The Cartesian Coordinate System

Plot the following points:

1. (-2,2) 6. (3,-5)

2. (0, 4) 7. (1, 0)

3. (-5,-3) 8. (0.0)

4. (2,-3) 9. (0,-8)

5. (1.5,3.5) 10. (-2.5,2 2/3)

Equation of a Line

Exercise: Convert the following equation of a line into standard form:

1. –3x – y + 4 = 0

2. y – 3x = 6

3. 0 = 3x + 2y

4. –3x + 3/2y = 1

5. 4x + 2y –5 =0

6. y =3x + 2

Slope

Exercise:

Find the slopes of the following lines formed by two given points:

a. (1,0) , (2,0)

b. (3,4) , (5,-3)

c. (-4,-2) , (2,4)

d. (5,1) , (-1,-5)

e. (2,1) , (1,4)

Solving the Equation of the Line Base on Slopes and y-intercept

Exercise: Draw the graph of a the following line by basing on the slope and y-

intercept.

1. y = 3x + 8

2. 3x + 2y = 18

3. 3x - 4y = 24

4. x + y = 8

5. x – 2y = 10

Find the equation of a line in standard form by basing on the slopes and y-intercepts.

1. m = 2/3 , b = 6

2. m = -2, b = -2

3. m = -1/2, b = 3

4. m = 2, b = 6

5. m = .8, b = 1/6

Exercise: Find the equations of the following lines with the given slope and passed

through the given points.

1. m = ½, (2,3)

2. m = -2, (1,-4)

3. m = 2/3, (3,6)

4. m = -4, (-1/2, -3/4)

5. m = - 2, (3/4, 0)

Parallel and Perpendicular Lines

Exercise: Predict which pair of the following pairs of lines is parallel.

1. 2x + 4y = 7, 2x = -4y + 11

2. 3x - 6y = 8, 3x = -6y + 14

3. 2x + 4y = 11, 4x + 8y = 19

4. 3x + 7y = 12, 9x + 21y = 2

5. 4x + 3y = 12, 4x = -3y + 15

Exercise: Which of the following lines are parallel.

1. 2x - 3y = 7, -2x + 3y =4

2. 4x -y = 1, -8x + 2y = -2

3. x - 2y = 1, 2x + y = 4

4. x/2 + y/2 = 5, 2x + 2y =3

5. x - y = 4, y- x = - 4

6. x/3 - y/3 = 2, 3x - 3y =1

Exercise: Which among the pairs of lines are perpendicular?

1. 2x + 3y = 7, 3x –2y = 4

2. 4x + y = 8 , x + 4y = 2

3. 3x – y = 2, x + 2y = 9

4. 5x – y=0, x + 5y = 2

5. 3x – 6y = 10, 6x + 3y = 8Exercise Graph the following linear equations; find the slopes and the intercept:

1. x = 4

2. x-y = 4

3. x+2y = 6

4. x-2y = 7

For each system, solve one of the equations for one of the variables.

1. y - 5x = 0 2. 2x - 3y =5 3. 5s + 3t =1

x + y = 12 x + y = 0 3s - t = 9

4.3a + 2b = 13 5. 2n +5m = 14 6. 3p - 10 = 4q

4a = 4b +19

Solve by the substitution method.

7. x = 3 8. y = x - 8 9. m = -2

x + y -2 y = 4 2n = m

10. y = 4x 11. y = 3x 12. y = x -2

x + y = 10 x + y = 20 x + y = 12

13. x + y = 1 14. x = 3y 15. a = b + 8

4y + x = 6 5y - x = 8 5b = 8

16. 2x - 3y = 5 17. 5v + u = -17 18. 4a = 3b + 15 x + 3y = -2

3u = 4v + 6 2b -a = 0

Solving Systems of Linear Equations in Two Unknowns

A. Graphical Solution

Exercise: Solve the following pairs of linear equations. Tell if there is no solution.

1. 3x + 2y = 12 and 4x + 3y = 12

2. 4x – 2y = 10 and 2x – 4y = 10

3. 5x – 2y = -10 and 2y + 5x = 5

4. 4x + 2y = 6 and x –y = 6.

5. 2x + 3y = 12, and 4x + 6y = 11

6. x = y, and x =3

p6+ q

3=0

n2=m

4

7. x = -2y, and 2x – y

8. y = 2x + 3 and 4x – 2y = 5

Exercise: Solve the following pairs of equations by addition method.

1. 3x + 2y = 12 and 4x + 3y = 12

2. 4x – 2y = 10 and 2x – 4y = 10

3. 5x – 2y = -10 and 2y + 5x = 5

4. 4x + 2y = 6 and x –y = 6.

5. 2x + 3y = 4 and 4x + 6y = 7

6. x = y, and x =3

7. 3x – 2y = 8 and 6x + 2y = 16

8. x + 1/2y = ½ and 1/2x + y = 2

Solutions Using Substitution

For each system, solve one of the equations for one of the variables.

1. y - 5x = 0 2. 2x - 3y =5 3. 5s + 3t =1

x + y = 12 x + y = 0 3s - t = 9

4.3a + 2b = 13 5. 2n +5m = 14 6. 3p - 10 = 4q

4a = 4b +19

Solve by substitution method.

7. x = 3 8. y = x - 8 9. m = -2

x + y -2 y = 4 2n = m

10. y = 4x 11. y = 3x 12. y = x -2

x + y = 10 x + y = 20 x + y = 12

p6+ q

3=0

n2=m

4

13. x + y = 1 14.x = 3y 15. a = b + 8

4y + x = 6 5y - x = 8 5b = 8

16. 2x - 3y = 5 17. 5v + u = -17 18. 4a = 3b + 15

x + 3y = -2 3u = 4v + 6 2b -a = 0

Exercise: Solve using elimination method. Check by graphical method.

1. 3x + 2y = 4 , 4x + 2y = 8

2. 4x + 3y = 6, 2x + 3x = 8

3. 5x – 2y = -10 and 2y + 5x = 5

4. 4x + 2y = 6 and x –y = 6.

5. x + y = 0 and 3x – y = 5.

6. x = y, and x =3

7. x = -2y, and 2x – y

Practice Problems: Solve the following systems of linear equations. Use any method.

1. x - y = -1, -x + 3y = -1

2. -x - 3y = 10, 5x + 2y = 45

3. 6x + 5y = 5, -x - y = -12

Solve by graphing

1. x + y = -3 2. -x + 2y = 4

-x + y = 5 x + y = 5

3. -x + y = -1 4. x + y = 5

3x + y = -5 5x + 6y = 29

5. 4x + 5y = 2 6. -x + 3y = -10

-5x + 3y = 7 x + y = 6

7. x + y = 6 8. 2x –6y =4

3x + 4y = -22 6x - 7y = -5

Solving age problems using two equations of two unknowns:

Exercises: Represent the following ages using algebraic expression.

1. Age of Sara 10 years from now

2. Sara is twice the age of John.

3. John is 5 years older than George.

4. Age of John 5 years ago.

5. Age of John is 3 times that of Sonny less 5 years.

6. John is 1 and a half times older than Sara.

7. John’s age is twice that of Joe ten years from now.

Exercise:

1. Mary is five years older than Sue. Ten years ago, she was twice as old as Sue. How

old are the two girls now?

2. Samantha is 8 and two-third years older than Jason. The sum of their ages is 23

years. How old is Jason?

3. Brian is three years older than Sydney. The sum of the ages of Brian and Sydney is

seven. How old is Brian?

4. Ryan was born 6 and one-fourth years after Alexander. The sum of their ages is 66

and eleven-twelfth years. How old is Alexander?

5. If Samantha were three times as old as she was five years ago, she will be sixty less

than six times her current age. How old is Samantha?

6. Al is 13 years older than Jack. Next year, the ratio of their ages will be 3:2. How old is

each now?

7. Last year Sue was three times as old as Pablo. Next year she will be twice as old as

Pablo is. How old is each now?

Number Problems:

Exercise: Solve the following.

1. The sum of two numbers is 41. IF their difference is 11, what are the numbers?

2. The units’ digit of a number is twice that of the tens’ digit. If the number is reversed

the difference is 36. Find the number.

3. The first digit of a number is 4 times that of the second digit. If the number is

reversed, the resulting number is 54 more than the original number. What is the

number?

4. The denominator of a fraction is 8 more than the numerator. If 3 is added to both

the numerator and the denominator, the value of the resulting fraction is ½. What

is the resulting fraction?

5. If 1 is subtracted from the numerator of a fraction, the value of the resulting

fraction is ½. However, if 7 is added to the denominator of the original fraction the

value of the resulting fraction is 1/3. Find the original fraction.

Speed Problem

Solve the following problems.

1. A car travelled at a steady speed for 120 km. Due to a mechanical problem, it returned at

half that speed. IF the total time for the round trip was 4 hour, 30 minutes, find the two

speeds.

2. Peter and Paul walked towards each other and after ten minutes the distance that

separate them is 1.2 kilometer. If they will walk, away from each other the distance that

will separate them is 1.4 kilometer. If Peter walk twice as fast as Paul what is the

distance that separate them if they walk towards each other for 15 minutes.

3. Erin run a certain distance at 0.4 km/min and returned to her starting point at 0.3 km/min.

If her total running time was 21 minutes, how far did she run at all..

4. Lynn swims at an average speed of 3.6 km/hr. If Becky swims 2.25 km in the same time

Lynn swims 1.8 km, what is Becky’s average speed?

5. A small plane had been flying 1 hour when a change of wind direction doubled its average

ground speed. IF the entire trip of 860 kilometer took 2.5 hour, how far did the plane

travel in the first hour?

Simple Interest and Investment Problems

Exercise: solve the following.

1. Mercedez receives 375 pesos per year from a 6000 pesos deposited in two banks. The

first bank pays 10% interest while the second bank pays only 5%. How much is invested

for each bank?

2. Gary invested one-third of his insurance settlement in a CD that yielded 12%. He also

invested one-third in Tara’s computer business. Tara paid Gary 15% on this investment.

If Gary’s total income from these investments was $10, 800 for a year, then what was

the amount of his insurance settlement?

Ratio Problem

Exercise: Solve the following ratio problems.

1. The ratio of the total number of animals in a zoo to the number of bears is 22:1. If

the total number of animals minus the number of bears is 315, find the number of

bears.

2. The ratio of boys to girls at a school dance was 5:3. There would have been an equal

number of boys and girls if there had been 3 more girls and 3 fewer boys. How many

boys and how many girls were at the dance?

3. There are 812 students in a school. There are 36 more girls than boys. How many

girls are there?

4. Since my uncle’s farmyard appeared to be overrun with chickens and dogs. I asked

him how many of each he had. Being a puzzler as well as a farmer, my uncle replied

that his digs and chickens had a total of 148 legs and 60 legs. How many dogs and

how many chickens does my uncle have?

5. A pond was stocked with 2000 fish, all tilapia and catfish. The ratio of tilapia and

catfish was 3 to 2. How many of each kind of fish were put in the pond?

6. In a pen containing pigs and chicken, it was found that there are 100 animals

present. IF there are 280 legs all in all, what is the number of chicken and pigs?

7. Three times a number increased by 3 is equal to 30. Find the number.

8. A plumber wishes to cut a 51-inch pipe into three pieces so that each

piece is 7 inches longer than the preceding one. Find the length of each piece.

9. The sum of two numbers is 25, and one number is four times the other number. Find

the numbers.

10. . A basketball team played 20 games and won two more games than it lost. Find the

number of games the team won.

11. If two times a number minus six is equal to 20, find the number.

12. Forty calculators are placed into two boxes so that one box has 4 more calculators

than the other box. How many calculators are in each box?

13. A man is six times as old as his son. In 9 years he will be three times as old as his

son. How old are they now?

14. A woman is twice as old as her daughter. Twenty years ago, she was four times as

old as her daughter. How old are they now?

15. Mark is 4 years older than his brother Mike. If the sum of their ages is 20, how old

are they now?

16. Marie is 12 years older than Mary. Nine years ago, Marie was twice as old as Mary.

Find their present ages.

17. Sam is 18 and Bill is 24. How many years ago was Bill three times as old as Sam?

18. Pat is five years older than her brother. Two years from now, the sum of their ages

will be 23. Find their present ages.

19. The sum of Tyler and Alane’s ages is 36. Twelve years ago, Alane was twice as old as

Tyler. Find their present ages.

20. Tara is two years older than Ashley. In 4 years from now, Tara will be twice as old as

Ashley was 4 years ago. Find their present ages.

21. A father is three times as old as his twin sons. If the sum of their ages in two years

will be 81, how old are they now?

22. How much cream that is 20% butterfat should be mixed with milk that is 5% butterfat

to get 10 gallons of cream that is 14% butterfat?

23. How much of a 90% alloy must be combined with a 70% gold alloy in order to make

60 ounces of an 85% gold alloy?

24. How much of an alloy that is 40% zinc should be added to 75 pounds of an alloy that

is 65% zinc to get an alloy that is 50% zinc?

25. How much of a solution that is 18% fertilizer must be mixed with a solution that is

30% fertilizer to get 50 gallons of a solution that is 27% fertilizer?

26. How much pure alcohol (100%) should be added to 40 ounces of a solution which is

20% alcohol to get 40 ounces of a solution which is 25% alcohol?

27. A merchant blends a $2.50 per pound tea with a $3.75 per pound tea to get 10

pounds of tea costing $3.00 per pound. How many pounds of each did the merchant

use?

28. Charles and Henry together have 49 marbles, and Charles has twice as many as

Henry and 4 more. How many marbles has each?

29. In an orchard containing 33 trees the number of pear trees is 5 more than three

times the number of apple trees. How many are there of each kind?

30. John and Mary gathered 23 quarts of nuts. John gathered 2 quarts more than twice as

many as Mary. How many quarts did each gather?

31. To the double of a number I add 17 and obtain as a result 147. What is the umber?

32. To four times a number I add 23 and obtain 95. What is the number?

33. From three times a number I take 25 and obtain 47. What is the number?

34. Find a number which being multiplied by 5 and having 14 added to the product will

equal 69.

35. I have five hundred pesos of which contain 50-peso and 100-peso bills. The number

of bills I have is seven. How many 50-peso bills I have?

36. A piggy bank contains 5-peso and 10-peso coins. The number of 10-peso coins is

twice that of 5-peso coins. The total amount of money in the piggy bank is 500 pesos.

37. A piggy bank contains 5-peso and 10-peso coins. The total amount of five-peso coin

is twice that of total amount of ten-peso coin. If the total money is 1000 pesos, how

many five peso coins is inside the piggy bank.

Inequalities

Exercise:

Find the interval (or intervals) of values of the variable that satisfy the inequalities:

 

1. x + 5 > 9

2. 4 <= b - 12

3.  x - 3.45 > 2.67

4.  4*y - 8 >= 10

5.  -3x - 5/ 6 < x/3

6.  (2/3)z-5 < 3z

7. (x + 2.3) / 3.2 >= 4.5

8. 4x - 8 >= 7x + 3

9. 3(y - 3) + (y - 3) >= (y - 3)

10. 3(t - 7) < 2 - 10t 

11. 0 < 2 - 3x < 5

12.  3(1 - x)(2 + x) > 0  

13. (3 - x)(4 + x) < 0

14. 3/8  < (3R - 4)/ 3

15.  0 < 1 - x/6 <= 1/6

16.  [7(3x - 5) + 3(x - 1)]/3 <= 1 - x/6

17. (x + 4)/ 3 + x/2 > 3(3 - 2x)

Linear Inequalities of Two Variables:

Exercises: Draw the graph of the following equations or equalities. Shade the areas if

applicable.

1. x > 50

2. y < 20

3. x+y > 40

4. x + 4y < 50

5. 2x + 3y > 6

6. y < 10

7. 4x + 3y < 12

8. 9x + 2y > 18

9. x-2y > 10

10. 7x + 4y =28

Exercise: Solve the following inequalities.

1. y < 50

x < 30

2. 2x +3y < 12

3x + 4y < 16

3. 4x + 5y < 20

y > 2

4. 3x + 10y < 30

2x +3y < 24

x < 8

5. x + y < 15

2x + 3y < 30

6. 4x + 3y = 36

y > 2

7. 7x + 9y < 63

x + y < 27

3x + 4y < 48

8. 3x + 4y > 24

y > 10

Solving Quadratic Equations

Exercise: Convert the following quadratic equation in a standard form, ax2 + bx + c = 0.

Identify also the value of a, b, and c.

1. x + 3x2 + 2 = 3

2. 2x2 = 3x

3. 3x2 + 2 + 3x = 0

4. 4x2 = 0

5. x2 + 3x + 4 = 0

6. x2 - 2x = 4

7. 5x2 = 9 – x

8. (3x - 2)2 = 2 (hint: Expand then rearrange.)

9. 2x2 -7x + 6 = 3

Exercise: Solve the roots of the equations

1. 4x2 - 25 = 0

2. 6x2 = 43x + 40

3. 3x2 - x - 14 = 0

4. 3x2-2x-1=0

5. x2 + 4x –5 =0

6. x2 + 13x + 36 = 0

7. a2 +a - 56 = 0

8. y2 - 64 = 0

9. p2 = 5p +24

10. r2 = 18 +7r

Finding Roots by Completing the Squares:

Solve by Completing the Square

1. 2s2 + 5s = 3

2. 3x2 = 3 - 4x

3. 9v2 - 6v - 2 = 0

4. x2 + 11x + 10 = 0

5. -12x2+ 22x =10

1. 6x2 + 15/2x + 3/2 = 0

2. 9x2 - 12x - 5 = 0

3. -x2 - 2x + 3 = 0

4. –2x2 – 3/2x + 5 = 0

5. –8x2 – x +19 =12

11. x2 + 13x + 36 = 0

12. a2 +a - 56 = 0

13. y2 - 64 = 0

Quadratic Formula

Determine the roots using quadratic formula:

1. x2 - 4x + 1 = 0

2. x2 + 4x = 12

3. d2 - 9 = 0

4. 4x2 = 35 - 4x

5. 3s2 + 2s + 4 =0

6. x2 - 14x - 15 = 0

7. a2 -5a + 4 = 0

8. b2 + 4 = 6b

9. y2 - 8y = 4

10. x2 - 10x = 23

11. x2 + 7x + 6 = 0

12. -r2 - 6r + 3 = 0

13. 2y2 + 3 = -7y

14. z2 -13z -32 =0

15. 3x2 + 2 = -8x

Solve for the equation

1. (n – 9)2 = 27

2. (m + 2/3)2 = 1/9

3. (n + 3/4)2 = 1/16

4. (r + 2/3)2 = 5/9

5. (y + 3/4)2 = 3/16

6. 2(5x + 1)2 = 50

7. 3(2x + 4)2 = 36

8. 5(z – 3)2 = 35

9. 7(m + 4)2 = 70

Solve the equation by completing the square.

10. x2 – 4x + 2 = 0

11. n2 – 3n + 5/4 = 0

12. r2 – 3r – 10 = 0

13. p2 + 5p + 6 = 0

14. 4y2 – 6y – 1/2 = 0

15. 3b2 – 12b – 9 = 0

16. 3n2 – 8n + 4 = 0

17. 2t2 – 5t = 4

18. 2c2 + c = 5

Use a calculator to solve the equation by completing the square.

Round your answers to the nearest hundredth.

19. t2 – 3.7t – 10 = 0

20. p2 – 5.2p + 6 = 0

21. y2 = 10 + 5.4y

22. b2 = 22b + 2

Scientific Notation

Exercise: Find the quotient.

1. 50,000,000 divided by 2050,000,000

2. .00000005 divided by 0.00002

3. .0004 divided by 50000

4. 10000 divided by .0005

5. .0004 divided by .000000006

Exercise: Add/Subtract the following:

1. 10 E4 + 12 E3

2. 105 E4 + 120 E3

3. 10 E64 - 15 E3

4. 19 E84 + 12 E103

5. 6 E4 + 12 E3

6. 110 E4 - 3 E3

7. 10 E4 + 12 E5

8. 10 E6 + 12 E3

Reducing Simplifying the Rational Expressions:

Exercise: Simply the following rational expressions:

Simplifying Polynomial / Monomial Algebraic Expressions

Exercise: Simplify the following rational expressions:

Multiplication of Rational Expressions

 Exercise: Multiply the following rational expressions:

1 .3 + 6a2b3a

2 .8x2 y3

32xy

3 .27x2 y 4

18x4 y2

1.x2+8x+15x+3

2.x2−5x−36x−4

3.x2+6x+10x+2

4 .x2−4x−6a+2

5.2x2−7x+52x−1

1 .y+3y

.y2

y2−9

2 .a+ba−b

. a2−b2

2a+2b

3 . x2+4x−21

x2−6x+−16.x2−8x+15

x2+9x+14

4 .4x2−2ax

x2−4ax+4a2.(2a−x )3

2x−a

5 .4z2−4

1+z2.1−z2z

.1−2z2+z4

2+2z

Division of Polynomial Functions

Find the quotient of the following:

1 .58

÷34

2 .x2

5÷x

15

3 .ab3

÷ba

4 .x2 y

÷xy4

5 .3n2

5÷

9n10

6 .4

b2÷a2

b

7 .4 x2

3 y÷(2 x )

8 .9a2

2b÷(6ab )

9 .1÷(3 x5

)2

10 . 4÷(2n

)3

11.a+b3

÷2a+2b6

12 .x2−12

÷x−14

13 .2n−54

÷6n−158

Addition and SubtractionExercise: Find the least common multiple of the following:

1. 35, 7

2. 3m, 6m2n

3. 2a, 8a3

4. (a-1), (2a-2)

5. (x + 1), (x2 –1)

6. (x +1),(x2+ 3x+2)

7. 2,3,4

8. (4x-2y), (y-2x)

Subtraction of Algebraic Fractions

Exercises:

Write each fraction in lowest terms.

1.   45j3

63j

2.   77d4

33d

3.   64h4

72h2

4.   a2 - 7a + 12

a2 + 7a - 44

5.   4i5 - 24i4

-9i2 + 54i

6.   c2 - 1

c - 1

7.   -13h2 + 195h - 702

-18h3 + 162h2

8.   3c7

-7c

9.   -10b + 90

2b2 - 162

10.   a + 9

-15a2 - 150a - 135

11.   d2 - 2d - 35

d2 - 49

12.   12g8

-16g4 + 12g3

13.   -10a5

-40a5 + 35a

14.   10c - 80

-19c2 + 1216

15.   c + 9

13c2 - 13c - 1170

16.   i2 - 8i + 7

i2 - 1

17.   b - 2

-4b2 + 16

18.   -5c

6c3

19.   d2 + 7d + 10

d2 - d - 30

20.   15h2

21h6

21.   13i2 + 78i - 520

42i3 + 420i2

 Write each fraction in lowest terms.

1.   54c3

72c4

2.   36f

48f4

3.   e2 - 7e - 44

e2 - 16

4.   24j3

15j4

5.   -3k2 + 48

10k2 - 40k

6.   h + 8

2h2 - 128

7.   k2 - 10k + 16 8.   -9b2 - 36b - 72 9.   -288f2 + 360f + 1188

k2 + 2k - 8 b3 + 7b2 + 20b + 24 -64f + 176

10.   16e3 + 192e2

20e2 + 276e + 432

11.   e - 3

4e2 - 36

12.   -3f2 - 18f - 3

2f2 + 12f + 2

13.   f2 - 6f - 55

f2 - 23f + 132

14.   b2 - 121

b2 + 18b + 77

15.   -35b + 140

-12b2 + 192

16.   e3 - 7e2 - 22e + 36

-10e2 - 20e + 40

17.   -8a2 + 26a - 20

-4a6 + 8a5

18.   12k2 - 768

k - 8

19.   -9a2 - 117a - 270

a + 3

20.   28k2 - 28k - 840

28k3 - 168k2

21.   -13b2 + 832

b - 8

Add the following fractions:

1.   8f

6

+

3f

3

2.   2k

7

+

7k

21

3.   5f - 9 + 2f - 11 4.   8 + 5

3 12 w g

5.   6c

3

+

2c

12

6.   4c - 5

16

+

8c

8

7.   3h

3

+

4h - 11

6

8.   9

18j

+

9j + 3

4j

9.   6

i

+

7

y

10.   6

24d

+

3u + 8

6d

11.   -9y + 5

5i

+

4i + 9

2y

12.   9c + 8

12

+

6c + 9

6

13.   4j + 12

6

+

3j

18

14.   7

v

+

5

i

15.   8

4g

+

6g + 3

36g

16.   3i + 5

9i

+

-4r + 6

36i

17.   -5w + 9

3g

+

6

6w

18.   8

e

+

4

w

Multiplying Fractions:

1.   g3

g5 - 2g3

×

28

12g - 36

2.   4u

-7j

×

-21j3

12u3 + 16u

3.   2s

9a

×

9a2

4s3

4.   3b3 - 7b2

3b2 - 7b

×

2u

4u3

5.   g2 - 7g

-3u

×

-6u3 + 15u

5g3 - 35g2

6.   72e6

24e4

×

8e3

36e5

 7.  5g4 + 45g3

g2 + 9g

×

-4g

-16g2 + 24g

8.   20k - 60

2k + 4

×

2k2 + 10k + 12

30

 9.  g2 + 8g + 12

9g + 63

×

g2 - 49

-7g4 - 42g3

10.   6g4 - 48g3

g2 - 64

×

g3 + g2 - 8g - 8

g2 - 1

11.   99b6

33b2

×

66b4

33b2

12.   -5p

-p3 - 3c

×

-p3c - 3c2

-5p2

13.   d2 - 9d + 8

-8d3 + 8d2

×

d2 - 16

d2 - d - 12

14.   -21g2 + 3pg

-7g + p

×

-4p

-8p2 - 20p

Subtracting Fractions:

1.   9b

7

-

3b

14

2.   7e

16

+

2e

4

3.   6e

8

-

5e + 14

16

4.   8h

28

+

3h

7

5.   4

b

+

3

s

6.   4i + 7

8

-

5i - 9

24

7.   9k + 11

4

+

3k + 10

12

8.   8e + 9

4e

-

2

24e

9.   -6j + 9

2j

+

-5p + 2

5p

10.   5h + 6

8

+

6h + 8

4

11.   8

d

-

6

t

12.   4j - 9

4

-

6j

16

13.   5

i

+

7

q

14.   9

12k

-

-7q + 4

24k

Express in the simplest form, noting any restrictions on the variable.

A. 1.

3a−6a−2 2.

4n+12n+3 3.

3n+19n+3 4.

6 x−6 y6 x+6 y

5.

2x−2

x2−1 6.

2 y+14

49− y2 7.

2 xy

x2 y− y2 x 8.

4 p3

4 p2−8 p

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22. 23. 24.

25. 26. 27. 28.

29. 30. 31.

6 x+8x

16 y2+9x2

4 y+20

4 y2−100

b2−25b2−12b+35

a2+8a+1616−a2

(4 n−3 )5

(3−4n )5(2 x− y )3

( y−2 x )4(a−5)2

25−a2

2− yy2−4 y+4

(3 r+7)2 (2 r−1(1−2 r )(7=3 r )

( x−2 )(2x+5 )(5+2 x )( x+2)

2ab+2ac+4 a2

4 b+4 c+8ax 2 +xyx2−xy

4n2−6n4n2+8n−21

2 z2+z−62 z+4

3 x2−15 x3x2−16 x+5

2n2−5n−34n2−8n−5

2 y2−9 y+42 y2−8 y

x2− y2

x2+2xy−3 y2

a2−b2

a2+5ab+4 b2

10−7a+a2

a2−4

6a3+10a2

36a3−100 a4 x2+16 xy+15 y2

2 x2+xy−10 y2

16 x2−49 y2

16 x2−56xy+49 y2

32. 33. 34.

C.

35. 36. 37.

38. 39. 40.

D. For which value(s) of x do the given fractions equal zero? (Hint: A fraction is equal to

zero only if its numerator equals zero.)

41. 42.

42. 44.

45. 46.

Simplifying Complex Fractions

Simplify the following complex fractions:

1. 2. 3.

a4−4a2 (a+2 )+2 (a+2 )

a−b+x (a−b )a−b

8a2+6ab−5b2

16a2−25b2

x4−10 x2+93−2 x−x2

a2−9b2−2a+6b(a+3b)2−4

2x3−13 x2+15 x15 x−7 x2−2x3

(a−b )3+4(b−a )(a−b )2+4(b−a )+4

a2−6ab+9b2−93a−9b−9

x3+6 x2−4 x−24x3−2 x2−36x+72

x2−3 xx2−2 x−3

x2−2 x−8x2−4 x

x4−x2

x3+x2−2xx2−2 x−15x2+3x−40

2x3−x2−10 xx3−2 x2−8x

3x 4+27 x3+60x2

6x2+6 x−72

Rational Exponents and Radical Expressions

Exercise: Simplify the following:

1. (x2y4)4

2. x3y3.x2y4

3.

x3

x2

4. ( x

2

y3)4

5.

−1

x2 y3

Simplify.

1. √75 2. √90 3. √294

4. √726g9 5. √384 6. √360

7. √0.64 8. √0.25 9. √0.0004

10. 5√1859 11. 12-√567c5e3 12. 6-√1008g7i1d0

13. 7√1176k8 14. 4√720 15. 2√360

16. 3-√3072h5j5i0 17. 8√588d4 18. -9√448e3g6i9

Simplify the following radicals

1. √0.81 2. √0.0004 3. √0.0009

4. √0.16 5. √0.0064 6. √0.25

7. √0.0049 8. √0.36 9. 12-√0.49

10. 2√0.25 11. 8-√0.0016 12. √0.0036

13. 6√0.04 14. 7√0.81 15. 10√0.0009

16. 9√0.04 17. 4-√0.0064 18. 3-√0.0036

19. 11√0.16 20. 8-√0.0049 21. 3√0.81

THE PRINCIPAL POWERS OF i

Simplify:

5 40254 483 15610 32 4) 16 3) 27 2) 1) yxyxyxb

Simplify:

1. √ x3 y6 z92. √60 xy7 z12

3. √−216 x5 y94.

4 √64 x8 y10

Simplify:

1) 3√ y+12√ y   2) 2√2x3+4 x √8 x 3)3√128 + 3√250

4.  √4 x7 y5+9 x2 √ x3 y5 − 5 xy √x5 y35) 6 y

4√48 x5−2 x4√246 xy4−4

4√3 x5 y 4

Simplify:

 1.) √14√35 2) √a3 b √ab4 3) 3√4 a2b3 3√8ab5 4) √ y (√ y−5 )

  5) (√2 x+4 )2 6) (√ y−2 ) (√ y+2 ) 7) 3√8ab

3√4a2b3 3√9ab4

Variation

Exercise:

The distance between you and lightning varies directly as the time interval between the lightning and the thunder. Suppose that thunder from lightning 5400 ft away takes 5 sec to reach you.

   a) Sketch the graph.  What is the constant of proportionality?

   b) If the time is 8 sec, how far is the lightning?

Exercise: Solve the following direct variation problems:

1. x varies directly with x. If x is 5 when y is 15, find the value of x when y is 25.

2. The salary of the person depends on the number of years he served on the company.

After five years the salary is 10, 100 pesos per month. What is his salary after 8

years?

Exercises:

1. The strength of a rectangular beam varies jointly as its width and the square of its

depth. If the strength of a beam 2 inches wide and 10 inches deep is 1000 pounds per

square inch, what is the strength of a beam 4 inches wide and 8 inches deep?

2. It is shown in engineering that the maximum load a cylindrical column of a circular cross-

section can hold varies directly as the fourth power of the diameter and inversely as

the square of the height. If a column 9 feet high and 3 feet in diameter will support a

load of 8 tons, how great a load will be supported by a column 12 feet high and 2 feet in

diameter?

3. Assume that y varies directly with x. If x = 2 when y = 12, find the value of y when x =

12.

4. Assume z varies jointly with x and y. Find the constant of variation when x = 3, y = 4, and

z = 24.

5. The width of a rectangle of fixed area varies inversely with the length. If the width of the

rectangle is 4 cm, its length is 10 cm. What is the length when the width is 8 cm?

6. Write a formula to express this law: at a constant temperature, the volume, V, of a gas

varies inversely with the pressure, P. Use k as the constant of variation.

7. Newton's Second Law states that the acceleration of an object is directly proportional to

the force acting on it and inversely proportional to the mass of the object. Write a

formula expressing this property using F as the force, m as the mass, and a as the

acceleration.

8. The amount of water emptied by a pipe varies directly with the square of the diameter of

the pipe. For a certain constant water flow, a pipe emptying into a canal will allow 200

gallons of water to escape in an hour. The diameter of the pipe is 6 inches. How much

water would a 12-inch pipe empty into the canal in an hour, assuming the same water

flow?

9. The illumination produced by a light source varies inversely as the square of the distance

from the source. If the illumination produced 1 meter from a certain light source is 768

footcandles, find the illumination produced 6 meters from the same source.

10. The volume of a gas varies inversely with the pressure and directly with the temperature

of the gas. If a certain gas occupies a volume of 1.3 liters at 300 degrees Kelvin and a

pressure of 18 Newtons per square centimeter, find the volume at 340 degrees Kelvin

and a pressure of 24 Newtons per square centimeter.

11. The frequency of a vibrating string varies inversely as its length. (This means that a

longer string vibrates fewer times in a second than a shorter string). Suppose a piano

string 2 feet long vibrates 250 cycles per second. What frequency would a string 5 feet

long have?

12. According to Hooke's Law, the force needed to stretch a spring is proportional to the

amount the spring is stretched.  If fifty pounds stretches a spring five inches, how much

will the spring be stretched by a force of 120 pounds? Ans. 12 in

13. Under certain conditions, the thrust T of a propeller varies jointly as the fourth power of

it diameter d and the square of the number n of revolutions per second.  Show that, if n

is doubled and d is halved, the thrust T is decreased by 75%.

14. The weight of a body varies inversely as the square of its distance from the center of the

earth.  If the radius of the earth is 4000 miles, how much would a 200-pound man weight

1000 miles above the surface of the earth?

Statistics and ProbabilitySolve the following:

n=5

1. Σ(XY – Z)

i= 2

n=6

2. Σ(2X + Y – Z)

i= 1

n=6

3. Σ(3X +2Y + Z +5)

i= 2

n=5

4. Σ(X2 + 2Y Z)

i= 2

n=5

5. (ΣXY)2 Hint: Solve the total value, then get its square.

i= 2

Permutations and Combinations

Try solving these exercises:

1. How many ways can we arrange three letters taken from the letters of the word

“variant”?

2. Three numbers in exact sequence has to be chosen from 47 numbers. What is the

sample space?

3. A CD key is compose of five letters and three digits. The letters and digits are not to

be repeated. What is the sample space?

Counting Sample Space

1. List down all the possible outcome in answering a three item survey with three

options. Count the elements of the sample space.

2. Determine the sample space if two letters are chosen without repetition from the

letters of the word “egypt”.

3. How many possible outcomes if you choose three digits from five digits if the

digits are not to be repeated?

Permutations, Combinations, Multiplication Rule:

4. In how many ways can the offices of president, vice president, and treasurer be

filled from a group of 8 people?

5. In how many ways can 6 girls be arranged in a straight line?

6. How many numbers between 10 and 20 can be formed by the digits 1, 2, and 3

if no repetition is allowed? How many can be formed if repetition is allowed?

7. How many odd-numbered integers can be formed by the digits 2, 3, 6, 5, 9, 7 if

each digit may be used only once?

8. In how many different ways can the letters of the word “number” be arranged if

each arrangement begins with a vowel?

9. A theater has 6 entrances. In how many ways can you enter and leave by a

different entrance?

10. In how many ways can you mail three letters in five letter boxes if no two letters

are mailed in the same box?

11. Archie has seven grocery stores and six meat markets. In how many ways can

you buy a pound of hot dogs and a bag of flour?

12. Four people enter a bus in which there are six empty seats. In how many ways

can the people be seated?