Math 110 online work that is due today at the start of class Section 1 2 Homework no worksheet to turn in on this one The Syllabus Quiz can still be redone if you haven t yet gotten 100 Any questions on the Section 1 2 homework that was due today Please CLOSE YOUR LAPTOPS and turn off and put away your cell phones and get out your notetaking materials Gateway Quiz Reminders You will be taking an 8 question Gateway Quiz in class at the next class session without a calculator The Gateway Quiz will consist of 8 questions similar to the ones on fractions and the order of operations that were due at the beginning of the previous class We will spend some time today on how to do these kinds of problems and you will have another set of eight practice Gateway problems to do at the end of the homework assignment due at the next class session You also have a Practice Gateway Quiz due at the start of next class session with a REQUIRED paper worksheet that must be turned in worksheet will be handed out now make sure you get one before you leave class You can try it up to 3 times and only your best score will count in the final grade Be sure to put the work on the worksheet for the Practice Quiz with the best score Section 1 3 Operations on Real Numbers Working with positive and negative signs A number line is a line on which each point is associated with a number 5 4 3 2 1 Negative numbers 0 1 2 3 4 Positive numbers 5 To add two numbers especially if one or both of them is negative it often helps to picture them on a number line Example 1 3 5 Locate the first number on the number line Starting from that number if the second number is positive move to the right by that many units If it s negative move to the left that many units 3 5 Count 5 units to the left from 3 5 4 3 2 1 Answer is 2 0 1 2 3 4 5 Example 2 2 3 Locate the first number on the number line Start from that first number The second number is negative so move to the left by that many units 3 Count 3 units to the left from 2 2 5 4 3 Answer is 5 2 1 0 1 2 3 4 5 Example from today s homework Answer 30 Subtracting Real Numbers Subtracting real numbers Substitute the opposite of the number being subtracted Add a b a b Example Subtract the following numbers 5 6 3 5 6 3 8 Multiplying or Dividing Real Numbers Multiplying or dividing two real numbers with same sign Result is a positive number Multiplying or dividing two real numbers with different signs Result is a negative number Multiplying or Dividing Real Numbers Examples Find each of the following products 4 2 3 24 4 5 20 Working with zero If b is a real number 0 b b 0 0 The quotient of any real number and 0 is undefined a 0 The quotient of 0 and any real number 0 0 a a 0 Sample problem from today s homework Answer 0 If this problem had been 7 0 the answer would be N undefined Exponents We may use exponential notation to write products in a more compact form Exponential notation for the product of five threes is 35 Base is 3 Exponent is 5 The notation means 3 3 3 3 3 or 243 Example Evaluate 43 43 4 4 4 64 Evaluating Exponential Expressions Example Evaluate each of the following expressions 34 3 3 3 3 5 2 5 5 62 6 6 2 4 3 81 25 36 2 4 2 4 2 4 3 42 3 4 4 48 8 8 8 512 The Order of Operations Order of Operations Simplify expressions using the order that follows If grouping symbols such as parentheses are present simplify expression within those first starting with the innermost set If fraction bars are present simplify the numerator and denominator separately 1 Evaluate exponential expressions roots or absolute values in order from left to right 2 Multiply or divide in order from left to right 3 Add or subtract in order from left to right Order of Operations Memory Device Please excuse my dear Aunt Sally 1 Please 2 Excuse 3 My Dear 4 Aunt Sally Parentheses Exponents Multiply and Divide Add and Subtract or just remember PEMDAS 18 Using the Order of Operations Example Evaluate 6 9 3 32 6 9 3 6 9 3 2 9 3 6 3 9 9 9 1 Write 32 as 9 Divide 9 by 3 Add 3 to 6 Divide 9 by 9 More examples Simplify the following expressions 6 2 2 25 6 2 2 32 6 4 32 2 32 34 3 6 8 5 3 6 3 2 4 2 3 18 16 2 21 3 7 7 16 2 18 3 6 6 Example Simplify the following 8 3 2 9 2 3 8 6 14 14 9 6 3 3 Sample problem from today s homework 324 7 Evaluating Algebraic Expressions To evaluate an algebraic expression substitute the numerical value for each variable into the expression and simplify the result using the order of operations Example Evaluate each expression for the given value a 5x 2 for x 8 5 8 2 40 2 38 b 3a2 2a 4 for a 4 3 4 2 2 4 4 3 16 8 4 44 NOTE Many of today s homework problems as well as Gateway problems 7 8 all require using the order of operations Order of operations 1 First calculate expressions within grouping symbols parentheses brackets braces absolute values fraction bars If there are nested sets of grouping symbols start with the innermost ones first and work your way out 2 Exponential expressions left to right 3 Multiplication and division left to right 4 Addition and subtraction left to right 24 Sample Gateway Problem 7 Strategy Calculate out the entire top expression and then the entire bottom expression using the order of operations on each part Then simplify the resulting fraction if necessary TOP EXPRESSION 24 4 7 2 Step 1 Parentheses 24 4 7 2 24 4 9 Step 2 Exponents 24 4 9 2 2 2 2 4 9 16 4 9 because 2 2 2 2 4 2 2 8 2 16 Step 3 Multiply Divide 16 4 9 16 4 9 16 36 Step 4 Add Subtract 16 36 20 25 Now calculate the bottom expression 2 6 2 4 Step 1 Parentheses 2 6 2 4 2 8 4 Step 2 Exponents There aren t any in this part Step 3 Multiply Divide 2 8 4 2 8 4 16 4 Step 4 Add Subtract 16 4 20 Now put the top over the bottom and simplify the resulting fraction TOP 24 4 7 2 20 1 1 BOTTOM 2 6 2 4 20 1 …