Vladimir Luzgin

Math Lessons for Gifted Students

Grade 8

Center Impulse

Week-end and evening classes for gifted students grades 5-9
Canada, ON, L4K 1T7, Vaughan (Toronto),
80 Glen Shields Ave., Unit #10,
Phone (416)826-7270
vluzgin@hotmail.com

---

Content

Click on the lesson!

Lesson 01.

Lesson 02.

Lesson 03.

Lesson 04.

Lesson 05.

Lesson 06.

Lesson 07.

Lesson 08.

Lesson 09.

Lesson 10.

Lesson 11.

Lesson 12.

Lesson 13.

Lesson 14.

Lesson 15.

Appendix 1. COORDINATE GEOMETRY.

Lesson 01

---

1. Solve the following ratio problems.

a) A cask is filled with 45 gallons of wine. Nine gallons are removed, and the cask is refilled with water. Then nine gallons of the mixture are removed and the cask is refilled with water again. Find the ratio of water to wine in the final mixture.

b) The ratio of legs in a right triangle is 5 : 12. Find the hypotenuse if the area of the triangle is 67.5 cm².

c) The legs of a right triangle are in the ratio 8 : 15. If the perimeter of the triangle is 50 cm, find its hypotenuse.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations:

1) 0.8 (9 + 2x) = 0.5 (2 - 3x) aaaaaaaaaaaaaaaaaaaaa 2) 5.4 (3x - 2) - 7.2 (2x - 3) = 1.2

3) - 3.7 (2.5x - 7.6) = - 3.66 + 2.1x aaaaaaaaaaaaaaa 4) 2.5 (0.2 + x) - 0.5 (x - 0.7) - 0.2x = 0.5

5) (2x - 1)(3 - 3x) = 10 + (x + 1)(1 - 6x) aaaaaaaaaaa 6) 5 (x + 1)(x - 1) - 2x = (5x - 3)(x + 1) + 10

4. A circle is inscribed in a given square and another circle is circumscribed about the same square. Find the area of the ring between the circles if the area of the square is 36.

5. Solve the following direct variation problems.

a) The volume of a sphere is directly proportional to the cube of its radius. If a tennis ball with a radius of 3.3 cm has a volume of 150 cm³, find the volume of a basketball, which has a radius of 12.2 cm.

b) For any planet, its year is the time that it takes to circle the sun once. Kepler's Third Law in astronomy states that, for any planet, the square of the number of Earth days in its year varies directly with the cube of its mean distance from the sun. Find the number of Earth days in the years of Venus and Mars if the mean distances from the sun in millions of kilometers for Venus, Earth, and Mars are 108, 149, and 228, respectively.

6. Solve the following word problems:

1) The price of oil is $150/m³ and the price is raised 10% every year. What will be its price at the end of three years?

2) A salesperson receives a salary of $100 a week and a commission of 5% on all sales. What must be the amount of sales for a week in which the salesperson's total weekly income is $360?

7. Solve the following word problems:

a) The sum of Julio and Ramona's ages is 29. Twice Ramona's age is 7 more than Julio's age. Find how old they are.

b) One number is 9 greater than the second, and 1/4 of the first increased by 1/5 of the second is 9. Find the numbers.

c) A father is 3 times as old as his son. Fifteen years from now he will be 9/5 times as old as the son. Find their ages now.

8. A triangle has vertices A(-4, 3), B(2, -5), and C(6,5). Find the lengths of its three medians.

9. Solve the following Number Theory problems.

a) N is a positive six-digit number whose unit's digit is 4. If 4 is moved to the front of the number a new integer is formed which is four times N. Determine N.

b) A four-digit number, which is a perfect square, is created by writing Anne's age in years followed by Tom's age in years. Similarly, in 23 years, their ages in the same order will again form a four-digit perfect square. Determine the present ages of Anne and Tom.

c) Prove that n³ - n is divisible by 6 for all integers n.

10. Simplify.

---

Answers

1. aaa a) 9 : 16. aaaaaa b) 19.5 cm. aaaaaa c) 21.25 cm.
2.

1) 0.5 aaaaaaaaaaaaa 2) - ²/₃ aaaaaaaaaa 3) ¹/₄

4) - ²²⁹/₂₁₀₀ aaaaaaaa 5) ¹/₄ aaaaaaaaaaaa 6) - 0.1

3.

1) - 2 aaaaaaa 2) - 5 ¹/₃ aaaaaaa 3) 2.8 aaaaaaa 4) - ⁷/₃₆ aaaaaaa 5) 1 aaaaaaa 6) -3

4. aaa 9 pi

5. aaa a) aaaaaaaa b)

6. aaa 1) $199.65 aaaaaaaaa 2) $ 5200

7. aaa a) Ramona 12, Julio 17 aaaaaaaa b) 24, 15. aaaaaaaa c) father - 30, som - 10.

9. aaa a) 102 564. aaaaaaaaa b) 15, 21. aaaaaaaaa

10. aa 1) 2b⁵ / (243a⁹) aaaaaaaaa 2) - y / (2x²) aaaaaaaaa 3) (a²c³) / (100b¹⁰) aaaaaaaaa 4) (8a) / (b²)

Return to the content

Lesson 2

---

1. Solve the following ratio problems.

a) The volume of a rectangular box is 7 500 cm³. If its dimensions are in the ratio 3 : 4 : 5, what are they?

b) The ratio of the length to the width to the height of a rectangular solid is 6 : 3 : 2. The surface area of the solid is 2 128 cm². Find the volume of the solid.

c) Find the ratio of the areas of two triangles, if the ratio of their heights is 5 : 3 and the ratio of bases is 3 : 2.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations.

1) ¹/₃ (4 - x) - 5¹/₂ = x aaaaaaaaaaaaaaaaaaaaaaaaa 2) ¹/₃ (x + 1) + ¹/₇ (x - 2) = 1

3) ¹/₅ (x + 5) = 4 - ¹/₃ (x + 1) aaaaaaaaaaaaaaaaaaa 4) ¹/₄ (14 - x) + ¹/₅ (3x + 1) = 3

5) ¹/₆ (5x - 7) - ¹/₇ (x + 2) = 2 aaaaaaaaaaaaaaaaaa 6) ¹/₅ (x - 4) = 9 + ¹/₉ (2x + 4)

4. In the figure below, four semicircles are drawn on the four sides of a rectangle. If the diagonal of the rectangle is 26 cm long, what is the total area of the shaded portion?

5. Solve the following direct variation problems.

a) The period of a pendulum, the time for a complete swing over and back, varies as the square root of its length. If the period is 2 seconds when the length is 1 m, find the period when the length is 4 meters.

b) The distance a body falls from rest in a vacuum varies as the square of the time falls. If it falls 64 feet in 2 seconds, find the distance it falls in 10 seconds.

6. Solve the following word problems:

1) The net price of a certain article is $306 after successive discounts of 15% and 10% off the marked price. What is the marked price?

2) If a merchant makes a profit of 20% based on the selling price of an article, what percent does the merchant make on the cost?

7. Solve the following word problems:

a) Five years ago a father's age was four times that of his son. Fifteen years from now father will be twice as old as the son. Find their ages now.

b) Boris caught a big fish. Its head was 5 inches long. The tail was as long as the head plus half the body. The body was as long as the head plus the tail. How long was the fish?

c) One mouse said to another, "If you give me one piece of cheese, then we will have an equal number of pieces". The other mouse replied, "If you give me one piece, then I will have double the number you have". What is the total number of pieces that the mice have?

8. On a grid, draw the triangle with these vertices and classify it as scalene, isosceles, or equilateral. State whether it is a right triangle. Find the perimeter and the area of the triangle.

a) A(7, -3), B(2, 6), C(-2, 2) aaaaaaaaaaa b) A(0, 3), B(5, 2), C(-1,0)

9. Solve the following Number Theory problems.

a) Show that, if n, m, and k are integer numbers, and n + m + k is divisible by 3, then n³ + m³ + k³ is also divisible by 3.

b) If n is an integer, prove that n³ + 6n² + 11n + 6 is always divisible by 6.

c) Prove that n⁴ + 4 is a composite number for all integers n > 1.

10. Evaluate.

Return to the content

Lesson 3

---

1. Solve the following ratio problems.

a) ABCD is a square. P, Q, and R are points on the sides AB, BC, and CD respectively. Find the ratio of the area of the triangle PQR to the area of the square ABCD if AP : PB = 1 : 3, BQ : QC = 2 : 3, and CR = DR.

b) A rectangle, twice as long as it is wide, is inscribed in a circle. Find the ratio of the areas of the circle and the rectangle.

c) Find the ratio of the surface area to the volume of a sphere of radius 2.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations.

1) ¹/₃ (4x - 8) - ¹/₅ (2x + 3) = 8 aaaaaaaaaaaaaaaaaaa 2) ¹/₄ (x - 2) - ¹/₂ = ¹/₆ (x + 7)

3) 2 (²/₅ x + 1) + 3¹/₃ = 4 - ¹/₂ (⁴/₅ x - 1) aaaaaaaaaaa 4) 3 (2¹/₂ x - 0.2) - 15¹/₁₅ = 6 - (²/₃ - 0.5 x)

4. Write algebraic expressions for the perimeter and area of the following figure:

5. Solve the following direct variation problems.

a) Under certain conditions the minimum distance for stopping a certain car with four-wheel brakes varies as the square of its initial velocity. If the minimum stopping distance for the car is 43 feet when the initial speed is 44 feet per second, find its stopping distance when going 80 feet per second.

b) If a certain sphere weighs 2 052 pounds find the weight of a shell of the same material between two concentric spheres having respective radii twice and three times that of the given sphere.

c) The weight of a shell of material between two concentric spheres of radii 2 feet and 3 feet weighs 1000 pounds. Find the weight of a shell of the same material between concentric spheres of radii 4 feet and 5 feet, respectively.

6. Solve the following word problems:

1) Ted borrowed $15 000.00 from the bank for a new car. What was the amount of interest for one year if the rate of interest was 13%?

2) Tom borrows $1 250 from the bank for 63 days at annum rate of 9%. Calculate the interest.

7. Solve the following word problems:

a) An 18 m log is cut into 2 pieces. The longer piece is 3 m shorter than twice the shorter piece. How long is the shorter piece?

b) Eric and Lois are trading hockey cards. If Eric gives four cards to Lois, they will have the same number. If Lois gives four cards to Eric, he will have twice as many as she. How many cards does each person have to start with?

c) A tennis club charges an annual fee and an hourly fee for court time. One year, Tony played for 39 h and paid $384. Sandra played for 51 h and paid $456. Find the annual fee and the hourly fee.

8. On a grid, draw the triangle with these vertices and classify it as scalene, isosceles, or equilateral. State whether it is a right triangle. Find the perimeter and the area of the triangle.

a) A(2, -3), B(6, 2), C(0, 3) aaaaaaaaaaa b) A(-6, 6), B(-3, -3), C(6, 0)

9. Solve the following Number Theory problems.

a) Prove that n⁴ + 2n³ - n² - 2n is divisible by 24 for all integers n.

b) Prove that n⁵ - 5n³ + 4n is divisible by 120 for all integers n.

c) Show that, if n and m are integer numbers, and n² + m² is divisible by 3, and then n and m are divisible by 3.

10. Graph these equations (use tables of values).

a) y = 2x aaaaaaaaaaaa b) y = 2x - 3 aaaaaaaaaa c) y = -3x + 2

d) y = -x + 4 aaaaaaaaa e) y = -0.5x + 1 aaaaaaa f) y = -1.5x - 2

Return to the content

Lesson 4

---

1. If x : y = 3 : 5, evaluate:

a) (2x + 3y) : (4x - y)

b) (3y² - 2x²) : (y² + 5x²)

c) (x² + 2xy + 3y²) : (2x² - 5xy + 2y²)

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations.

1) ²/₅ (3x - 1) = 4 - ¹/₂ (x + 2) aaaaaaaaaaaaaaaaaaaaa 2) ¹/₆ (8 - x) + ¹/₃ (5 - 4x) = ¹/₂ (x + 6)

3) ¹/₃ (x - 1) + ¹/₁₂ (5x + 2) = ¹/₄ (3x + 5) aaaaaaaaaaa 4) ¹/₃ (4x - 15) - ¹/₄ (17 - 3x) = ¹/₂ (x + 5)

4. ABC is a right-angle triangle with vertex C on the semicircle drawn, with AB as diameter. Semicircles are also constructed, as shown, with diameters BC = 6 cm and AC = 8 cm. Find the area of the shaded region.

5. Solve the following inverse variation problems.

a) Boyle's Law states that if temperature is kept constant, the volume of a gas inversely proportional to the pressure applied to it: V ~ 1/P [constant T]. A tank contains 10 L of hydrogen at a pressure of 500 kPa. If the hydrogen is released into the atmosphere where the pressure is 100 kPa, what volume would it occupy?

b) Boyle's Law states the volume of an enclosed gas at constant temperature varies inversely as the pressure. If the volume of the gas is 5 cubic feet when the pressure is 15 pounds per square inch, find the volume when the pressure is 45 pounds per square inch.

6. Solve the following word problems:

1) Lydia deposits $2 400 at the Credit Union and earns interest at the annual rate of 5.5%. What is the interest after 6 months?

2) Andrew had a student loan of $4 500. The simple interest on the loan was charged at a rate of 6.25%. What is the interest for 1 day?

7. Solve the following word problems:

a) A lifeguard earns an hourly rate for 20 h work in one week and an increased rate for overtime. One week Theresa worked 24 h and received $166.40. The next week she worked 27.5 h and received $200.00. Find her hourly rate and her overtime rate of pay.

b) The ratio of girls to boys in a certain class was 4 : 7. Then 3 boys left the class leaving a ratio of 2 : 3. How many of each was in the class originally?

c) From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. Find the number of girls at the beginning.

8. On a grid, draw the triangle with these vertices and classify it as scalene, isosceles, or equilateral. State whether it is a right triangle. Find the perimeter and the area of the triangle.

a) A(1, -2), B(9, 5), C(-3, 3) aaaaaaaaaaa b) A(-3, -1), B(6, -4), C(6, 2) aaaaaaaaaaa c) A(-2, 4), B(4, 1), C(2, -3)

9. Solve the following Number Theory problems.

a) Place digits in the squares to get the numbers that are perfect squares.

aaa 6  _  6 aaaaaaaaaaaaaaa 4  _  _  1 aaaaaaaaaaaaaaa 8  _  _  9

b) The number 1 is both the square of an integer and the cube of an integer. What are the next three larger integers, which are both a square and a cube of a positive integer?

c) The sum of five consecutive whole numbers is 195. Find the numbers.

10. Graph these equations (use the slope and y-intercept; do not use tables of values).

a) y = 0.5x aaaaaaaaaaaa b) y = ²/₃x - 2 aaaaaaaaaa c) y = ³/₅x + 2

d) y = -2x + 1 aaaaaaaaa e) y = -1.5x + 3 aaaaaaaaa f) y = -³/₄x + 5

Return to the content

Lesson 5

---

1. Solve the following ratio problems.

a) A cone has the same height as a cylinder but twice the diameter. Find the ratio of their volumes.

b) Find the ratio of the surface area of the cone to the surface area of the cylinder with the same base and equal heights if the ratio of the radius to the height is 3 : 4.

c) A closed cylinder, with a height equal to its diameter has the same diameter as a sphere. Find the ratio of (1) their volumes; (2) their surface areas.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations.

1) ¹/₂ (9x -5) - ¹/₃ (3 + 5x) - ¹/₄ (8x - 2) = 2 aaaaaaaaaaaa 2) ⁴/₅ (7 - 8x) + 3 (1 + 4x) = ¹/₂ (12x + 17)

3) 2 (5x - 24) - ¹/₁₁ (16 + x) = ¹/₄ (7x - 2) aaaaaaaaaaaaaa 4) ¹/₂ (4x - 3) - ¹/₃ (5 - 2x) - ¹/₃ (3x - 4) = 5

4. Determine the area of the triangle ABC if:

a) AB = 41 cm, BC = 15 cm, BHa | aAC, BH = 9 cm;

b) AB = 14 cm, BC = 13 cm, AC = 15 cm;

c) AB = 15 cm, BC = 10 cm, AC = 17 cm.

5. Solve the following inverse variation problems.

a) The force of attraction between two electric charges varies inversely as the square of the distance between them. If the force is 0.5 ounce when two electric charges 6 inches apart, find the force when they are 12 inches apart.

b) The electrical resistance of a wire varies directly with its length and inversely as the square of the diameter of its cross section. If 500 m of 3 mm diameter wire have a resistance of 35 ohms, what is the resistance of 12 km of 5 mm wire of the same material?

6. Solve the following word problems:

1) What is the simple interest charged on a loan of $2 500 for 90 days at 11%?

2) The yearly interest paid on a loan of $1 200 is $180. What is the annual rate of interest, expressed as a percent?

7. Solve the following word problems:

a) A big burger, an order of fries, and a soft drink cost $2.90. Two big burgers, an order of fries, and a soft drink cost $4.40. A big burger with a soft drink costs $2.10. If Eddie orders three big burgers, two orders of fries, and a soft drink, determine how much it will cost him.

b) A burger, an order of fries and a soft drink cost $2.80. A burger and an order of fries cost $2.05. Two burgers and an order of fries cost $3.30. How much will a burger, two orders of fries and a drink cost?

c) If three positive integers are added two at a time, the sums are 180, 208, and 222. Find the integers.

8. aa a) Find the coordinates of the points on the x-axis which are 5 units from A(5, 4).

b) Find the coordinates of the point on the y-axis, which is equidistant from the points A(4, 0) and B(2, 6).

9. Solve the following Number Theory problems.

a) If m and n are positive integers such that m² - n² = 29, find the product mn.

b) At present, the sum of the ages of a father and his son is 33 years. Find the smallest number of years until the father's age is four times the son's age.

c) Determine the number of positive integral solutions of the equation a² - 7a + b² - 7b + 2ab = 0.

10. State the x- and y-intercepts of each of the following lines. Graph these lines.

a) 2x - 3y = 6 aaaaaaaaaaaa b) 3x + 4y = - 12 aaaaaaaaaa c) -3x + 5y = 15

Return to the content

Lesson 6

---

1. Solve the following ratio problems.

a) A cylinder, with a height equal to its diameter, is inscribed in a sphere. Find the ratio of: (1) their volumes; (2) their surface areas.

b) A sphere and a cone are each designed to fit snuggly (at separate times) inside a cylinder of radius R and height 2R. Write the ratio of the volume of the sphere to the volume of cone to the volume of the cylinder.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following equations.

1) ¹/₆ (8x + 7) - ¹/₂ (- 2 + 5x) = 3 - ¹/₄ (3 - 2x) aaaaaaaaaaaaa 2) ²/₃ (x - 4) + ¹/₈ (3x + 13) = ³/₅ (2x - 3) - 7

3) ¹/₅ (2x + 3) - [¹/₂ (3 - x) - 7x] = ¹/₃ (7x + 11) + 1 aaaaaaaaa 4) ¹/₂ (6x + 5) - [¹/₂ (2x + 1) + 2x] = ¹/₄ (10x + 3)

5) ⁸/₁₅ (x + 10) - 24 ¹/₂ = ⁷/₁₀ x - ²/₅ (11x - 5)

4. Determine the area of the parallelogramm ABCD (AB || CD, BC || AD) if:

a) AB = 10 cm, AC = 17 cm, BHa | aAD, BH = 8 cm;

b) AB = 14 cm, AD = 18 cm, / BAD = 30^o (Hint: prove that in a right-angled triangle with a 30^o acute angle the side opposite to that angle is half the hypotenuse).

5. The speed of a satellite varies inversely as the square root of its distance from Earth's center. When the space shuttle is in orbit at an altitude of 200 km, its speed is 28 000 km/h. If the radius of Earth is 6 370 km, find

a) the speed of the shuttle at an altitude of 500 km.

b) the distance between the moon and Earth. (384 700 km)

6. Solve the following word problems:

1) In a certain army post, 35% of the enlisted are from New York State, and 24% of these are from New York City. What percentages of the enlistees in the post are from New York City?

2) Calculate the final cost of a new stove: the original selling price was $789, but there was a discount of 15%. Sales tax is 7% and GST is 8%. Why isn't the original selling price equal to the sale price?

7. Solve the following word problems:

a) When each of the three numbers is added to twice the sum of the other two, the results are 64, 62, and 59. Find the numbers.

b) When three numbers are combined in pairs, their sums are 33, 39, and 42. What are the numbers?

c) Divide 120 into three parts so that three-quarters of the greatest exceeds the middle number by 5, and three-quarters of the middle number exceeds the least by 10.

8. Solve the following Coordinate Geometry problems.

a) A(8, -1) is an endpoint and M(5, -5) is the midpoint of a line segment. Locate A and M on a grid and then find the coordinates of the other endpoint of the segment.

b) Two vertices of an isosceles triangle are (-5, 4) and (3, 8). The third vertex is on the x-axis. Find the possible coordinates of the third vertex.

9. Solve the following Number Theory problems.

a) If m and n are positive integers such that m² - n² = 29, find the product mn.

b) At present, the sum of the ages of a father and his son is 33 years. Find the smallest number of years until the father's age is four times the son's age.

c) Determine the number of positive integral solutions of the equation a² - 7a + b² - 7b + 2ab = 0.

10. Newtonbrook Rent-A Car rents luxury cars based on a fixed charge plus a cost per kilometre driven. It costs $93.40 to drive from Toronto to Montreal, a distance of 255 km. It costs $335.0 to drive from Toronto to Thunder By, a distance of 1600 km.

a) Determine the cost per kilometer driven. Show your work.

b) Determine the fixed cost to rent a luxury car. Show your work.

c) State the equation for the rental cost in terms of the distance driven.

Return to the content

Lesson 7

---

1. Solve the following ratio problems.

a) A cone with radius r and height h = 0.75r is inscribed in a sphere. Find the ratio of: (1) their volumes; (2) their surface areas.

b) A sphere is inscribed in a cone with radius r and slant height l = 2r. Find the ratio of: (1) their volumes; (2) their surface areas.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve equations.

4. Deternine the area of the trapezoid ABCD (BC || AD) if

1) AB = 10 cm, BC = 7 cm, CD = 17 cm, AD = 28 cm;

2) AB = 15 cm, BC = 17 cm, CD = 13 cm, BHa | aAD, BH = 12 cm;

3) AB = 25 cm, BC = 18 cm, CD = 25 cm, AD = 32 cm;

5. The volume of a fixed quantity of gas varies directly with the absolute temperature and inversely with the pressure: V ~ T/P [constant m]. 25 L of oxygen at 104 kPa pressure and 5^oC is released into the atmosphere where the pressure and temperature are 100 kPa and 25^oC. What volume does it occupy?

6. Solve the following word problems.

1) Andrew deposited $100 in a saving account at 10% per annum. He withdrew the $100 and the interest earned after 6 month to purchase a gift at $79.95. If the tax on the gift was 6%, how much money did Andrew have left?

2) Sam's age is 125% Of Mary's age. What is the percentage Mary's age of Sam's age?

7. Solve the following word problems.

a) The sum of two numbers is five times their difference. What is the quotient when the larger number is divided by the smaller?

b) If 1 is added to the numerator of a fraction, the result is equivalent to 3/4. If 1 is added to the denominator of the same fraction, the result is equivalent to 2/3. Find the fraction.

c) If 1 is added to the numerator of a fraction and 2 is subtracted from the denominator, the result is equivalent to 2. If 1 is subtracted from the numerator of the same fraction and 2 added to the denominator, the result is equivalent to 1/3. What is the fraction?

8. Solve the following Coordinate Geometry problems.

1) The diagonals of a parallelogram bisect each other. Verify this statement for the parallelogram whose vertices are A(-1, -3), B(4, -2), C(5, 3), and D(0, 2).

2) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. Verify this statement for the quadrilateral whose vertices are A(-2, 4), B(0, 8), C(6, 6), and D(4, -2).

9. Factor by grouping:

1) a³ - 2a² + 2a - 4 aaaaaaaaaaaaaaaaaaaa 2) x³ - 12 + 6x² - 2x

3) c⁴ - 2c² + c³ - 2c aaaaaaaaaaaaaaaaaaa 4) - y⁶ - y⁵ + y⁴ + y³

5) a²b - b²c + a²c - bc² aaaaaaaaaaaaaaaa 6) 2x³ + xy² - 2x²y - y³

7) 16ab² - 21a²b + 2ac² - 5b²c aaaaaaaaa 8) 6a³ - 21a²b + 2ab² - 7b³

9) ma - mb + na - nb + pa - pb aaaaaaaaa 10) ax - bx - cx + ay - by - cy

11) x² + ax² - y - ay + cx² - cy aaaaaaaaa 12) ax² - 2y - bx² + ay + 2x² - by

10. Solve the following problems.

a) Given the lines y = 2x + 4 and y = -x +7. Graph the lines and find the coordinates of their point of intersection. Find the area of the triangle formed by the two lines and the x-axis.

b) The equations of the three sides of a triangle are y = 2x - 4, y = -0.5x + 6, and y = -3x + 6. Graph the lines on the same axes and find the coordinates of the vertices of the triangle.

Return to the content

Lesson 8

---

1. Solve the following problems.

1) If x + 2y = 5, z + 2x = 9, and y + 2z = 10, find the value of x + y + z.

2) If a + b + c = 26, a + b = 15, and b + c = 20, find the value of b.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following inequalities and graph the solution set.

1) 2 (3 - x) - 3 (2 + x) < x aaaaaaaaaaaaaaaaaaaaa 2) 4 (2 - 3x) - (5 - x) > 11 - x

3) 10 (3x - 2) - 3 (5x + 2) + 5 (11 - 4x) > 25 aaaa 4) 25x - 17 < 4 (x - 1) + 2 (17x + 7) - (13x + 1)

5) x - 2 > 4.7 (x + 2) - 2.7 ( x - 1) aaaaaaaaaaaaaa 6) 3.2 (x - 6) - 1.2x < 3 ( x - 8)

4. Solve the following problems:

1) D is a point on the side AC of a triangle ABC such that AD = 5 cm and DC = 7 cm. Determine the area of the triangle ABC if the area of the triangle ABD equals 12.5 cm².

2) If the sides of a triangle have lengths 30 cm, 40 cm, and 50 cm, what is the length of the shortest altitude?

5. A jar contains 4 red candies, 6 orange candies, 3 purple candies, and 7 yellow candies. You select one candy from the jar without looking.

1) What is the probability that you will select an orange candy?

2) What is the probability that you will select either a yellow or red candy?

6. Solve the following word problems.

1) If a worker receives a cut of 25% in salary, what percent increase must he get to regain his original salary?

2) A man has rectangular patio in his garden. He decides to enlarge it by increasing both length and width by 10%. What is the percentage increase in area?

7. Solve the following word problems.

a) A fraction has a value of 1/2. A fraction obtained by subtracting 8 from the denominator and adding 8 to the numerator of the given fraction has value 5/6. Find the fraction.

b) Find two numbers such that three times the second exceeds twice the first by 1, and the ratio of their sum to their difference is 9 : 2.

c) The sum of the digits of a two-digit number is 7. The number formed by reversing the digits is 45 less than the original number. What is the original number?

8. Solve the following Coordinate Geometry problems.

1) One endpoint of a line segment is A(4, 6), and the other is on the x-axis. Find the coordinates of the endpoint on the x-axes for the slope of the segment to be -0.5.

2) Two line segments, with a common endpoint A(0, 4), have slopes 2 and -0.5. If the other endpoints are on the x-axes, find their coordinates.

9. Factor.

1) x² + ax - a²x - axy aaaaaaaaaaaaaaaaaaaaaa 2) a²n + x² - anx - ax

3) 5a³c + 10a² - 6bc - 3abc² aaaaaaaaaaaaaaa 4) 21a + 8xy³ - 24y² - 7axy

5) ac² - ad + c³ - cd - bc² + bd aaaaaaaaaaaaa 6) ax² + ay² - bx² - by² + b - a

7) an² + cn² -ap + ap² - cp + cp² aaaaaaaaaaaa 8) xy² - by² - ax + ab + y² - a

9) x²y + x + xy² + y + 2xy + 2 aaaaaaaaaaaaaa 10) x² - xy + x - xy² + y³ - y²

10. Solve the following problems.

a) The equation of a line is y = 3x + b. Find the value of b if the line passes through the point A(-1, 4).

b) The equation of a line is y = mx + 2. Find the value of m if the line passes through the point B(-2, 6).

Return to the content

Lesson 9

---

1. Solve the following problems.

1) If   ¹/_{(x + 5)} = 4, find the value of   ¹/_{(x + 6)}.

2) Find the value of a + d if a + b = 12, b + c = 15, and c + d = 19.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following inequalities and graph the solution set.

1) ¹/₃ (x - 1) - ¹/₂ (x - 4) < 1 aaaaaaaaaaaaaaaa 2) ¹/₅ (x + 4) - ¹/₄ (x - 1) > 1

3) 2 - ¹/₄ (3x - 7) + ¹/₅ (x + 17) > 0 aaaaaaaaaa 4) ¹/₂ - ³/₄ x < ²/₅ (3 - x)

4. Solve the following problems.

1) A trapezoid ABCD (BC || AD) has three equal sides AB = BC = CD and the base AD is two units less than the sum of these three sides. The distance between the parallel sides is 5 units. Determine the area of the trapezoid, is square units.

2) In a trapezoid ABCD (BC || AD), BC = 6 cm, AD = 18 cm, E is a midpont of CD, and the distance between BC and AD equals 12 cm. Find the area of the triangle ABE.

5. Given two 5's and three 6's, calculate the probability of randomly drawing two numbers from a box to form

1) the two-digit number 55;

2) the number 56.

6. Solve the following word problems.

1) The length of a rectangle is increased by 15% and the width is decreased by 20%. Find the percentage change in the area of the rectangle.

2) If the radius of a circle is decreased by 10%, by what percent is its area decreased?

7. Solve the following word problems.

a) The sum of the digits of a two-digit number is 7. The number formed by reversing the digits is two more than double the original number. Find the original number.

b) The ones digit of a two-digit number is 5 more than the tens digit. The number formed by reversing the digits is eight times the sum of the digits. Find the number.

c) The sum of the digits of a three-digit number is 14. The number obtained from the original by interchanging the last two digits is 9 less than the original. The sum of the first and last digits is the middle digit. Find the number.

8. Solve the following Coordinate Geometry problems.

1) A line segment has length 10, and its endpoints are on the coordinate axes. If the slope of the line segment is ³/₄, find the possible coordinates of its endpoints.

2) The points A(6, 3), B(2, 6), and C(2, 3) are given. Find the coordinates of a point D such that CD is parallel to AB when D is on the y-axis.

9. Factor.

1) x² - 10x + 24 aaaaaaaaaaaaaaaaaaaaaa 2) x² + 8x + 7

3) x² + x - 12 aaaaaaaaaaaaaaaaaaaaaaaa 4) x² - 13x + 40

5) x² + 15x + 54 aaaaaaaaaaaaaaaaaaaaaa 6) x² - 2x - 35

7) 70a - 84b + 20ab - 24b² aaaaaaaaaaaa 8) 21bc² - 6c - 3c² + 42b

9) 30a³ - 18a²b - 72b + 120a aaaaaaaaaa 10) 12y - 9x² + 36 - 3x²y

10. Determine the equation of each line in the the slope and y-intercept form (y = mx + b); then rewrite the equation in standard form (Ax + By + C = 0).

1) The slope is ¹/₂ and y - intercept is - 5.

2) The line passes through the points (-3, -5) and (-19, 3).

3) The slope is 2 and the line passes through the point (-5, 2).

4) The line is parallel to the line y = -5x + 1 and passes through the point (-2, -1).

5) The line is perpendicular to y = -10x + 3 and passes through the point (10,1).

6) The line is perpendicular to y = -10x + 3 and passes through the point (10,1).

7) The line is parallel to the line y = 5x - 1 and its y-intercept is the same as y = 0.5x + 7.

Return to the content

Lesson 10

---

1. Solve the following problems.

1). If a + ¹/_a = 3, determine a² + ¹/_a².

2). The valus of a, b, and c are such that a - b = b - c = 3. Determine the value of a² - 2b² + c².

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following inequalities and graph the solution set.

1) ²/₃ (3x - ¹/₂) + x > 1 - ¹/₂ (¹/₃ - 10x) aaaaaaaaaaaaa 2) 2 (3x - 1) - ¹/₂ (4x + 1) < ²/₃ (x - 3) + ¹/₃

3) (2x - 5)² - 0.5x < (2x - 1)(2x + 1) - 15 aaaaaaaaaaa 4) (12x - 1)(3x + 1) < 1 + (6x + 2)²

4. Find the perimiter and the area of the figure below.

5. A bag contains 2 yellow marbles, 3 red marbles and 5 white marbles.

1) If one marble is selected at random, what is the probability that it is yellow?

2) If two marble are selected at random, what is the probability that both marbles are yellow?

3)If two marble are selected at random, what is the probability that one marble is yellow and the other is red?

6. Solve the following word problems.

1) If each edge of a cube is increased by 150%, what is the percentage increase in the surface area?

2) A 10 kg watermelon is 95% water by mass. It is then partially dehydrated so that it is now 90% water. What is the mass of the dehydrated watermelon?

7. Solve the following word problems.

a) The sum of the digits of a three-digit number is 12. If the units and the tens digits are interchanged the number is increased by 36, and if the units and hundreds digits are interchanged, it is increased by 198. Find the number.

b) In a three-digit number, the hundreds digit is equal to the tens digit, and is 2 more than the ones digit. The number formed by reversing the digits is 19 times the sum of the digits. Find the original number.

c) Denise invested $2000, part at 7% per annum and the rest at 8% per annum. After one year, the interest earned on the 7% investment was $50 more than the interest on the 8% investment. How much did she invest at each rate?

8. Solve the following Coordinate Geometry problems.

1) Determine if the three points are collinear: (a) (8, 5), (3, 2), (-3, -1); aaaaaa (b) (-3, 3), (2, 1), (12, -3).

2) A triangle has vertices A(-2, 3), B(3, -2), and C(4, 6). Determine whether or not it is a right triangle.

9. Expand.

1) (x + y)² aaaaaaaaaaaaaaa 2) (a - b)² aaaaaaaaaaaaaaa 3) (x - 9)² aaaaaaaaaaaaaaa 4) (a + 13)²

5) (7 - y)² aaaaaaaaaaaaaaa 6) (2a + 3b)² aaaaaaaaaaaaa 7) (9n - 8m)² aaaaaaaaaaaa 8) (0.2p - 0.3q)²

9) (0.5x³ + 6x)² aaaaaaaaa 10) (4a³ - 11a²)² aaaaaaaaa 11) (3x³ - 5a⁴)² aaaaaaaaaaa 12) (¹/₃n - ³/₂m)²

10. Solve the following problems.

a) Determine the value of k so that the lines kx + 2y - 3 = 0 and 2x - 3y + 5 = 0 have the same x-intercept.

b) A line passes through A(-4, 5) and B(2, 2.5). Find the length of the segment between the two axes.

Return to the content

Lesson 11

---

1. Solve the following problems.

1) Find the value of A + B, if   ^{(3x + 5)}/_{(x - 3)} = A + ^B/_{(x - 3)}.

2) Find the value of A + B + C, if    ^{(2x2 - 3x - 8)}/_{(x - 3)} = Ax + B + ^C/_{(x - 3)}.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following double inequalities and graph the solution set.

1) - 3 < 2x - 1 < 3 aaaaaaaaaaaaaaaaaaaa 2) 2 < 6 - 2x < 5

3) - 6.5 < ¹/₂ (7x + 6) < 20.5 aaaaaaaaaaa 4) - 1 < ¹/₃ (4 - x) < 5

5) 1.5 > ¹/₂ (1 - 3x) > - 2.5 aaaaaaaaaaaa 6) 0 > ¹/₂ (4x - 1) > - 2

4. A certain triangle has sides that are, respectively, 12 cm, 35 cm, and 37 cm long. An isosceles trapezoid with an area equals to that of the triangle has two bases of 6 cm and 54 cm. What is the perimeter of the trapezoid?

5. Solve the following word problems.

1) An integer is chosen at random from 1 to 60 inclusive. What is the probability the integer chosen contains the digit 4?

2) Using each of the digits 4, 5, 6, 7, and 8 only once, create a three-digit number. What is the probability that it has only even digits?

6. Solve the following word problems.

1) In the first year of the United States Stickball League, the Bombers won 50% of their games. During the second season of the league the Bombers won 65% of their games. If the ratio of the numbers of the games played in the first and in the second seasons is 2:3, what percentage of the games did the Bombers win in the first two years of the league?

2) Energy-conservation experts report that there is a 4.5% reduction in home-heating costs for 1oC reduction in house temperature. A house is kept at a constant 20o C day and night. What percent reduction in heating costs should occur if the house were kept at 18o C from 07:00 to 22:00 and 15o C from 22:00 to 07:00?

7. Solve the following word problems.

a) A candy store merchant sells 25 kg of jujubes for $3.00/kg. If he uses one brand selling for $2.25/kg and a second brand selling at $3.75/kg, how much of each brand is required?

b) A grocer desires 50 pounds of nuts to sell at 55 cents a pound. He mixes nuts selling at 40 and 65 cents per pound. How many pounds of each kind should he use?

c) A vinegar-water solution is used to wash windows. If 1200 L of a 28% solution are required, how much of 16% and 36% solutions should be used?

8. Solve the following Coordinate Geometry problems.

1) The diagonals of a rhombus are perpendicular to each other. Verify this statement for the quadrilateral whose vertices are A(-3, -2), B(-1, 3), C(4, 5), and D(2, 0).

2) A line segment has endpoints A(3, 2), and B(5, 5). C is a point such that line segment AC is perpendicular to AB. Find the coordinates of C if C is on the x-axis.

9. Complete the square.

1) x² + 2xy + y² aaaaaaaaaaaaaaa 2) a² - 2ab + b² aaaaaaaaaaaaaaa 3) 1 - 2z + z² aaaaaaaaaaaaaaaaaaaaa 4) 64 + 16b + b²

5) n² + 4n + 4 aaaaaaaaaaaaaaaa 6) 4x² + 12x + 9 aaaaaaaaaaaaaaa 7) 28xy + 49x² + 4y² aaaaaaaaaaaaaa 8) 0.25x² + 3x + 9

9) ¹/₉x² + ²/₁₅xy + ¹/₂₅y² aaaaaaa 10) ¹/₁₆a² - ab + 4b² aaaaaaaaaaa 11) ¹/₁₆x⁴ + 2x²a + 16a² aaaaaaaaaaa 12) ⁹/₂₅a⁶b² - a⁴b⁴ + ²⁵/₃₆a²b⁶

10. A triangle has vertices A(-5, 2), B(9, 6), C(1, -4). Find the equations of the three medians and the coordinates of its centroid.

Return to the content

Lesson 12

---

1. Solve the following problems.

1) Prove the identity: (a + b + c)² = a² + b² + c² + 2(ab + ac + bc).

2) a, b, and c are real numbers such that a + b + c = 3 and a² + b² + c² = 5. Find the value of ab + ac + bc.

3) a, b, and c are real numbers such that a² + b² + c² = 2. Find the mimimum value of ab + ac + bc.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following systems of inequalities and graph the solution set.

1) 1 - 5x > 11 aaaaaaaaaaaaa and aaaa 6x - 18 > 0

2) 2(x + 1) < 8 - x aaaaaaaaa and aaaa - 5x - 9 < 6

3) 5 (x - 2) - x > 2 aaaaaaaaa and aaaa 1 - 3 (x - 1) < - 2

4) 7x + 3 > 5 (x - 4) + 1 aaaa and aaaa 4x + 1 < 43 - 3 (7 + x)

5. Pierre has two sets of twelve cards each numbered 1 to 12. One set of cards is red and the other is blue. If one card selected from each deck, what is the probability of the two cards summing to 15?

4. Solve the following problems.

1) M and N are points on the sides AB and BC of a triangle ABC such that AM : MB = 2 : 3, AN : NC = 4 : 3. Find the area of the triangle AMN if the area of the triangle ABC is 70 cm².

2) In a triangle ABC, D divides AB in the ratio 1 : 2, and E divides BC in the ratio 3 : 4.If the area of the triangle BDE is 6 cm², find the area of the triangle ABC.

6. Solve the following word problems:

1) A company's profit is 5.4% of its sales. It must pay 48% corporate taxes on its profit. It always pays 60% of its after-taxes profit to its stockholders as dividends, and retains the balance as a reserve. How much did the company retain as a reserve in a year when its sales were $10000000.

2) A fertilizer contains nitrogen, phosphorus and potassium in the ratio 1:2:4 by mass. If 58% of the mass consists of materials other than these nutrients, what percent of the total mass of the fertilizer is phosphorus?

7. Solve the following word problems:

a) Cream 38% butter fat is to be mixed with milk 3% butter fat to obtain 100 gallons of milk of 5% butter fat. How many gallons of each should be mixed?

b) A 21% oil-gas mixture is to be formed by mixing 80% and 20% oil-gas mixture. If 35 L are required, how much of each should be used?

c) Brand A fertilizer is 32% phosphorus, while Brand B is 18% phosphorus. How much of each must be used to produce 56 t of a 24% mixture?

8. Solve the following Coordinate Geometry problems.

1) Right triangle ABC has vertices A(1, 4), B(9, 3), and C on the x-axes. If the side AB is the hypotenuse, find the coordinates of C.

2) The coordinates of the midpoints of the sides of a triangle are given: A(-3, -3), B(-3, 8), and C(5, 5). Find the coordinates of the vertices of the triangle.

9. Factor.

1) x² - y² aaaaaaaaaaaaaaaaaaa 2) 16 - a² aaaaaaaaaaaaaaaaaaaaa 3) ⁹/₁₆ - n² aaaaaaaaaaaaaaaaaaa 4) 81x²y² - 0.36z⁴

5) 1 - (2x - 1)² aaaaaaaaaaaaaa 6) (3x + 2y)² - (2x - 3y)² aaaaaaaa 7) a⁴ - (9b + a²)² aaaaaaaaaaaaa 8) 81a⁴ - 16b⁴

9) (2x + y)² - (2x - y)² aaaaaaa 10) (a + b)² - (b + c)² aaaaaaaaaaa 11) 9a⁶x⁴ - 25b²y⁸ aaaaaaaaaaa 12) 9 - a² + b² - 6b

10. Solve the following problems.

a) The inhabitants of Xenor use two scales for measuring temperature. On the A scale, water freezes at 0^o and boils at 80^o; on the B scale, water freezes at - 20^o and boils at 120^o. What is the equivalent on the A scale of a temperature of 29'aon the B scale?

b) If two poles 20' and 80' high are 100' apart, determine the height of the point of intersection of the lines, which run from the top of each pole to the foot of the other pole.

Return to the content

Lesson 13

---

1. Solve the following problems.

1) If x - y = 5 and x + 3xy + y = - 20, find xy.

2) x and y are real numbers and x² + 3xy + y² = 5. Determine the maximum possible value of xy.

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following systems of inequalities and graph the solution set.

1) 3 (2 - 3x) - 2 (3 - 2x) > x aaaaaaaaaaa and aaaa 6 < x² - x(x - 8)

2) 3.3 - 3 (1.2 - 5x) > 0.6 (10x + 1) aaaa and aaaa 1.6 - 4.5 (4x - 1) < 2x + 26.1

3) 5.8 (1 - x) - 1.8 (6 - x) < 5 aaaaaaaaaa and aaaa 8 - 4 (2 - 5x) > - (5x + 6) < - 2

4) 2.5a - 0.5 (8 - a) < a + 1.6 aaaaaaaaaa and aaaa 1.5 (2a - 1) - 2a < a + 2.9

5. Solve the following problems.

1) If two dice are rolled, what is the probability that the sum of the spots showing is 9?

2) What is the probability of rolling a sum of 11 with two dice? What is the probability of getting any sum other than 11?

4. Nine playing cards from the same deck are placed as shown in the figure below to form a large rectangle of area 180 sq. in. How many inches are there in the perimeter of this large rectangle?

6. Solve the following word problems.

1) How many kilograms of 35% salt solutions and 45% salt solution should be mixed to make 500 kg of 43% solution?

2) How many kilograms of 9% silver alloy and 12% silver alloy are combined to make 500 kg of 10.8% silver alloy?

7. Solve the following word problems:

a) 5 kg of tea and 8 kg of coffee cost $58. If the price of tea increases 15% and that of coffee 10%, the new total is $65.30. What are the new prices for 1 kg of tea and 1 kg of coffee?

b) It is estimated that to complete an excavation would take 2 days using 4 bulldozers and 3 steam shovels, and 3 days using 5 bulldozers and one stream shovel. How many days would one bulldozer and one steam shovel take?

c) The isosceles triangle ABC has the base AB 10 feet long. The rectangle PQRS is inscribed into the triangle ABC, so that Q is on AC and R is on BC, and the base PS is 6 feet long. The sum of the altitudes of the triangle and the rectangle is 14 feet. Find the altitude PQ of the rectangle.

8. Solve the following Coordinate Geometry problems.

1) From the coordinates of the vertices given, determine if the quadrilateral is a rectangle. (a) (-4, 3), (-2, -7), (8, -3), (6, 1); aaaaaa (b) (-2, 6), (-5, -1), (2, -4), (5, 3).

2) Find the slope of OA.

9. Prove the following formulas:

a³ + b³ = (a + b)(a² - ab + b²) [Sum of Qubes] aaaaaaaaaaa a³ - b³ = (a - b)(a² + ab + b²) [Difference of Qubes]
Use these formulas to factor the following algebraic expressions:

1) 8 + a³ aaaaaaaaaaaaaaaaaaa 2) 1 + x³ aaaaaaaaaaaaaaaaaaaaa 3) 27a³ - 8b³ aaaaaaaaaaaaaaaaaaaaa 4) ¹/₈a³ + ²⁷/₁₂₅b³

5) p⁶ + q⁶ aaaaaaaaaaaaaaaaaa 6) p⁶ - q⁶ aaaaaaaaaaaaaaaaaaaaa 7) 1 + a³b³ aaaaaaaaaaaaaaaaaaaaaaa 8) (2x + y)³ - (2x - y)³

9) (a + b)³ + (b + c)³ aaaaaaa 10) (3a + 4b)³ - (3x - 4b)³ aaaaaaa 11) (3a² + b³)³ + (a² - 2b²)³ aaaaaaaa 12) ¹/₂₇(x + y)³ - ⁸/₂₇(x - y)³

10. Solve the following problems.

1) Find the equation of the line with y-intercept 3 that is parallel to the line 2x - y + 7 = 0.

2) Find the equation of the line parallel to the line 4x - 3y + 12 = 0 that passes through (3, -2).

3) Find the equation of the line with y-intercept 3 that is perpendicular to the line 3x + 2y + 8 = 0.

4) Find the equation of the line through (4,1) that is perpendicular to 5x - 3y + 12 = 0.

Return to the content

Lesson 14

---

2. Find the value of the following (do without a calculator and show all your work).

3. Solve the following systems of inequalities and graph the solution set.

1) x (x - 1) - (x² - 10) < 1 - 6x aaaaaaaaaaaaa and aaaa 3.5 - (x - 1.5) < 6 - 4x

2) (3x +2)² > (3x - 1)(3x + 1) - 31 aaaaaaaaa and aaaa (2x - 3)(8x + 5) < (4x - 3)² - 14

3) ¹/₃ (5x + 8) - x > 2x aaaaaaaaaaaaaaaaaaaa and aaaa 1 - ¹/₄ (6 - 15x) > x

4) 2x > 3 - ¹/₁₁ (13x - 2) aaaaaaaaaaaaaaaaaa and aaaa ¹/₆ x + ²/₃ (x - 7) < ¹/₉ (3x - 20)

5. You are rolling two 6-sided dice, but one die is numbered from 3 to 8 and the other die is numbered from 4 to 9. What is the probability of rolling a sum of 14 with these two dice?

6. Solve the following word problems:

1) A chemist mixes hydrochloric acid solution of 30% strength and 40% strength to get 100 kg of hydrochloric acid solution of 34% strength. How many kilograms of each should be used?

2) Ian invested $6 000, part at 7.5% per annum and the remainder at 8.5% per annum. The total interest, after one year, from these investments was $480. How much was invested at each rate?

9. Factor completely.

1) 7x⁴ - 56x aaaaaaaaaaaaaa 2) 6a² + 24b² + 24ab aaaaaaaaaaa 3) x² - y² - 10y - 25 aaaaaaaaaaaaaaaaa 4) a² - b² - a + b

5) ab² - a - b³ + b aaaaaaaaa 6) x³ - 3y² + 3x² - xy² aaaaaaaaaa 7) 49(x - 4)² - 9(x + 2)² aaaaaaaaaaaaaa 8) (x + 1)³ + x³

9) 8x³ + (x - y)³ aaaaaaaaaa 10) 21bc² - 6c - 3c³ + 42b aaaaaa 11) 30a³ - 18a²b - 72b + 120a aaaaaaaa 12) x³ + y³ + 2xy(x + y)

8. The diagram shows a 5 by 13 rectangle divided into five parts. The lower diagram shows the same rectangle with the parts rearranged. Explain why there is an empty space.

10. Graph the triangle ABC with vertices A(-3, 1), B(5, -1), and C(9, 9). Draw:

a) median from A to the midpoint of BC;

b) altitude from C to AB;

c) perpendicular bisector of AC.
Find the equations of the lines drawn.

Return to the content

Lesson 15

---

2. Find the missing value (do without a calculator and show all your work).

3. Solve the following systems of inequalities and graph the solution set.

1) ¹/₃ (3x - 2) + ¹/₆ (12x + 1) > 0 aaaaaaaaaaaaa and aaaa ¹/₇ (14x - 21) + ²/₉ (9x - 6) < 0

2) 3y - ¹/₃ (2y + 1) > 4 - ¹/₃ (2 - y) - y aaaaaaaaa and aaaa ¹/₃ (5y - 1) - (y - 1) > 3y

3) (2y - 1)(3y + 2) - 6y(y - 4) < 48 aaaaaaaaaaaa and aaaa ¹/₈ (y - 1) - ¹/₄ (6y + 1) - 1 < 0

5. Given one each of 3, 4, 5, 6, and 7, you are asked to create a two-digit number. What is the probability that it is odd?

6. Solve the following word problems:

1) A man with $40 000 invests some money at 6%, twice as much at 5%, and the remainder at 4%. His income from the investments is $2 000. Find the amount invested at each rate.

2) A car's cooling systems contains a 25% solution of antifreeze. Half the system is drained and then topped up with pure antifreeze. What is the strength of the antifreeze in the system now?

9. Factor completely.

1) x³ - 2x² + 2x - 1 aaazaaaaaaaaaaaaa 2) 8x³ + 6x² + 3x + 1 aaaaaaaaaaaaaaa 3) a³ - b³ + 5a²b - 5ab² aaaaaaaaaaaaa 4) a³ - b³ + 3a² + 3ab + 3b²

5) a⁴ + ab³ - a³b - b⁴ aaaaaaaaaaaaaaa 6) (x² + x + 2)(x² + x + 1) - 12 aaaaaaa 7) x⁴ + x² + 1 aaaaaaaaaaaaaaaaaaaaaa 8) x⁵ + x + 1

9) (xy + xz + yz)(x + y + z) - xyz aaaaa 10) (x + y + z)³ - x³ - y³ - z³ aaaaaaaaa 11) x³(y - z) + y³(z - x) + z³(x - y) aaaaa 12) x³ + y³ + z³ - 3xyz

8. Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to one half of it.

10. A(-6, -4), B(6, 2), C(-2 ,8) are the vertices of a triangle. Find:

a) equation of the three sides;

b) equation of the perpendicular bisector of each side;

c) point of intersection, P, of the perpendicular bisectors, and the length of the segments PA, PB, and PC.

Return to the content

Appendix 1. COORDINATE GEOMETRY

---

1. THE LENGTH AND THE MIDPOINT OF A LINE SEGMENT

The Cartesian coordinate system is useful for describing the positions of points in the plane. In this system, the distance between any two points A(x₁, y₁) and B(x₂, y₂) is easily found. In the diagram, the triangle ABC is a right triangle.

The coordinates of C are (x₂, y₁). Therefore, the lengths of the sides AC and BC are AC = |x₂ - x₁| and BC = |y₂ - y₁|.

By the Pythagorean Theorem:

AB² = AC² +BC² = (x₂ - x₁)² + (y₂ - y₁)² AB = SQUARE ROOT [AC² + BC²]

The distance between any two points A(x₁, y₁) and B(x₁, y₁) is given by the formula: AB = SQUARE ROOT [(x₂ - x₁)² + (y₂ - y₁)²] It is possible to calculate the coordinates of the midpoint of a line segment when the coordinates of the endpoints are known. If M is the midpoint of a line segment having endpoints A(x₁, y₁) and B(x₂, y₂), then the coordinates of M are: x = (x₁ +x₂)/2, y = (y₁+ y₂)/2.

Return to the content