Math Basics for the Health Care Professional Copyright © 2014, 2009 by Pearson Education, Inc. All Rights Reserved FOURTH EDITION UNIT Pre-Algebra Basics 7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Student Learning Outcomes 1.Define basic terms in algebra: integer, number statement, expression, and coefficient 2.Learn the relationships between positive and negative numbers 3.Solve problems that include absolute value

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Student Learning Outcomes 4.Complete the integer operations of addition, subtraction, multiplication, and division 5.Calculate the square root 6.Use the order of operations to solve problems

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Student Learning Outcomes 7.Write expressions for word problems 8.Solve equations

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Overview Algebra uses  Symbolic reasoning  Mathematical properties  Processes  Calculations to solve for “unknowns” (4 + x) × ÷ 3 − 2 UNKNOWN 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Integers Positive whole numbers (1, 2, 3, …) Their negatives (−1, −2, −3, …) Zero (neutral) ZERO (neutral) 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Number Line Labeled with integers Extends in both directions  − means negative  + means positive Any integer on the RIGHT is greater than any integer on the LEFT 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Number Line Number line ZERO (neutral) 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Absolute Value Uses straight lines for notation Indicates how far from zero Never negative |−3| = 3 away |+3| = 3 away 7-1, 7-2, 7-3

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Adding Integers with like Signs Add two positive numbers.  = |2| + |7| = +9  The numbers move right on the number line , 7-3, 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Add two negative numbers.  −4 + −3 = |4| + |3| = −7  The numbers move left along the number line 3 Adding Integers with like Signs 4 7-2, 7-3, 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Adding Integers with Unlike Signs Take absolute values Subtract smaller from larger Use sign of larger − = |8| − |5| = +3 (or 3) −5− , 7-3, 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Subtracting Integers Find its “opposite,” and then add it Example: (+5) − (−9) Step 1: Change math function sign from – to +. (+5) + (−9)

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Subtracting Integers Example: (+5) − (−9) Step 2: Change number sign to its opposite. (+5) + (+9) Step 3: Then add to find solution. (+5) + (+9) = OR COUNT THE SPACES ON THE NUMBER LINE! 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Multiplying Integers Whole number format  2 × 4 = 8 Algebra format  (2)(4) = 8 or 2 ∙ 4 = 8 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Multiplying Integers Rules to determine sign of the product  Positive × positive = positive (+2)(+3) = (+6) or 6  Negative × negative = positive (−7)(−6) = (+42) or 42  Negative × positive = negative (−5)(+6) = (−30) 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Dividing Integers Whole number format  28 ÷ 7 Algebra format  28/7 or 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Dividing Integers Rules to determine sign of the quotient  Positive / positive = positive (+108)/(+3) = (+36) or 36  Negative/negative = positive (−51)/(−3) = (+17) or 17  Negative/positive = negative; Positive/ negative = negative (-63)/(+3) = (−21) and (+63)/ (−3) = (−21) 7-4

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Exponential Notation Base: Number being multiplied (3) Exponent: Number of times base is being multiplied (5) 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Exponential Notation Exponential notation  Three forms Exponential form (exponential notation) 3 5 Factor form (repeated multiplications) 3 × 3 × 3 × 3 × 3 or (3)(3)(3)(3)(3) or 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 Standard form

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Exponents: Rules Number raised to 1st power = itself  7 1 = 7 Number raised to 2nd power is called “squared”  5 2 is “5 squared” Number raised to 3rd power is called “cubed”  4 3 is “4 cubed” 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Exponents: Rules Number raised to 0 power = 1 (except 0)  9 0 = 1, 15 0 = 1, but 0 0 = 0 Handling zero  0 0 = 0, 0 1 = 0, 0 2 = 0, 0 3 = 0, 0 4 = 0, etc. 7-1

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Square Roots Sign = Factor × factor = the number Example: Why? 10 × 10 = 10 2 = 100 More examples: 7-5

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Order of Operations Perform operations in this order  Step 1: Perform any calculations involving parentheses, fraction bars, exponents, and square roots.  Step 2: Perform all multiplication and division, working from left to right. 7-6

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Order of Operations Perform operations in this order  Step 3: Perform all addition and subtraction, working from left to right 7-6

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Meet my Aunt Sally! “Please Excuse My Dear Aunt Sally” (PEMDAS)  P:Parentheses and fraction bars  E: Exponents and roots  M:Multiplication  D:Division  A: Addition  S: Subtraction 7-6

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Example: Order of Operations Having a rule ensures the correct answer! 7-6

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Example: Order of Operations Problem: × 8  Following order of operations (PEMDAS)  correct answer Step 1: 7 × 8 = 56 Step 2: = 61 (correct answer)  Not following order of operations (PEMDAS)  incorrect answer Step 1: = 12 Step 2: 12 × 8 = 96 (incorrect answer) 7-6

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Algebraic Expressions Variable (unknown number)  Examples: p, t, x, y, s  Practical example: s represents for number of students 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Algebraic Expressions Coefficient (known number)  Example: −8x (−8 is known) (x is unknown)  Example: x is the same as 1x  Example: xy is the same as 1xy KNOWN COEFFICIENT  5x  UNKNOWN VARIABLE 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Expressions Expressions contain numbers and/or variables.  Examples: 9 + z, xy, 9(3 − x), 20 + (9 − x) 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Expressions Word problem:  An LPN spends $240 on books (known amount).  Dental assistant spends y on books (unknown amount).  Write an expression for their combined $ spent on books. Combined spending is: $240 + y 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Common Algebraic Expressions “Some number n increased by 2.7”  n “9 less than some number x”  x − 9 “Sum of two numbers a and b”  a + b 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Common Algebraic Expressions “Some number q multiplied by 5”  5q “Product of x and −9”  −9 x “4 times sum of 3 numbers x, y, z”  4(x + y + z) 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Evaluating Expressions Evaluating (not “solving”)  Given: value of the variable  Wanted: the evaluation with this given value 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Evaluating Expressions Evaluate 5y + 12 when y = 11  Step 1: Replace y with 11. 5∙  Step 2: Follow the order of operations (PEMDAS). 5∙ 11 = 55 (multiply first) = 67 (then add)  The algebraic expression evaluated with y = 11 is

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Writing Expressions from Words Key words you can use: 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Eliminating Confusion Example of possible confusion  “five more than x”  x + 5 (an expression)  “five is more than x”  5 > x (an equation or inequality) Example of possible confusion  “four less than z”  z – 4  “four minus z”  4 – z  “four is less than z”  4 < z 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Eliminating Confusion Correct example: 12 points higher on this test than on the last  Let x = score on last test  Then twelve more is x

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Solving Equations Get variable by itself on one side by using inverses  Addition inverse of subtraction (and vice versa)  Multiplication inverse of division (and vice versa) 7-8

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Solving Equations Example: Solve x = −5.2 x = −5.2 x − 1.5 = −5.2 − 1.5 x = −5.2 − 1.5 x = −

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Writing Equations from Words Equation: mathematical sentence with an “=” sign Problem: If a patient’s current temperature decreases by 2 degrees, it will be 99.4 degrees. What is the patient’s current temperature? 7-7

Math Basics for the Health Care Professional, Fourth Edition Michele Benjamin Lesmeister Writing Equations from Words temp decreased by 2 degrees is 99.4 degrees t – 2 = 99.4 t - 2 = 99.4 t − = t =