Any questions on the Section 2 1 homework that was due today Comment on the 2 1 homework Types of outcomes when solving linear equations in one variable 1 One solution nonzero most problems in 2 1 Example 2x 4 4 x 3 Solution x 4 2 One solution zero as in problems 6 20 Example 2x 4 4 x 1 Solution x 0 3 Solution All real numbers as in problems 15 17 21 Example 2x 4 2 x 2 Solution All real numbers R on computer 4 No solutions as in problems 16 18 23 Example 2x 4 2 x 3 Solution No solution N on computer Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 2 2 An Introduction to Problem Solving Reminder This homework assignment on section 2 2 is due at the start of next class period Make sure you turn in the worksheet showing all your work for problems 5 19 of this assignment If you don t turn this in or if you don t completely show your work on any problem s your online score will be reduced for those 15 problems out of the 24 total problems in the online assignment Section 2 2 General strategy for problem solving 1 Understand the problem Read and reread the problem Choose a variable to represent the unknown Construct a drawing whenever possible 2 Translate the problem into an equation 3 Solve the equation 4 Interpret the result Check solution State your conclusion Example 1 The product of twice a number and three is the same as the difference of five times the number and Find the number Understand Read and reread the problem If we let x the unknown number then twice a number translates to 2x the product of twice a number and three translates to 2x 3 five times the number translates to 5x and the difference of five times the number and translates to 5x Example cont Translate The product of twice a and 3 number 2x 3 is the same as the difference of 5 times the number 5x and Example cont Solve 2x 3 5x 6x 5x simplify left side 6x 5x 5x 5x add 5x to both sides x simplify both sides Now CHECK your answer Left side 2x 3 2 3 4 3 6 4 3 3 2 3 9 2 Right side 5x 3 4 5 3 4 3 4 15 4 3 4 18 4 9 2 You can perform this check quickly by using your calculator Sample problem from today s homework Answer 6 Example 2 A car rental agency advertised renting a Buick Century for 24 95 per day and 0 29 per mile If you rent this car for 2 days how many whole miles can you drive on a 100 budget Understand Read and reread the problem If we let x the number of whole miles driven then 0 29x the cost for mileage driven Example cont Translate Daily costs mileage costs plus 2 24 95 maximum budget are equal to 0 29x 100 Example cont Solve 2 24 95 0 29x 100 49 90 0 29x 100 simplify left side 49 90 49 90 0 29x 100 49 90 0 29x 50 10 0 29 x 50 10 0 29 0 29 subtract 49 90 from both sides simplify both sides divide both sides by 0 29 x 172 75 simplify both sides Example cont Interpret Check Recall that the original statement of the problem asked for a whole number of miles If we replace number of miles in the problem with 173 then 49 90 0 29 173 100 07 which is over our budget However 49 90 0 29 172 99 78 which is within the budget State The maximum number of whole number miles is 172 MATH 110 Section 2 2 Homework Problem Tip If you re having trouble doing percent problems that give you a new value after a certain percent increase or decrease from an old value such as sales tax problems try thinking about it this way Think about when you go shopping to buy say a TV Usually you know how much the TV costs for example 400 and the percent tax rate for example 5 5 Normally what you do or the salesclerk s computer does is calculate the TOTAL COST by taking 5 5 of 400 then adding that amount back onto the 400 price of the TV to get the total cost to you The working equation is PRICE TAX TOTAL COST In words here s what you did after writing the 5 5 as a decimal 0 055 PRICE 055 times PRICE TOTAL COST Plugging in the numbers we get 400 055 x 400 400 22 422 Notice that you ve multiplied the OLD VALUE the price before tax by the 055 The same basic format applies to anything with a percent increase or decrease from an original amount Old amount of old amount new amount Remember to write the percent as a decimal This equation works for raises in pay population increases or decreases and many other percent change problems especially where you re given the new amount and the percent change and you need to work backwards to find out the old amount Example After a 6 pay raise Nora s 2005 salary is 39 703 What was her salary in 2004 Round to the nearest dollar Solution Recall the equation Old amount of old amount new amount The old amount is her 2004 salary which is unknown so we ll call it X This gives us the equation X 0 06X 39703 Example cont After a 6 pay raise Nora s 2005 salary is 39 703 What was her salary in 2004 Round to the nearest dollar X 0 06X 39703 This simplifies to 1 06X 39703 Divide both sides X 39703 by 1 06 to get X 1 06 Answer Her 2004 salary was 37 456 Now check your answer 37456 06 x 37456 37456 2247 39703 NOTE that this DOES NOT give you the same answer as if you subtracted 6 of the new salary 39703 from the new salary Try it and you ll see that it doesn t work It s not real far off but enough to give you the wrong answer and the bigger the percentage the farther off you ll be Sample problem from today s homework Answer 30 23 Reminder This homework assignment on section 2 2 is due at the start of next class period Make sure you turn in the worksheet showing all your work If you don t turn this in your online score will be reduced If you don t completely show your work on any problem s your online score will be reduced for those problems You may now OPEN your LAPTOPS and begin working on the homework assignment