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Equations and Inequalities 1

Prerequisite Problems

1. Simplify the following fractions :

(a). [pic] (b). [pic]

2. (a). 7.24 + 0.04 (b). [pic]

3. Calculate the differences between the following numbers :

(a). −7 − (−10) = (b). 25 – 6.28 =

4. Calculate the products of the following numbers :

(a). (−1.2) × 2.32 = (b). [pic]

5. Calculate the quotients of the following numbers :

(a). 1.86 ÷ 2.3 = (b). [pic]

6. Calculate the results of the following mathematical operations by applying the distributive property.

(a). – 6 (4 + (– 3)) = (b). [pic]

7. Compare the following numbers, using the symbols , or = .

(a). – 4 – 2 (b). 1.4 0.5 (c). [pic] [pic]

8. Draw two number lines and graph the following numbers on them. N.B. The distance between consecutive integers must be 1 cm.

(a). 3 ; – 2 ; [pic] ; [pic]

(b). – 1; 0.4 ; 3.2 ; 2

9. Translate the following statements into ratios.

(a). Sixteen of the thirty-three students in Joan’s class are of Canadian origin.

(b). Paul answered eight questions out of ten correctly on his mathematics exam.

10. A runner went around a track four times : the first time in 3.82 minutes, the second time in

3.5 minutes, the third time in 3.94 minutes and the fourth time in 3.39 minutes. How much time on average did he take to go once around the track? Describe the steps in the solution and give the answer.

(1). Identify what is required.

(2). Translate the statement into mathematical language.

(3). Estimate the result.

(4). Solve the problem.

(5). Verify the result.

1.2 Practice Exercises

Simplify the following algebraic expressions.

1. 8y + 3 – 5y =

2. 9z – 4 + 3z – 7 =

3. – 2z + 6 + 2 – 5z =

4. – 12b + 23b – 9 =

5. [pic]

6. 0.5d – 0.9d – 0.2d =

7. – 0.8a – 2.1 – 3a + 6 =

8. 18y – 26y – 31y =

9. 42 – 40z – 31z +7 =

10. 0.89x + 0.92x + 8 + 18 =

11. 18xy + 22x – 8xy – 6x =

12. – 7z – 22yz + 18yz – 9z =

13. 8bc – 12cb + 7c – 9 – 10 =

14. 0.3d +0.7d – 0.5 b – 1.2 b =

15. [pic]

16. – 3.2xy – 4.8x + 5xy – 8 =

17. 8z – 6y – 3z – 2z + 19y =

18. – 29bc + 13c – 8b – 12bc – 5c =

19. – 18xy – 9y – 2 + 22xy – 7y =

20. 4.5ab + 8 + 6.2ab – 7 + c =

21. – xy + 8 + xy – 2x =

2.2 Practice Exercises

Perform the following multiplications by applying the distributive property of multiplication over addition.

1. 5(6x + 8) =

2. – 3(– 2y + 12) =

3. – 2(4z + 6) =

4. 5(– 6y – 2) =

5. – 4(– 8x + 10) =

6. – (2x – 3y + 8) =

7. 5(– 3 + 3z) =

8. – 6(6x – 9) =

9. – (3z – 6) =

10. [pic]

11. [pic]

12. [pic]

13. 0.5(– 7x + 2) =

14. − (− 0.2z – 6) =

15. − 0.3(− 0.7z +5) =

16. 1.6 + (6z – 3.2) =

17. 0.2(4x + 3y – 6) =

18. − 3(− 0.4z – 9) =

19. − 0.6(1.2y + 0.6) =

20. − (−0.5b + 3a – 4) =

3.2 Practice Exercises

1. Given the universe N, solve each of the following equations. Verify each solution and graph it on a number line.

(a). x + 7 = 13

(b). 3x – 5 = 22

(c). − x + 5 = 2

(d). 2x – 8 = x + 12

(e). 7y – 2y = 20 – 5

2. Given the universe Z, solve the following equations and verify the solution obtained. Then graph the solution on a number line.

(a). −4x = 16

(b). −3y – 5 = 13

(c). 4x – 6x = −18 + 20

(d). 3y – 15 = 4y + 5

(e). 8x – 6x + 3 = −5 − 2x

3. Given the universe Q, solve each of the following equations. In addition, for items (a) to (e) , verify the solution obtained and graph this solution on a number line.

(a). −4x − 1 = 16

(b). [pic]

(c). [pic]

(d). 0.8x – 2 = 4.4

(e). 0.2 – 3x = 5.2

(f). 3(2x + 5) = 12

(g). 8x + 2(−3x – 4) = −6

(h). 6y – 12 = 9 – (4y + 2)

(i). 2.4z – 3.2z = 16

(j). 28 – 2(5y – 6) = 18 – 4y

(k). [pic]

(l). 5(x + 3) = 6(x + 2)

(m). 6(y + 1) + 8(y – 2) = 3

(n). 0.8 – (0.4x – 0.2) = 0.6x – 1.4

(o). 1.25x – 2(−2.5x + 3) = 9

4.2 Practice Exercises

1. Given the universe N, solve the following inequalities. Verify each solution and graph it on a number line.

(a). 4x – 3 ≤ 5

(b). 3x + 1 ≥ 2x + 7

(c). 7y + 2 > 5y + 12

(d). −18 + 7z < 32 – 3z

(e). 4(2x – 3) ≤ 3x + 8

2. Given the universe Z, solve the following inequalities. Verify each solution and graph it on a number line.

(a). −5x +2 ≤ 17

(b). −6x + 4x ≤ 3x + 20

(c). 4y + 9 > −5y – 18

(d). 6x + 3 ≥ 5x – 1

(e). 6 – (4y + 3) < 6y + 13

3. Given the universe R, solve the following inequalities and graph the solutions on a number line.

(a). 18 + x ≤ 2x + 5

(b). 11– 6x < 24 + 6x

(c). [pic]

(d). 3(z – 9) > 2z + 5z + 24

(e). 0.8 – (6x + 1.2) ≤ 2x – 2.25

(f). 7y + 3y – 6 < 9

(g). 3x – 5 > 7x + 6

(h). 11x – 1 > 8x + 8

(i). 2.25x – 3.75x ≥ 2

(j). 7x ≤ 2(x – 4)

Unit 5 Ratio and Proportion Problems

1. To bake 24 cookies, Ms Crocker usually uses 500 mL of flour, 375 mL of brown sugar, 125 mL of shortening and other ingredients. To bake 72 cookies, what quantity of flour, brown sugar and vegetable shortening must she use? What should she do to modify her recipe?

2. Michele is doing her income tax return. She notes that $5000 was withheld at source for income tax. Her salary was $20,000. Her friend, Ginette, earned a salary of $25,000. At the same tax rate, what is the income tax Ginette should expect to pay?

3. Louise pays $1,200 in taxes on a property evaluated at $60,000. What amount of tax should a cottage owner pay whose property is evaluated at $50,000?

4. Mark borrows a certain sum of money on which he must pay $135 interest over 12 months, plus capital. How much interest will he have to pay if he manages to reimburse the total capital within

9 months.

5. Louis travels 164 km by car in 2 hours. How much time will he take to travel the remaining 410 km to reach his destination if he continues his trip at the same speed?

6. This week, 3 cans of tomato soup are selling for $1.92. How much will 5 cans of tomato soup cost?

Unit 6 Formulas Problems

1. [pic] is the formula used to calculate the normal blood pressure of a person according to his or her age. Using this formula, calculate the normal blood pressure of (a) a person who is 20 years old (b) a person who is 40 years old and (c) a person who is 60 years old. What is the relation between normal blood pressure and age?

2. Given the formula for calculating the perimeter of a rectangle to be P = 2(L+W) where P = the perimeter, L = the length and W = the width, calculate the length of a triangle whose perimeter is 64 cm and whose width is 12 cm.

3. Given the formula for calculating the area of a trapezoid to be [pic] where B = 0.70 cm,

b = 0.50 cm and A = 24 cm2, find the value of the height and verify your solution.

4. A dishwasher is connected to a 220 volt (V) circuit. What will be the intensity of the current ( I ) in amperes if the resistance ( R ) of the dishwasher is 70 ohms ? Use the formula V = RI. Verify your solution.

5. Using a calculator where necessary, determine the following :

(a). 252 (b). 72 (c). 112 (d). 132 (e) 10002 (f). 43.32

(g). [pic] (h). [pic] (i). [pic] (j). [pic]

(k). [pic] (l). [pic] (m). [pic] (n). [pic]

6. Calculate the numerical value of R in the formula P = I2 R , where P = 12 and I = 2.

7. Calculate the volume (V) of a cube whose side (s) measures 5 cm , given that V = s 3.

8. Knowing that V = π r 2 h , find the volume of a cylinder whose height (h) and radius r measure 6 cm and 2 cm respectively.

9. Find the area (A) of a circle whose radius (r) measures 3 cm , given that the area of a circle

A = π r 2 (π = 3.1416).

10. Calculate the height (h) of a cylinder whose radius (r) measures 5 cm and whose volume (V) is 785 cm2 . Use the formula V = π r 2 h with π = 3.14.

Unit 7 Problems involving equations in one variable

1. Two children are given 60 marbles. One of the children receives twice as many marbles as the other. What is each child’s share?

2. Jake is 30 years old. How old will he be in x years ?

3. Twice a number decreased by 8 gives 30. What is this number?

4. In 6 years, Michelle will be twice her present age. What is her present age?

5. Three times a number minus 4 is equal to 20. What is the number?

6. Twice a number plus 8 is equal to 3 times this number minus 29. What is the number?

7. 5 years ago, Paul was one third as old as he will be in 5 years time. How old is he now?

8. A garage mechanic buys a lathe, a drill, and a grinder for his workshop at a total cost of $6,400. The drill is worth 5 times the price of the grinder and the price of the lathe is double that of the drill. Calculate the respective prices of the lathe, the drill, and the grinder.

9. Lisa calculates the calories in the food that makes up the main course of her lunch. The main course consists of ground-beef patty (260 calories) and a cup of broccoli (45 calories). She wants her main course to contain 500 calories. If one fried potato contains 15 calories, how many fries could she eat? Verify your solution.

10. Determine two consecutive odd numbers whose sum is equal to 68.

11. The sum of the measures of the three angles of a triangle is equal to 180°. In a given triangle, the second angle measures 20 degrees more than the first, and the third angle measures 35 degrees more than the second. What is the measure of each of these angles?

5.2 Practice Exercises

Calculate the value of the variable in each of the following proportions. Verify your solutions.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. An automobile travels 200 km on 16 litres of gasoline. How many litres will it consume to travel 550 km.

21. You draw a scale diagram in which 1 cm represents 20 m. On this diagram, what length will you allow for a lane which measures 250 m?

22. Water is composed of hydrogen and oxygen in the proportion of 2 to 1. How many parts oxygen are there in a quantity of water which contains 260 parts hydrogen?

23. Sonia uses 4 litres of paint to cover 26 square metres of wall. She has 6 litres of paint left. How many square metres can she cover with the remaining paint?

24. The ratio of the number of teeth in each of two gears is 8 : 5. The bigger gear has 96 teeth. What is the number of teeth in the smaller gear?

25. A 250 g container of yogurt sells for $0.89. How much should a 1 kg container sell for ?

(1 kg = 1000 g)

26. A block of copper has a 20 cm3 volume. Its mass is 176 grams ; what will be the volume of an

800 gram block?

27. A bank charges a customer who borrows $500, $45 interest per year. How much interest will it charge on an $800 loan for the same period?

6.2 Practice Exercises

Calculate the value of the unknown variable in each of the following formulas and verify your solution. Round your answer to the nearest tenth if necessary.

1. [pic] where V = 266 and B == 19. Calculate the value of h.

2. [pic] where W = 45 and l = 4. Calculate the value of k.

3. [pic] where K = 30 , m = 202 and L = 101. Calculate the value of R.

4. [pic] where m = 4 , v = 30 and r = 8. Calculate the value of F.

5. [pic] where R = 5. Calculate the value of V . [pic]

6. [pic] where T = 875 and r = 5. Calculate the value of a.

7. [pic] where R = 6 and I = 2.5. Calculate the value of P.

8. [pic] where V = 141.3 , r = 6 and h = 4. Calculate the value of b. [pic]

9. [pic] where a = 3 and b = 4. Calculate the value of C.

10. [pic] where S = 222 , l = 35 and a = 2. Calculate the value of n .

11. [pic] where g = 9.8 and t = 0.6. Calculate the value of d .

12. [pic] where P = 150. Calculate the value of a .

13. What will be the height (h) of a wall that is 40 cm thick ( T ) , given that [pic] ?

14. Determine the height (h) of a triangle if the base ( b ) is 8 cm and the area ( A ) is 40 cm2. Use the formula[pic].

15. What is the intensity ( I ) of a current in amperes used by a toaster whose resistance ( R ) is

22 ohms, when it is connected to a 120 volt ( V ) circuit ? Use the formula V = IR .

16. The area ( A ) of a trapezoid is 54 cm2 , the longer base ( B ) measures 11 cm and the height ( h ) is 6 cm. What is the length of the shorter base (b) ? Use the formula[pic].

17. Determine the volume (V) of a cube with 9 cm sides ( s ). Use the formula [pic].

18. An aeroplane travels 1,800 kilometres (d) by flying at a velocity (v ) of 600 km / h. How much time (t) does it take to cover this distance? Use the formula [pic].

19. What is the coefficient of resistance ( r ) in a metal cylinder 2 cm thick ( T ), where the pressure

( p) is 100 kg per cm2 and the interior diameter ( d ) is 20 cm. Use the formula [pic].

20. Calculate the volume ( V ) of a sphere whose radius ( r ) measures 9 cm. Use the formula [pic]

21. Calculate the height ( h ) of a cylinder whose radius ( r ) measures 20 cm and whose area ( A ) is 7540 cm2. Use the formula [pic]

22. The intensity (I ) in amperes of an electric current in a circuit produced by a series of batteries can be calculated by the formula [pic]. Find this intensity when V = 2 volts , R = 10 ohms ,

r = 2 ohms, n = 3. ( V is the voltage , R the resistance in the circuit , r the internal resistance in a battery and n the number of batteries)

23. Sebastian’s house, evaluated ( v ) at $72,000 , was damaged by fire. His home is insured ( I ) for $54,000 through an insurance policy containing an 80% reimbursement clause. The losses ( L ) are evaluated at $8,500. How much will the insurance company pay ( P ) Sebastian ? Use the formula [pic].

7.2 Practice Exercises

Solve the following problems by completing all the steps for solving problems which contain an equation in one variable.

1. What is the number whose triple plus 5 is equal to its double plus 12?

2. In 6 years, Nina’s age will be double what it was 6 years ago. Find Nina’s present age.

3. In the same amount of time, an electrician wired 15 more solenoids than an apprentice; together they wired 61. How many did the apprentice rewire?

4. A mother is 40 years old and her daughter is 10 years old. In how many years will the mother’s age be three times that of her daughter?

5. Patrick walked a distance of 6 kilometres; he then travelled a certain distance by train and in the last lap, covered twice this distance by car. If the total length of his trip is 126 kilometres, how far did he go by train?

6. Adding 40 to a number or multiplying it by 5 gives the same result. What is this number?

7. Find a number that the sum that the sum of one-sixth and one-ninth of this number is equal to 5.

8. A rowboat is tied to a post; one fourth of the post is buried in the ground, one third is submerged under water and the part which is exposed measures 5 metres. How many metres long is the post?

9. In 5 years, triple Martina’s age will be 75. What is Martina’s present age?

10. Determine three consecutive numbers whose sum is 102.

11. A language school has 500 students. The number of students studying French is double that studying Spanish; the latter accounts for three times the number of students in the German course. How many students are taking each of the courses?

12. There are 520 people in a concert hall. The number of women is 6 times that of the men, and the number of children is three times that of the men. How many spectators are there in each category?

13. Arthur sells a horse for $280 more than half of what it cost him. He thereby makes an $88 profit. How much did he pay for the horse?

14. Marianna gives apples to her children. If she gives 5 apples to each child, she will have 4 left, but if she gives 8 apples to each child, she will be 8 short. How many children does she have?

15. Marilyn has $2.25 in change. The number of dimes ($0.10) is double that of the quarters ($0.25), and the number of nickels ($0.05) is triple the number of dimes. How many nickels, dimes and quarters does she have?

Equations and Inequalities 1 Self-Evaluation Test Name:………………………………..

1. Given the universe Z, solve the equation 3(2y + 8) – 5 = 3y – 2 . (The detailed solution and the verification are required)

2. Given the universe R , solve the inequality 6 – (2x – 4) < x + 8 and graph your solution on a number line. (The detailed solution and the verification are required)

3. Given the universe Q , solve the equation [pic]. (The detailed solution and the verification are required)

4. Calculate the height ( h ) of a cone whose radius ( r ) is 5 cm and whose volume ( V ) is 210 cm3. Use the formula [pic] where π = 3.1416. (The detailed solution and the verification are required)

5. Louis travelled 180 kilometres by bicycle in 3 days. On the second day of his trip, he covered

9 km more than on the first day and on the third day, he covered 18 km more than on the second. How far did he travel each day? (The steps in the solution and the verification are required)

Solve the following problems by completing all the steps for solving problems which contain an equation in one variable.

1. What is the number whose triple plus 5 is equal to its double plus 12?

2. In 6 years, Nina’s age will be double what it was 6 years ago. Find Nina’s present age.

3. In the same amount of time, an electrician wired 15 more solenoids than an apprentice; together they wired 61. How many did the apprentice rewire?

4. A mother is 40 years old and her daughter is 10 years old. In how many years will the mother’s age be three times that of her daughter?

5. Patrick walked a distance of 6 kilometres; he then travelled a certain distance by train and in the last lap, covered twice this distance by car. If the total length of his trip is 126 kilometres, how far did he go by train ?

6. Adding 40 to a number or multiplying it by 5 gives the same result. What is this number?

7. Find a number that the sum that the sum of one-sixth and one-ninth of this number is equal to 5.

8. A rowboat is tied to a post; one fourth of the post is buried in the ground, one third is submerged under water and the part which is exposed measures 5 metres. How many metres long is the post?

9. In 5 years, triple Martina’s age will be 75. What is Martina’s present age?

10. Determine three consecutive numbers whose sum is 102.

11. A language school has 500 students. The number of students studying French is double that studying Spanish; the latter accounts for three times the number of students in the German course. How many students are taking each of the courses?

12. There are 520 people in a concert hall. The number of women is 6 times that of the men, and the number of children is three times that of the men. How many spectators are there in each category?

13. Arthur sells a horse for $280 more than half of what it cost him. He thereby makes an $88 profit. How much did he pay for the horse?

14. Marianna gives apples to her children. If she gives 5 apples to each child, she will have 4 left, but if she gives 8 apples to each child, she will be 8 short. How many children does she have?

15. Marilyn has $2.25 in change. The number of dimes ($0.10) is double that of the quarters ($0.25), and the number of nickels ($0.05) is triple the number of dimes. How many nickels, dimes and quarters does she have?

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