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 • Understand the problem • Read and reread the problem • Choose a variable to represent the unknown • Construct a drawing, whenever possible • Translate the problem into an equation • Solve the equation • Interpret the result • Check solution • State your conclusion
Example 1: Understand The product of twice a number and three is the same as the difference of five times the number and ¾. Find the number. 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 ¾” translates to 5x - ¾.
Example (cont.) Translate The product of the difference of is the same as twice a number 5 times the number and 3 and ¾ 2x · 3 = 5x – ¾
Example (cont.) Solve 6x + (-5x) = 5x + (-5x) – ¾(add –5x to both sides) 2x· 3 = 5x – ¾ 6x = 5x – ¾(simplify left side) x = - ¾(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.)
Example 2: Understand 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? 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 maximum budget are equal to mileage costs plus 2(24.95) + 0.29x = 100
Example (cont.) Solve 49.90 – 49.90 + 0.29x = 100 – 49.90 (subtract 49.90 from both sides) (divide both sides by 0.29) 2(24.95) + 0.29x= 100 49.90 + 0.29x = 100(simplify left side) 0.29x = 50.10(simplify both sides) 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 +/- % ofold 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 + % ofold 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 checkyour 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.)
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.