Math M111: Lecture Notes For Chapter 2 Section 2.1: For the following exercises, Solve for x: One solution or Conditional: 24.

-[6x - (4x + 8)] = 9 + (6x + 3)

28.

2[ -(x - 1) + 4] = 5 + [ -(6x - 7) + 9x]

30.

- (-2 + 4x) - (3 - 4x) + 5 = - (-3 + 6x) + x + 1

Special Cases: • Case 1: Fractions (multiply by the least common denominator or LCD): 44.

2x + 5 3x + 1 − x + 7 = + 5 2 2

• Case 2: Decimals : 46.

0.09k + 0.13(k + 300) = 61

• Case 3: No Solution or Contradiction: 56.

-4(x + 2) = -3(x + 5) - x

• Case 4: solution is all real numbers or Identity: 60.

4[6 - (1 + 2m)] +10m = 2(10 - 3m) + 8m

Section 2.2: ab x 1 Ex 2: Solve for b (isolate b) x = y( a − b ) 2

Ex 1: Solve for b (isolate b) y =

Ex 3: Solve for L (isolate L) P= 2L + 2W More examples from the book: 8. Solve d = rt for t 16. Solve S = 2π rh + 2π r2 for h 5 18. Solve C = (F − 32 ) for F 9

The following are applications on motion and percentage problems: 26. In 1975, rain shortened the Indianapolis 500 race to 435 mi. It was won by Bobby Unser, who averaged 149.213 mph. What was his time to the nearest thousands? 38. A mixture of acid and water is 35%. If the mixture contains a total of 40L, how many liters of pure acid are in the mixture? How many liters of pure water? 40. A certificate of deposit for 1 yr pays $221 simple interest on a principal of $3400. what is the interest rate being paid on this deposit? 1

Section 2.3: Word more, increased less, decreased product, twice, times quotient is, are of

Symbol

+ . ÷

= .

Example 3 more than a number a number is increased by 5 3 less than a number a number is decreased by 5 twice a number, 7 times a number quotient of a number and 5 5 is 2 more than a number 20% of a number

Translation x+3 x+5 x–3 x–5 2x, 7x x/5 5= x + 2 0.2x

In each of the following, (a) translate as an expression and (b) translate as an equation or inequality. Use x to represent the number 8. The product of 6 and a number, decreased by 12 10. 12 more than one-half of a number 12. The product of 9 more than a number and 6 less than the number 14. The quotient of 6 and five times a nonzero number Solve the following: 16. The sum of a number and -4 is 12. Find the number. 18. If the quotient of a number and 6 is added to twice the number, the result is 8 less than the number. Find the number 20. When 75% of a number is added to 6, the result is 3 more than the number. Find the number.

The perimeter is the distance around or = 2 length + 2 width or P = 2L + 2W 30. The John Hancock Center (Exercise 29) tapers as it rises. The top floor is rectangular and has perimeter 520 ft. The width of the top floor measures 20 ft more than one-half its length. What are the dimensions of the top floor? 32. The Vietnam Veterans Memorial in Washington, D.C., is in the shape of two sides of an isosceles triangle. If the two walls of equal length were joined by a straight line of 438 ft, the perimeter of the resulting triangle would be 931.5 ft. Find the lengths of the two walls.

34. In a recent year, video rental revenue was $.27 billion more than twice video sales revenue. Together, these revenues amounted to $9.81 billion. What was the revenue from each of these sources? 36. Ted Williams and Rogers Hornsby were two great hitters. Together they got 5584 hits in their careers. Hornsby got 276 more hits than Williams. How many base hits did each get?

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Percentage and Mixture Problems Ex 1: The price of a book was increased by 6%. If the origina l price is $20, find the new price. Ex 2: The price of a book was decreased by 6%. If the original price is $20, find the new price. More examples from the book: 38. In 1998, the number of participants in the ACT exam was 995,000. Earlier, in 1990, a total of 817,000 took the exam. What percent increase was this? 40. In 1990, the average tuition for private 4-year universities in the United States was $10,348 for fulltime students. By 2000, it had risen approximately 86.6%. To the nearest dollar, what was the approximate cost in 2000? 46. Holly Rioux invested some money at 3.5% simple interest, and $5000 more than 3 times this amount at 4%. In one year, she earned $1440 in interest. How much did she invest at each rate? 48. Ron Hampton placed $15,000 in an account paying 6%. How much additional money should he deposit at 4% so that the total return on the two investments is 5.5%? 50. How many liters of a 14% alcohol solution must be mixed with 20 L of a 50% solution to get a 30% solution? 52. How many liters of a 10% alcohol solution must be mixed with 40 L of a 50% solution to get a 40% solution? 54. How much water must be added to 6 gal of a 4% insecticide solution to reduce the concentration to 3%? (Hint: Water is 0% insecticide.) 56. Lee Ann Spahr wants to mix tea worth 2 cents per oz with 100 oz of tea worth 5 cents per oz to make a mixture worth 3 cents per oz. How much 2 cents tea should be used?

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Section 2.4: 10. Nana Nantambu found some coins while looking under her sofa pillows. There were equal numbers of nickels and quarters, and twice as many half-dollars as quarters. If she found $2.60 in all, how many of each denomination of coin did she find? 12. John Joslyn has a jar in his office that contains 39 coins. Some are pennies, and the rest are dimes. If the total value of the coins is $2.64, how many of each denomination does he have? 14. In the nineteenth century, the United States minted two-cent and three-cent pieces. Frances Steib has three times as many three-cent pieces as two-cent pieces, and the face value of these coins is $2.42. How many of each denomination does she have? 16. The Delgado Community College production of The Music Man was a big success. For opening night, 410 tickets were sold. Students paid $3 each, while nonstudents paid $7 each. If a total of $1650 was collected, how many students and how many nonstudents attended? 22. A train leaves Kansas City, Kansas, and travels north at 85 km per hr. Another train leaves at the same time and travels south at 95 km per hr. How long will it take before they are 315 km apart? 24. Lois and Clark are covering separate stories and have to travel in opposite directions. Lois leaves the Daily Planet at 8:00 A.M. and travels at 35 mph. Clark leaves at 8:15 A.M. and travels at 40 mph. At what time will they be 140 rni apart? 26. When Dewayne drives his car to work, the trip takes 30 min. When he rides the bus, it takes 45 min. The average speed of the bus is 12 mph less than his speed when driving. Find the distance he travels to work. 28. On an automobile trip, Aimee Cardella maintained a steady speed for the first two hours. Rush-hour traffic slowed her speed by 25 mph for the last part of the trip. The entire trip, a distance of 125 mi, took 2.5 hr. What was her speed during the first part of the trip?

The sum of all angles in any triangle is 180 30. Find the measure of each angle in the triangle with the following angles: (x + 15), (10x – 20) and (x + 5) 42. Find four consecutive integers such that the sum of the first three is 54 more than the fourth.

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Section 2.5: Solve and graph your solution: 26. 30 . 34.

− 2( x + 4) ≤ 6 x + 16

3 1 (k − 2 ) − (2 k − 7) ≤ 3 5 4 10 x + 2( x − 4 ) < 12 x − 10

Example :

4 < 2x + 6 < 12

46.

− 15 ≤ 3 p + 6 < −12

50.

−3≤

3m + 1 ≤3 4

64. Jacques d’Hemecourt scored 92 and 96 on his first two tests in Methods in teaching Mathematics. What score must he make on his third test to keep an average of 90 or greater?

Section 2.6: A Set is a collection of objects, the objects are the elements. Example:

A = { 1, 2, 3, 4} ;

B = { 3, 4, 5, 6, 7}

• Intersections I of A and B: elements that are in A and B = { 3, 4} • Unions U of A and B: elements that are either in A or in B or both = { 1, 2, 3, 4, 5, 6, 7} Solve and graph your solution: 26.

x < -1 and x > 3

30.

-3x < 3 and x + 2 < 6

32.

7x + 6 < 48 and -4x > -24

44.

x + 1 > 3 or x + 4 < 2

Express in the simplest interval form: 52.

[-9, 1] U (- ∞ , -3)

54.

[-1, 2] U (0, 5)

Decide whether intersection or union should be used. Solve and graph: 60.

-2x – 6 < -18 and 2x > -18

62.

-8x < -24 or -5x > 15

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Section 2.7: • Case 1:

f ( x ) = ....... ; one absolute value. Two cases to try:

- f(x) and + f(x)

Solve: 16.

|2x – 9| = 18 2−

20.

• Case 2:

5 m = 14 2

f ( x ) < ....... ;

f ( x ) > .......

Graphing the solution is important

Solve: 30.

|5 - x | > 3

32.

|-2x - 4| > 5

42.

|4x + 1| < 21

46.

|-2x - 4| < 5

64.

|x + 5| - 2 = 12

66.

|6x - 1| - 2 > 6

68.

|r - 2| - 3 < 4

• Case 3: f ( x ) = g ( x )

two cases to try:

; two absolute values, one in each side

Different signs and same signs

Solve: 72.

|7x + 12 | = |x – 8|

74.

2 1 r− 2 = r +3 3 3

78.

|3x - 1 | = |3x + 9|

• Special Cases: Solve: 84.

|8n + 4 | = -4

86.

|x + 9 | > -3

88.

|4x - 1| < 0

92.

|4x + 1| - 2 < -5 6