PowerPoint PresentationAny questions on the Section 1.2 homework that was due today?Slide 3Gateway Quiz Reminders:Section 1.3Working with positive and negative signsSlide 7Slide 8Slide 9Subtracting Real NumbersMultiplying or Dividing Real NumbersSlide 12Slide 13Sample problem from today’s homework:ExponentsEvaluating Exponential ExpressionsSlide 17Slide 18Slide 19Slide 20Example:Slide 22Evaluating Algebraic ExpressionsNOTE: Many of today’s homework problems as well as Gateway problems 7 & 8 all require using the order of operations.Sample Gateway Problem # 7:Now calculate the bottom expression: 2(6+2) + 4Full Solution to Sample Problem #7:Sample Gateway Problem # 8:Full Solution to Sample Problem #8:Quick review of fractions:If you want more tips on Gateway Problems 1-6 (and today’s last 8 homework problems) click on the PowerPoint Slides item in today’s Assignment folder for Section 1.3, then click on the link for Gateway Problems. (click here to show link)Reminders:You may now open your laptops to work on the homework for section 1.3 and/or the practice Gateway Quiz. If you have finished the homework and practice quiz already, or if you get them both finished before the end of the class period, show your on-screen 100% score to the teacher or TA and you may then leave class early.Slide 34Math 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?PleaseCLOSE YOUR LAPTOPS,and turn off and put away your cell phones,and get out your note-taking 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 1.3Operations on Real NumbersWorking with positive and negative signsA number line is a line on which each point is associated with a number.2– 2 0 1 3 4 5– 1– 3– 4– 5Negative numbersPositive numbers2– 2 0 1 3 4 5– 1– 3– 4– 5To 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)Answer is: -22– 2 0 1 3 4 5– 1– 3– 4– 5Example 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.2- (-3)(Count 3 units to the left from -2)Answer is: -5Example from today’s homework:Answer: -30Subtracting 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 =– 8Subtracting Real NumbersMultiplying or dividing two real numbers with same sign•Result is a positive numberMultiplying or dividing two real numbers with different signs•Result is a negative numberMultiplying or Dividing Real NumbersExamples:Find each of the following products.4 · (–2) · 3 =–24 (–4) · (–5) = 20Multiplying or Dividing Real Numbers•If b is a real number, 0 · b = b · 0 = 0.•The quotient of any real number and 0 is undefined.• The quotient of 0 and any real number = 0.0aa0a  0Working with zero:Sample problem from today’s homework:Answer: 0If this problem had been 7/0, the answer would be “N” (undefined)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 243ExampleEvaluate 43. 43 =4 • 4 • 4 = 64 ExponentsWe may use exponential notation to write products in a more compact form.Evaluate each of the following expressions.34= 3 · 3 · 3 · 3 = 81(–5)2= (– 5)(–5) = 25–62= – (6)(6) = –36(2 · 4)3= (2 · 4)(2 · 4)(2 · 4) = 8 · 8 · 8 = 5123 · 42= 3 · 4 · 4 = 48Evaluating Exponential ExpressionsExample:The Order of OperationsOrder of OperationsSimplify 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 (Parentheses)2. Excuse (Exponents)3. My Dear (Multiply and Divide)4. Aunt Sally (Add and Subtract) … or just remember PEMDAS18Using the Order of OperationsEvaluate:23396 23396 )9(396 9)3(6 991Write 32 as 9. Divide 9 by 3.Add 3 to 6.Divide 9 by 9.Example:More examplesSimplify the following expressions.52226 32226 3246 3223424)58(632216)3(632161831821637367Simplify the following.)3(29)2(386968 314314Example:Sample problem from today’s homework:3247-Evaluating Algebraic Expressions(a) 5x – 2 for x = 8Evaluate each expression for the given value.(b) 3a2 + 2a + 4 for a = – 4 5(8) – 2 = 40 – 2 = 383(– 4)2 + 2(– 4) + 4 = 3(16) + (– 8) + 4 = 44 To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result using the order of