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)

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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 solving1) 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 ¾. 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 ¾” translates to 5x - ¾.

Example (cont.)

Translate

The product of

·

twice a number

2x

and 3

3

is the same as

=

5 times the number

5x

and ¾

¾

the difference of

–

Example (cont.)

Solve

2x · 3 = 5x – ¾

6x = 5x – ¾ (simplify left side)

6x + (-5x) = 5x + (-5x) – ¾ (add –5x to both sides)

x = - ¾ (simplify both sides)

Now CHECK your answer:Left side: 2x·3= (2·-3/4)·3 = -6/4·3 = -3/2·3= -9/2Right 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

2(24.95)

mileage costs

0.29x

plus

+

are equal to

= 100

maximum budget

Example (cont.)

Solve

2(24.95) + 0.29x = 100

49.90 + 0.29x = 100 (simplify left side)

0.29x = 50.10 (simplify both sides)

49.90 – 49.90 + 0.29x = 100 – 49.90 (subtract 49.90 from both sides)

29.0

10.50

29.0

29.0

x(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 = 39703This simplifies to 1.06X = 39703Divide 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 LAPTOPSand begin working on the

homework assignment.