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Algebra - Mixed Practice (Up to Rational Expressions)

Lesson Plans > Mathematics > Algebra > Mixed Content
 

Algebra - Mixed Practice (Up to Rational Expressions)

This is a fairly lengthy unit in most algebra textbooks.  In these worksheets we'll review much of the content up to this point in the curriculum, and also cover the following topics: reducing algebraic fractions; multiplying and dividing algebraic fractions; finding least common multiples of algebraic expressions; adding and subtracting algebraic fractions; complex fractions; solving fractional equations.

Some algebra one curricula do not include fractional equations that result in quadratics. However, since we have already completed the quadratic unit, we will include some quadratic fractional equations here. Those will be notated below, so if you don't want to have your students do those problems (or if you want them to be "bonus"), you may do so.

Worksheet 7.1

  • Finding LCMs
  • Converting phrases to algebraic expressions
  • Reducing algebraic fractions

Worksheet 7.2

  • Combining like terms
  • Multiplying and dividing numeric fractions
  • Multiplying and dividing fractions

Worksheet 7.3

  • Solving linear equations
  • Factoring polynomials
  • Finding the LCM of algebraic expressions

Worksheet 7.4

  • Solving quadratic equations
  • Adding and subtracting numeric fractions
  • Adding and subtracting algebraic fractions

Worksheet 7.5

  • Solving word problems
  • Complex fractions

Worksheet 7.6

  • Factoring
  • Simplifying and factoring
  • Solving fractional equations (the last three problems are quadratics)

Worksheet Sets in this Series

Worksheets for this section are below the index. Click the "Overview" link to get a more detailed view of the entire series.

Lesson by Mr. Twitchell

Handouts/Worksheets

Mixed Practice 7.1

  1. Find the LCM of 32 and 24.
     
  2. Find the LCM of 20, 28, and 35.
     
  3. Find the LCM of 9, 28, and 70.
     
  4. Find the LCM of 2, 3, 4, 5, and 6.
     
  5. Find the LCM of 7, 8, 9, and 10.

    Write and simplify these algebraic expressions
     
  6. Two more than three times a number
     
  7. Five times seven less than a number
     
  8. The sum of two more than a number and three less than a number
     
  9. The product of a number and two less than the number
     
  10. Five less than a number, multiplied by seven more than twice the number

    Reduce the following fractions
     
  11. 12xy3
    9xy4

     
  12. x + 1
    x2 - 1

     
  13. 2x - 4
    8

     
  14. x2 + 5x + 6
    x + 2

     
  15. 3x - 9
    7x - 21

     
  16. x2 - 4
    x2 + 4x + 4

     
  17. x2 + 7x + 12
    x2 + 5x + 4

     
  18. 9x2 + 18x + 9
    4x2 - 4

Mixed Practice 7.1: Answer Key

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Mixed Practice 7.2

  1. 3x + 2 - 4x +5 =
     
  2. (5x + 2) - (2x - 3) =
     
  3. (x2 - 3x + 2) - (x2 + 4x - 11) =
     
  4. -4(2x - 1) - 3(2 - 5x + x2) =
     
  5. 1
    3
    (6x - 3) + x(x + 4) =
     
  6. 2
    3
     · 
    9
    12
    =
     
  7. 1
    7
     · 
    14
    3

     
  8. 9
    2
    ÷
    3
    4

     
  9. x
    12
     ÷
    3
    4
    =  
     
  10. 3x
    4
     · 
    2x
    5
    =
     
  11. x2 - 9
    x + 1
     · 
    x2 - 1
    x - 3

     
  12. 2x + 12
    x - 6
     · 
    5x - 30
    4x + 24

     
  13. 4x2 + 4x + 1
    x2 + 3x + 2
     · 
    x2 - 1
    4x2 - 1

     
  14. 5x - 1
    12
     ÷
    25x2 - 1
    6

     
  15. 1
    10x
     ÷
    3
    15x

     
  16. x2 + 2x + 1
    x2 - 4x + 4
     ÷
    x + 1
    x - 2

     
  17. 2 ÷
    3
    5

     
  18. x
    y2
     · 
    x + 1
    y - 1
      ÷
    xy
    y - 1

Mixed Practice 7.2: Answer Key

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Mixed Practice 7.3

Solve the equations

  1. 3x - 12x = x + 15
     
  2. 5(x - 12) = 3(x + 4) + 10
     
  3. x(x + 1) = (x - 1)(x - 2) + 20
     
  4. 3x + 10 - 2(x + 1) = 15
     
  5. 2(x - 3(x - 1)) + 2 = 32 - 2x

    Factor the following expressions
     
  6. x2 - 18x + 17
     
  7. 3x2 - 12
     
  8. 4x2 - 4xy + y2
     
  9. 16x4 - 81

    Find the least common multiple of the expressions
     
  10. 14x3y; 21xy4z
     
  11. x; 4x; 6x2
     
  12. 2x; 6x - 6
     
  13. 2x + 1; 2x2 - x - 1
     
  14. x2 - 9; x2 + 6x + 9
     
  15. x2 + 3x + 2; x2 + 4x + 3
     
  16. 3x + 6; 4x + 8
     
  17. 2; x + 2; 2x - 4
     
  18. 9x2 - 16; 3x2 + 2x - 8
     
  19. x2 + 8x + 16; x2 + 7x + 12 
     
  20. x3 - 4x2 + 3x; 5y

Mixed Practice 7.3: Answer Key

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Mixed Practice 7.4

Solve the following equations

  1. x2 - 13x + 30 = 0
     
  2. 2x2 - 7x = -6 
     
  3. 2(x + 4) = x(x + 1) - 22
     
  4. 10x2 = 21x + 27
     
  5. (x + 2)(x + 3) = (x - 1)(x + 1) + 32

    Simplify the fractional expressions
     
  6. 1
    2
    +
    7
    3
    =
     
  7. 3 +
    1
    3
    +
    1
    4

     
  8. 9
    2
    + 2 = 
     
  9. 5
    6
    -
    1
    4

     
  10. 5 -
    11
    3

     
  11. x
    3
    +
    3x
    2

     
  12. 1
    x
    +
    1
    y

     
  13. x + 1
    4
    +
    x - 1
    3
    =
     
  14. 1
    x + 1
    +
    2
    x2 + 2x + 1
    =
     
  15. x + 1
    x2 - 1
    -
    1
    x

     
  16. 1
    2x + 3
    -
    2
    4x + 6

     
  17. 1 -
    2
    x + 1

     
  18. 1
    x
    +
    1
    x2
    -
    1
    2x

Mixed Practice 7.4: Answer Key

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Mixed Practice 7.5

  1. An item's price is increased by $10, and then the new price is increased by 25%. The new price is $45. What was the original price?
     
  2. An item's price is increased by 25%, and then $10 is added to the price. The new price is $45. What was the original price?
     
  3. The sum of three consecutive odd integers is 165. What is the largest of the numbers?
     
  4. The product of two consecutive positive even integers is 168. What is the smaller of the two numbers?
     
  5. Bob can drive 20 miles in a certain number of minutes. He drove 30 more miles (at the same speed) in 28 minutes. How long did his total trip take?
     
  6. The number of feet in a measurement is 160 more than the number of yards in the same measurement. How many feet long is the object being measured?
     
  7. Two numbers add to 15. The smaller number is 3 less than the larger. What are the two numbers?

    Simplify the following fractional expressions
     
  8. abc
    def
    bc
    ef

     
  9.  
    a
    b
     
    b
    c

     
  10. 1 +
    1
    2
    1 -
    1
    2

     
  11. x +
    1
    x
    x -
    1
    x

     
  12. a + b
    c
    a - b
    c

     
  13. 1 +
    2
    x
    +
    1
    x2
    1 -
    1
    x2

     
  14. x2 - 9
    x2
    x2 + 4x + 3
    x

Mixed Practice 7.5: Answer Key

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Mixed Practice 7.6

Simplify/factor the following expressions

  1. x2 + 18x + 45 
     
  2. 2x2 - 5x - 12
     
  3. 12x2 - 75
     
  4. x(x + 2) + 3(x + 2) 
     
  5. (x + 2)2 - 1
     
  6. a2 + 2ab + b2 - c2

    Solve the following equations
     
  7. x
    2
    +
    2x
    3
    = 14
     
  8. x + 1
    2
    -
    x + 1
    3
    = 2
     
  9. x + 1
    5
    -
    x
    10
    =
    8
    5

     
  10. 2
    x + 1
    +
    3
    x - 1
    =
    36
    x2 - 1

     
  11. 1
    a
    -
    3
    a2
    = -
    2
    a3

     
  12. 1 +
    1
    x
    +
    1
    2x
    =
    17
    14

     
  13. 1 -
    1
    x2
    1 -
    1
    x
    = 12
     
  14. 1 +
    5
    x
    +
    6
    x2
    = 0
     
  15. x
    x - 1
    -
    x
    x + 1
    =
    20
    99

     
  16. 3x
    2
    +
    2
    3x
    =
    82
    9

Mixed Practice 7.6: Answer Key

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