Intermediate Algebra Questions
With Solutions and Explanations - Sample 1
Solutions and full explanations of intermediate algebra questions in sample 1 are presented.

  1. Write 230,000,000,000 in scientific notation.

    Solution

    Write the given number in the form

    a × 10 n , where a is a real number such that 1 ≤ |a| < 10 and n is an integer.

    230,000,000,000 = 2.3 × 100,000,000,000 = 2.3 × 10 11

  2. Evaluate: 30 - 12÷3×2 =

    Solution

    According to order of operations, 12÷3×2 (division and multiplication) is done first from left to right

    12÷3×2 = 4 × 2 = 8

    Hence

    30 - 12÷3×2 = 30 - 8 = 22

  3. Evaluate: |4 - 8(3 - 12)| - |5 - 11| =

    Solution

    According to order of operations, inner brackets first. Hence

    |4 - 8(3 - 12)| - |5 - 11| = |4 - 8*(-9)| - |5 - 11|

    According to order of operations, multiplication within absolute value signs (which may be considered as brackets when it comes to order of operations) next. Hence

    = |4 + 72| - |5 - 11|

    = |76| - |-6|

    = 76 - 6 = 70

  4. Evaluate: -18 + 4(6 ÷ 2)2

    Solution

    According to order of operations, inner brackets first. Hence

    -18 + 4(6 ÷ 2)2 = -18 + 4(3)2

    According to order of operations, power next. Hence

    = -18 + 4*9

    According to order of operations, multiplication next. Hence

    = -18 + 36

    = 18

  5. Evaluate: 11 + sqrt(- 4 + 6×4÷3)

    Solution

    According to order of operations, inner brackets first where 6×4÷3 is first calculated since it has a multiplication and a division.

    6×4÷3 = 24÷3 = 8

    Hence

    11 + sqrt(- 4 + 6×4÷3) = 11 + sqrt(- 4 + 8)

    = 11 + sqrt(4) = 11 + 2 = 13

  6. Simplify: 12x3 - 3(2x3 + 4x -1) - 5x + 7

    Solution

    First expand the term - 3(2x3 + 4x -1)

    12x3 - 3(2x3 + 4x -1) - 5x + 7 = 12x3 - 6 x3 - 12 x + 3 - 5x + 7

    Group like terms

    = 6 x3 - 17 x + 10

  7. Simplify: (x4 / x3)3

    Solution

    Use quotient of powers formula xm / xn = xm - n to simplify (x4 / x3).

    (x4 / x3)3 = (x)3

    = x3

  8. Simplify: (3x2y-2)3 / (9xy3)3

    Solution

    Use power of quotient formula (am) / (bm) = (a/b)m to rewrite the given expression as follows.

    (3x2y-2)3 / (9xy3)3 = [ (3 x2y-2) / (9 x y3) ]3

    = [ (x2 - 1) / 3 (y3 + 2) ]3

    = [ x / 3 y 5 ]3

    = x3 / (27 y15) = (1/27)(x3 / y15)

  9. Simplify: (2x-3y4)3(x3 + y)0 / (4xy-2)3

    Solution

    Note that the above expression is defined when neither x nor y is equal to zero and therefore (x3 + y)0 = 1. Hence

    (2x-3y4)3(x3 + y)0 / (4xy-2)3 = (2x-3y4)3 / (4xy-2)3

    = [ (2x-3y4) /(4 x y-2) ] 3

    = (1/2)3 [ y4 + 2 / x1 + 3 ]3

    = (1/8) [ y6 / x4 ] 3

    = (1/8) ( y18 / x12 )

  10. Write as a mathematical inequality:"9 is less than the product of M and N".

    Solution

    9 < M × N

  11. Find the slope of the line perpendicular to the line y = (1/3)x - 7

    Solution

    Two lines are perpendicular if the product of their slopes is equal to -1. The slope of the given line is equal to 1 / 3. If m is the slope of the line perpendicular to the given line, then

    m × (1/3) = -1

    Solve for m

    m = - 3

  12. Write an equation of the line with slope -3 and y-intercept (0 , -5).

    Solution

    y = m x + b is the general form of the equation of a line in slope intercept form. Hence for m = -3 and b = -5, we have the equation

    y = - 3 x - 5

  13. Solve the equation: -5x + 20 = 25

  14. Solve the inequality: -3x + 4 < -8

  15. Solve the equation: 2x2 - 32 = 0

  16. Solve the equation: -0.25x + 1.3 = -0.55x - 0.2

  17. Solve the equation: -0.25x2 + 1.5 = -10.75

  18. What is the slope of a line perpendicular to the line x = -3?

  19. What is the slope of a line parallel to the line x = 5?

  20. What is the slope of a line perpendicular to the line y = 6?

Answers to the Above Questions
  1. 2.3 × 1011
  2. 22
  3. 70
  4. 18
  5. 13

  6. 6x3 -17x + 10
  7. x3
  8. (1/27)(x3 / y15)
  9. (1/8)(y18 / x12)
  10. 9 < M × N

  11. -3
  12. y = -3x - 5
  13. -1
  14. x > 4
  15. -4 , 4

  16. -5
  17. -7 , 7
  18. 0
  19. undefined
  20. undefined

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