x 2 – 2x – 6  Construct T-table xy(x, y) 33 9 (  3, 9) 22 2 (  2, 2) 0 66(0,  6) 1 77(1,  7) 2 66(2,  6) 3 33(3,  3) 42(4, 2) 59(5, 9)  Plot-Pts & Connect-Dots

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 56 Bruce Mayer, PE Chabot College Mathematics GRAPH BY PLOTTING POINTS  Step1. Make a representative T-table of solutions of the equation.  Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.  Step 3. Connect the solutions in Step 2 by a smooth curve

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 57 Bruce Mayer, PE Chabot College Mathematics Domain & Range by Graphing  Graph y = x 2. Then State the Domain & Range of the equation  Select integers for x, starting with −2 and ending with +2. The T-table:

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 58 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Now Plot the Five Points and connect them with a smooth Curve (−2,4)(2,4) (−1,1)(1,1) (0,0)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 59 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  The DOMAIN of a function is the set of ALL first (or “x”) components of the Ordered Pairs that appear on the Graph  Projecting on the x-axis ALL the x-components of ALL POSSIBLE ordered pairs displays the DOMAIN of the function just plotted

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 60 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Domain of y = x 2 Graphically  This Projection Pattern Reveals a Domain of

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 61 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  The RANGE of a function is the set of all second (or “y”) components of the ordered pairs. The projection of the graph onto the y-axis shows the range

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 62 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  The projection of the graph onto the y-axis is the interval of the y-axis at the origin or higher, so the range is

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 63 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §1.4 Solve Linear Eqns

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 64 Bruce Mayer, PE Chabot College Mathematics Solution For an Equation  Any NUMBER-REPLACEMENT for the VARIABLE that makes an equation true is called a SOLUTION of the equation.  To Solve an equation means to find ALL of its Solutions.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 65 Bruce Mayer, PE Chabot College Mathematics Equivalence and Addition Principle  Equivalent Equations Equations with the SAME SOLUTIONS are called EQUIVALENT equations  Addition Principle For any real numbers u, v, and w, u = v is equivalent to u + w = v + w – e.g.; u = 3, v = 3, w = 7 → 3 = 3 and 3+7 = 3+7

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 66 Bruce Mayer, PE Chabot College Mathematics Multiplication Principle  The Multiplication Principle: For any real numbers r, s, and t with t ≠ 0, r = s is equivalent to rt = st  Example Solve for x  Solution Multiply Both Sides by 4/3 

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 67 Bruce Mayer, PE Chabot College Mathematics Linear Equations  A linear equation in one variable, such as x, is an equation that can be written in the standard form  where a and b are real numbers with a ≠ 0

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 68 Bruce Mayer, PE Chabot College Mathematics Solution to Linear Equations  Procedure for solving linear equations in one variable 1.Eliminate Fractions: if Needed, Clear Fractions by Multiplying both sides of the equation by the least common denominator (LCD) of all the fractions 2.Simplify: Simplify both sides of the equation by removing parentheses and other grouping symbols (if any) and combining like terms

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 69 Bruce Mayer, PE Chabot College Mathematics Solution to Linear Equations 3.Isolate the Variable Term: Add appropriate expressions to both sides, so that when both sides are simplified, the terms containing the VARIABLE are on ONE SIDE and all constant terms are on the other side. 4.Combine Terms: Combine terms containing the variable to obtain one term that contains the variable as a factor.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 70 Bruce Mayer, PE Chabot College Mathematics Solution to Linear Equations 5.Isolate the Variable: Divide both sides by the coefficient of the variable to obtain the solution. 6.Check the Solution: Substitute the solution into the original equation

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 71 Bruce Mayer, PE Chabot College Mathematics Example  Solve Linear Eqn  Solve for x:  SOLUTION (No Fractions to clear) Step 2

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 72 Bruce Mayer, PE Chabot College Mathematics Example  Solve Linear Eqn  Solve for x:  SOLUTION Step 3 Step 4 Step 5

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 73 Bruce Mayer, PE Chabot College Mathematics Example  Solve Linear Eqn  Solve for x:  SOLUTION Step 6 Check x = 3:  State  The Solution is x = 3 

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 74 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §1.5 Problem Solving

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 75 Bruce Mayer, PE Chabot College Mathematics Mathematical Model  A mathematical model is an equation or inequality that describes a real situation.  Models for many APPLIED (or “Word”) problems already exist and are called FORMULAS  A FORMULA is a mathematical equation in which variables are used to describe a relationship

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 76 Bruce Mayer, PE Chabot College Mathematics Formula Describes Relationship RelationshipMathematical Formula Perimeter of a triangle: a b c h Area of a triangle:

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 77 Bruce Mayer, PE Chabot College Mathematics Example  Geometry of Cone RelationshipMathematical Formulae h r Volume of a cone: Surface area of a cone: (slant-sides + bottom)

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 78 Bruce Mayer, PE Chabot College Mathematics Example  °F ↔ °C RelationshipMathematical Formulae Celsius to Fahrenheit: Fahrenheit to Celsius: CelsiusFahrenheit

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 79 Bruce Mayer, PE Chabot College Mathematics Example  Mixtures RelationshipMathematical Formula Percent Acid, P : Base Acid

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 80 Bruce Mayer, PE Chabot College Mathematics Solving a Formula  Sometimes the formula is solved for a Different variable than the one we need  Example  A mathematical model tell us that voltage, V, in a circuit is equal to current, I, times resistance, R: V = I R  To determine the amount of resistance in a circuit, it would help to first solve the formula for R.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 81 Bruce Mayer, PE Chabot College Mathematics Example  Solve V = IR for R  We solve this formula for R by treating V and I as CONSTANTS (having fixed values) treating R as the only variable.  Begin by writing the formula so that the variable for which we are solving, R, is on the left side. I R = V

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 82 Bruce Mayer, PE Chabot College Mathematics Example  Solve V = IR for R  Finally, use Algebra properties to isolate the variable R. Divide both sides by I. Isolated R → Done

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 83 Bruce Mayer, PE Chabot College Mathematics Example  Trapezoid Base, B B b h This formula gives the relationship between the height, h, and two bases, B and b, of a trapezoid and its area, A.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 84 Bruce Mayer, PE Chabot College Mathematics Example  Trapezoid Base, B Mult. Prop. of Equality. Assoc. Prop. Inverse Prop. Identity Prop.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 85 Bruce Mayer, PE Chabot College Mathematics Example  Trapezoid Base, B Add. Prop. Of Equality Divide by h. Distributive Prop. Inverse Prop.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 86 Bruce Mayer, PE Chabot College Mathematics Example  Prismatic Volume h b l The volume of a triangular prism is given by: If the volume of a triangular cylinder is 880 cm 3, the base is 10 cm, and the length is 22 cm, then find the height.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 87 Bruce Mayer, PE Chabot College Mathematics Example  Prismatic Volume First, solve the equation for h.

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 88 Bruce Mayer, PE Chabot College Mathematics Example  Prismatic Volume  Second, find the height, h, by substituting the given values of V, b, and l into this formula:  State  The height of the triangular prism is 8 cm

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 89 Bruce Mayer, PE Chabot College Mathematics Example  % from a Pie Chart  The pie chart shown at right represents the distribution of grades in MTH2 (Calculus) last year. Use the information in the chart to estimate how many B’s will be given in a new class of size 70 students. F 8% A 16% B 28% C 36% D 12%

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 90 Bruce Mayer, PE Chabot College Mathematics Example  % from a Pie Chart  According to the chart, 28% OF the students should get a grade of B. Let x represent the number of students getting a B. F 8% A 16% B 28% C 36% D 12%

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 91 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §1.n Exercise Sets NONE Today → Lecture PPT took Entire Class Time  Identity Symbol Difference of Two Squares Identity

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 92 Bruce Mayer, PE Chabot College Mathematics All Done for Today The “Defined as” Symbol

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 93 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 94 Bruce Mayer, PE Chabot College Mathematics Set Membership Notation

MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 95 Bruce Mayer, PE Chabot College Mathematics Tool For XY Graphing  Called “ Engineering Computation Pad” Light Green Backgound Tremendous Help with Graphing and Sketching Available in Chabot College Book Store I use it for ALL my Hand-Work Graph on this side!