Intermediate Algebra Questions
With Solutions and Explanations - sample 2
Solutions and full explanations of intermediate algebra questions in sample 2 are presented.

  1. (True or False)     The inequality |x + 1| < 0 has no solution.

    Solution

    The absolute value of a real expression is either positive or equal to zero. Therefore there is no value of x that makes |x + 1| negative and therefore |x + 1| < 0 is never true and the statement "The inequality |x + 1| < 0 has no solution" is TRUE.

  2. (True or False)     If a and b are negative numbers, and |a| < |b|, then b - a is negative.

    Solution

    Since a and b are both negative, they are positioned to the left of zero on the number line. Since |a| < |b|, a is closer to zero than b and therefore a is greater than b which written as

    a > b

    Subtract a to both sides and simplify

    a - a > b - a

    0 > b - a

    Hence the statement "b - a is negative" is TRUE.

  3. (True or False)     The equation 2x + 7 = 2(x + 5) has one solution.

    Solution

    Let us solve the given equation

    2x + 7 = 2(x + 5)

    2x + 7 = 2x + 10 , expand right hand term

    2x + 7 - 2x = 2x + 10 - 2x , subtract 2x from both sides

    7 = 10 , simplify

    The above statement is never true and therefore the given equation has no solutions. The statement "The equation 2x + 7 = 2(x + 5) has one solution" is FALSE.

  4. (True or False)     The multiplicative inverse of -1/4 is -1/8.

    Solution

    If a real number x is not equal to zero, its multiplicative inverse is equal to 1/x. Hence the inverse of -1/4 is equal to

    1 / (-1/4) = (1/1) / (-1/4) = (1/1)*(-4/1) = - 4

    and therefore the statement "The multiplicative inverse of -1/4 is -1/8" is FALSE.

  5. (True or False)     x ÷ (2 + z) = x ÷ 2 + x ÷ z

    Solution

    let us use the values x = 8 and z = 2 and evaluate the values of the left side and right side expressions.

    Left side: x ÷ (2 + z) = 8 ÷ (2 + 2) = 2

    Right side: x ÷ 2 + x ÷ z = 8 ÷ 2 + 8 ÷ 2 = 4 + 4 = 8

    Since x ÷ (2 + z) = x ÷ 2 + x ÷ z is not true for one value of x and one value of z, the statement is FALSE.

  6. (True or False)     |-8| - |10| = -18

    Solution

    Evaluate left side.

    |-8| - |10| = 8 - 10 = - 2

    hence the statement |-8| - |10| = -18 is FALSE.

  7. (True or False)     (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2)

    Solution

    Evaluate left side.

    (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1

    Evaluate right side.

    8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4

    hence the statement (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2) is FALSE.

  8. (True or False)     31.5(1.004)20 < 31.6(1.003)25

    Solution

    Use calculator and calculate left and right sides of inequality.

    31.5(1.004)20 = 34.118 (rounded to 3 decimal places)

    31.6(1.003)25 = 34.057 (rounded to 3 decimal places)

    since 34.118 is greater than 34.057 the statement 31.5(1.004)20 < 31.6(1.003)25 is FALSE.

  9. (True or False)     The graph of the equation y = 4 has no x-intercept.

    Solution

    The line with equation y = 4 is a horizontal line parallel to the x axis and hence cannot intersect the x axis. The statement " the graph of the equation y = 4 has no x-intercept" is TRUE.

  10. (True or False)     The value of n(n + 3)/2 = 3/2 when n = 0.

    Solution

    Evaluate n(n + 3)/2 = 3/2 for n = 0.

    n(n + 3)/2 = 0(0 + 3) / 2 = 0 / 2 = 0

    The statement "the value of n(n + 3)/2 = 3/2 when n = 0" is FALSE.

  11. (True or False)     The distance between the numbers -9 and 20 is equal to the distance between 9 and -20 on the number line.

  12. (True or False)     If f(x) = sqrt(1 - x), then f(-3) = 2.

  13. (True or False)     The slope of the line 2x + 2y = 2 is equal to 2.

  14. (True or False)     |x + 5| is always positive.

  15. (True or False)     The distance between the points (0 , 0) and (5 , 0) in a rectangular system of axes is 5.

  16. (True or False)     1 / (2x - 4) is undefined when x = -4.

  17. (True or False)     (-1/5)-2 = 25.

  18. (True or False)     The reciprocal of 0 is equal to 0.

  19. (True or False)     The additive inverse of -10 is equal to 10.

  20. (True or False)     1 / (x - 4) = 1/x - 1/4.

Answers to the Above Questions
  1. TRUE
  2. TRUE
  3. FALSE (2x + 7 = 2x + 10 , 7 = 10 no solution)
  4. FALSE ( multiplicative inverse of -1/4 is -4)
  5. FALSE (try the values x = 8 and z = 2)

  6. FALSE ( = 8 - 10 = -2)
  7. FALSE (left side = 1 , right side = 4)
  8. FALSE
  9. TRUE
  10. FALSE ( =0 )

  11. TRUE
  12. TRUE
  13. FALSE ( = -1 )
  14. FALSE ( = 0 for x = -5)
  15. TRUE

  16. FALSE (undefined when 2x - 4 = 0 , x = 2)
  17. TRUE
  18. FALSE (the reciprocal of 0 is undefined)
  19. TRUE
  20. FALSE (try x = 2)

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